



































THE 


CIVIL ENGINEER’S POCKET-BOOK. 


MR. TRAUTWINE’S ENGINEERING WORKS. 


The Field Practice of Laying out Circular Curves for Bail 
roads. By John C. Trautwine, Civil Engineer. Twelfth 
edition, 12mo, gilt edge.$2 50 

The Measurement and Cost of Earthwork. By John C. 
Trautwine, Civil Engineer, Ninth edition, 8vo, cloth . . 2 00 

The Civil Engineer’s Pocket-Book of Mensuration, Trigo¬ 
nometry, Surveying, Hydraulics, Hydrostatics, Instruments and 
their adjustments, Strength of Materials, Masonry, Principles 
of Wooden and Iron Roof and Bridge Trusses, Stone Bridges 
and Culverts, Trestles, Pillars, Suspension Bridges, Dams, 
Railroads, Turnouts, Turning Platforms, Water Stations, Cost 
of Earthwork, Foundations, Retaining Walls, etc., etc., etc. 

In addition to which the elucidation of certain important 
Principles of Construction is made in a more simple manner 
than heretofore. By John C. Trautwine, C. E. 12mo, 832 
pages, illustrated with about 700 wood-cuts. Morocco, flap, 

gilt edge. Twelfth edition, twenty-eighth thousand . . . 5 00 

* 

Any of the above books will be sent to any part of the United 
States or Canada on receipt of list price. 

Send money in Registered Letter, P. O. Order , or Check . 

JOHN WILEY & SONS, 

Scientific Publishers, 

15 Astor Place, New York City. 







THE 


/S' 


CIVIL ENGINEER’S POCKET-BOOK, 


OF 


Mensuration, Trigonometry, Surveying, Hydraulics, Hydrostatics, 
Instruments and their Adjustments, Strength of Materials, 
Masonry, Principles of Wooden and Iron Roof and 
Bridge Trusses, Stone Bridges and Culverts, 

Trestles, Pillars, Suspension Bridges, Dams, 

Bailroads, Turnouts, Turning-Plat¬ 
forms, Water Stations, Cost of 


Earthwork, Foundations, 
Ketaining Walls, 
Etc., Etc., Etc. 


IN ADDITION TO WHICH THE ELUCIDATION OF CERTAIN 
IMPORTANT PRINCIPLES OF CONSTRUCTION IS 
MADE IN A MORE SIMPLE MANNER 
THAN HERETOFORE. 


' By JOHN C. TRAUTWINE, C.E., 

* ^ * ' 


AUTHOR OF “ THE MEASUREMENT AND COST OF EARTHWORK,” “ THE FIELD 
PRACTICE OF LAYING OUT CIRCULAR CURVES FOR RAILROADS,” ETC. 


COPIOUSLY ILLUSTRATED. 



REVISED 


By JOHN C. TRAUTWINE, Jn., C. E. ? r 

• ^ vi' 


NOV 3 188^' 

NEW YORK: 

JOHN WILEY & SON 



15 ASTOR PLACE. 

LONDON: E. & F. N. SPON. 

1887. 


< 




$’ * 


Entered, according to Act of Congress, in the year 1882, by 
JOHN C. TRAUTWINE, 

in tne Office of the Librarian of Congress at Washington. 


Copyright by John C. Trautwine, Jr., 1887. 


\VM. RUTTER A CO., 

BOOK MANUFACTURERS, 
SEVENTH A CHERRY STS., PH I LA. 



• • 







THE AUTHOR 



lljb 



TO THE MEMORY OF HIS FRIEND, 

THE LATE 


BENJAMIN H. LATROBE, Esq., 

CIVIL ENGINEER. 


V 


No pains have been spared to maintain the position of 
this as the foremost Civil Engineer’s Pocket-Book, not only 
in the United States, but in the English language. 

JOHN WILEY & SONS, 

Scientific Publishers, 

15 Astor Place, New York City. 


vi 



PREFACE TO FIRST EDITION. 


QHOULD experts in engineering complain that they do not find 
anything of interest in this volume, the writer would merely 
remind them that it was not his intention that they should., The 
book has been prepared for young members of the profession; and 
one of the leading objects has been to elucidate in plain English, a 
few important elementary principles which the savants have envel¬ 
oped in such a haze of mystery as to render pursuit hopeless to any 
but a confirmed mathematician. 

Comparatively few engineers are good mathematicians; and in the 
writer’s opinion, it is fortunate that such is the case; for nature rarely 
combines high mathematical talent, with that practical tact, and 
observation of outward things, so essential to a successful engineer. 

There have been, it is true, brilliant exceptions; but they are very 
rare. But few even of those who have been tolerable mathematicians 
when young, can, as they advance in years, and become engaged in 
business, spare the time necessary for retaining such accomplish¬ 
ments. 

Nearly all the scientific principles which constitute the founda¬ 
tion of civil engineering are susceptible of complete and satisfactory 
explanation to any person who really possesses only so much element¬ 
ary knowledge of arithmetic and natural philosophy as is supposed 
to be taught to boys of twelve or fourteen in our public schools.* 


* Let two little boys weigh each other on a platform scale. Then when they 
balance each other on their board see-saw, let them see (and measure for themselves) 
that the lighter one is farther from the fence-rail on which their board is placed, in 
the same proportion as the heavier boy outweighs the lighter one. They will then 
have learned the grand principle of the lever. Then let them measure and see that 
the light one see-saws farther than the heavy one, in the same proportion; and they 
will have acquired the principle of virtual velocities. Explain to them that equality 
tf momenta means nothing more than that when they seat themselves at their metis- 

vii 




vm 


PREFACE. 


The little that is beyond this, might safely be intrusted to tli 
savants. Let them work out the results, and give them to the engi 
neer in intelligible language. We could afford to take their words fo 
it, because such things are their specialty ; and because we know thai 
they are the best qualified to investigate them. On the same princi¬ 
ple we intrust our lives to our physician, or to the captain of the 
vessel at sea. Medicine and seamanship are their respective special¬ 
ties. 

If there is any point in which the writer may hope to meet the 
approbation of proficients, it is in the accuracy of the tables. The 
pains taken in this respect have been very great. Most of the tables 
have been entirely recalculated expressly for this book; and one of 
the results has been the detection of a great many errors in those in 
common use. He trusts that none will be found exceeding one, or 
sometimes two, in the last figure of any table in which great accuracy 
is required. There are many errors to that amount, especially where 


ured distances on their see saw, they balance each other. Let them see that the weight 
of the heavy boy, when multiplied by his distance in feet from the fence-rail 
amounts to just as much as the weight of the light one when multiplied by his dis¬ 
tance. Explain to them that each of the amounts is in foot-pounds. Tell them that 
the lightest one, because he see-saws so much faster than the other, will bump 
against the ground just as hard as the heavy one; and that this means that their 
momentums are equal. The boys may then go in to dinner, and probably puzzle their 
big lout of a brother who has just passed through college with high honors. They 
will not forget what they have learned, for they learned it as play, without any ear¬ 
pulling, spanking, or keeping in. Let their bats and balls, their marbles, their 
swings, Ac, once become their philosophical apparatus, and children may be taught 
(really taught) many of the most important principles of engineering before they 
can read or write. 

It is the ignorance of these principles, so easily taught even to children, that con¬ 
stitutes what is popularly called “The Practical Engineer;” which, in the great 
majority of cases, means simply an ignoramus, who blunders along without knowing 
any other reason for what he does, than that he has seen it done so before. And it 
is this same ignorance that causes employers to prefer this practical man to one who 
18 conversant with principles. They, themselves, were spanked, kept in, &c, when 
boys, because they could not master leverage, equality of moments, and virtual velo¬ 
cities, enveloped in x’s, p’s, Greek letters, square-roots, cube-roots, &c, and they 
naturally set down any man as a fool who could. They turn up their noses at science, 
not dreaming that the word means simply, knowing why. And it must be confessed 
that they are not altogether without reason ; for the savants appear to prepare their 
books with the express object of preventing purchaser 1 :, (they have but few readers,) 
from learning why. 






PREFACE. 


IX 


the recalculation was very tedious, and where, consequently, interpo 
lation was resorted to. They are too small to be of practical import¬ 
ance. He knows, however, the almost impossibility of avoiding larger 
errors entirely ; and will be glad to be informed of any that may be 
detected, except the final ones alluded to, that they may be corrected 
in case another edition should be called for. Tables which are abso- 
lutely reliable, possess an intrinsic value that is not to be measured by 
money alone. With this consideration the volume has been made a 
trifle larger than would otherwise have been necessary, in order to 
admit the stereotyped sines and tangents from his book on railroad 
curves. These have been so thoroughly compared with standards 
prepared independently of each other, that the writer believes them 
to be absolutely correct. 

In order to reduce the volume to pocket-size, smaller type has been 
used than would otherwise have been desirable. 

Many abbreviations of common words in frequent use have been 
introduced, such as abut, cen, diag, hor, vert, pres, &c, instead of 
abutment, center, diagonal, horizontal, vertical, pressure, &c. They 
can in no case lead to doubt; while they appreciably reduce the 
thickness of the volume. 

Where prices have been added, they are placed in footnotes. They 
are intended merely to give an approximate or comparative idea of 
value; for constant fluctuations prevent anything farther. 

The addresses of a few manufacturing establishments have also 
been inserted in notes, in the belief that they might at times be found 
convenient. They have been given without the knowledge of the 
proprietors. 

The writer is frequently asked to name good elementary books on 
civil engineering; but regrets to say that there are very few such in 
our language. “ Civil Engineering,” by Prof. Mahan of West Point; 
“ Roads and Railroads,” by the late Prof. Gillespie; and the “ Manual 
for Railroad Engineers,” by George L. Vose, Civ. Eng, and Professor 
of Civil Engineering in Bowdoin College, Brunswick, Maine, are 



X 


PREFACE. 


the best. The last, published by Lee & Shepard, Boston, 1873, is the 
most complete work of its class with which the writer is acquainted. 

Many of Weale’s series are excellent. Some few of them are 
behind the times; but it is to be hoped that this may be rectified in 
future editions. Among pocket-books, Haswell, Hamilton’s Useful 
Information, Henck, Molesworth, Nystrom, Weale, &c, abound in 
valuable matter. 

The writer does not include Rankine, Moseley, and Weisbach, 
because, although their books are the productions of master-minds, 
and exhibit a profundity of knowledge beyond the reacli of ordinary 
men, yet their language also is so profound that very few engineers 
can read them. The writer himself, having long since forgotten the 
little higher mathematics he once knew, cannot. To him they are but 
little more than striking instances of how completely the most simple 
facts may be buried out of sight under heaps of mathematical rubbish. 

Where the word “ ton ” is used in this volume, it always means 
2240 lbs, because that is its meaning in U. S. law. 

John C. Trautavine. 

Philadelphia, September, 1876. 









PREFACE TO THE TWENTY-SECOND THOUSAND. 


OINCE the appearance of its last edition (the twentieth thousand) in 
^ 1883, the “ Pocket-Book ” has been thoroughly revised, and many 
important additions and other alterations have been made. These 
necessitated considerable change in the places of the former matter, 
and it was deemed best to turn this necessity to advantage, and to make 
a thorough re-arrangement, putting all of the articles, as far as possi¬ 
ble, in a rational order, so that the reader might, in many cases, be able 
to readily find a desired subject without the use of the index, which, 
notwithstanding, has been greatly enlarged. All of the articles for¬ 
merly in the appendix, and such of those in the glossary as seemed out 
of place there, have been transferred to their proper places in the body 
of the work. 

In making the re-arrangement, the rule has been to proceed from the 
abstract to the concrete, from the theoretical to the practical, from the 
general to the particular. All this will appear from a glance at the new 
and very full table of contents on pages xxiii to xxxii. Thus, beginning 
with Arithmetic, we proceed to Mensuration of lines, angles, surfaces, 
solids, the simple, in each case, preceding the complex. Next come the 
various branches of Surveying, with surveying and engineering instru¬ 
ments. Then Sound, Heat, Air and Water, Hydrostatics, Hydraulics, 
Water Supply, and Mechanics. Next, under Materials and their 
Properties, we have Specific Gravity, Weight, Weights and Measures, 
Weights, Dimensions, Prices, etc, of many manufactured and other arti¬ 
cles, and Strength of Materials. This last leads naturally to strengths 
of beams and trusses, after which follow Suspension Bridges. Proceed¬ 
ing next to subjects more closely connected with actual construction, 
and in which theory plays a less important part, we come to Well 
Boring, Dredging, Foundations, and Stonework, the latter including 
Quarrying and Masonry, Bricks, Mortar, Cements and Concretes, Re¬ 
taining Walls, and Arches of brick and stone. Then follows the arti¬ 
cle on Railroads, under which head have been grouped, in order, 
Railroad Construction (including Grades, Curves, Earthwork, Tunnels, 
Trestles, Roadway, and Turnouts), Railroad Equipment (including 
Turntables, Water Stations, &c), Railroad Rolling-Stock and Operation, 
and Railroad Statistics. By the transfer to the body of the book of the 
various articles in the appendix, the glossary comes to its proper place 
at the end of the volume, just before the index. 

It is believed that this thorough re-arrangement will, in many cases, 
enable the reader to find a subject more readily by the aid of the table 






Xll 


PREFACE TO THE TWENTY-SECOND THOUSAND. 


of contents than by that of the index; or, indeed, as already suggested, 
without the aid of either, and by the natural place of the subject in the 
order of arrangement. 

The following are some of the principal changes of subject matter 
in this edition. (The former pages of revised articles are giveu in 
parentheses.) On page 34, a table of Fractions reduced to exact deci¬ 
mals has been added. On pages 128 to 140, are two new Tables of 
Circumferences and Areas of Circles, to not less than four places of deci¬ 
mals. These tables have been calculated independently by two persons, 
expressly for this work, and the two results compared after the casting 
of the plates. They are therefore believed to be correct.'* The rules 
for finding chords, radii, &c, of Circular Arcs, p 141 (16, 17) have been 
greatly extended, re-modeled, re-illustrated, and systematized. Under 
Thermometers, pp 213, &c (309, &c), the rules for conversion of Fahren¬ 
heit to Centigrade readings, and vice versa , have been put into more con¬ 
venient shape, and others added for the conversion of both of these 
scales to that of Reaumur and vice versa. In consonance with this 
change, and to make this subject complete, three new Conversion Tables 
have been substituted for the one formerly given. The article on Flota¬ 
tion, &c, p 235 (635) has been corrected and amplified. The opening 
articles in Hydraulics, pp 237, &c (535, &c), in which are explained the 
divisions of the total head and their several offices, have been re-written, 
and it is believed they will be found to give a fuller and more satisfac¬ 
tory explanation than before, of the various phenomena connected with 
this subject. On p 254 will be found a new formula for the discharge 
through a pipe of varying diameter. The remarks on City Water-Pipes, 
Valves and Fire-IIydrants, pp 293 to 305 (572 to 577) have been re-writ¬ 
ten, enlarged, and modernized. New forms of Pipe-Joint, Stop-Valve, 
and Fire-Hydrant, and several modern appliances, are described and 

illustrated; and recent data are given for the Cost of Pipe and Laying. 

■----- 1 

* And us nearly accurate as their number of decimal places permits, except that ir X be added to 

the Anal decimal of the circumferences and areas corresponding to the following diameters, their 
error in excess will be a little less than their present error in deficiency. (Here followed, in the Tld 
thousand, a list of these diameters.) The greatest error in these is now less than 1 in the final dec¬ 
imal. It will then be less than %, as in all the others. (These trifling inaccuracies have been cor¬ 
rected for the 'Ibth thousand.) 















PREFACE TO THE TWENTY-SECOND THOUSAND. xiii 


On p 364 (587) the Velocities, &c, of Falling Bodies are tabulated in con¬ 
venient form. The remarks on Centrifugal Force, p 366 (494), have 
been revised, and the explanation made clearer. On p 396 will be found a 
concise but full account of the arrangement of the new Standard Rail¬ 
way Time. The various articles on pp 398 to 433 (357 to 382) in which 
are given the Dimensions, Weights, Prices, «fcc, of various manufactured 
articles in common use, have been thoroughly overhauled, and corrected 
up to the present time. Among these may be noted especially Iron 
Pipes, p 405 (364) to which Boiler Tubes have been added; Bolts, Nuts, 
and Washers, pp 406 to 408 (374 to 376); a new list of Wire Gauges, giv¬ 
ing the new British Standard Gauge, p 410 (368); Wire Ropes, p 413 
(380); Tin Plates, p 419 (379); and Window-Glass, p 432 (514). 

On p 435 (174) is a new table of crushing loads of American woods, 
deduced from the last census. The former table was based upon Hodg- 
kinson’s experiments, made chiefly with foreign woods. The coefficients 
for transverse strengths of timber, p 493 (185), have also been changed, 
where necessary, to correspond with results of the census experiments. 

On p 449 (234) is a revised and complete list of Phoenix Segment Col¬ 
umns, illustrated. A number of changes have been made in Rivets 
and Riveting, pp 468, &c (653, &c). 

Under Transverse Strength, pp 478, &c (183, &c), have been grouped 
the several articles on Moments of Rupture, of Inertia, and of Resist¬ 
ance, Open and Closed Beams, Shearing of Beams, <tc, which were for¬ 
merly scattered through widely separated parts of the book (183, &c, 
194, &c, 217, &c, 642 to 650). These have been carefully revised and 
amended, and arranged into a systematic whole, from which the student 
can now obtain a clear and correct notion of this somewhat troublesome 
subject. On pp. 521 to 523 (211 to 213) are new rules and tables for I 
and channel beams; and, on pp 525 to 527 (373) similar rules and tables 
for angle and T iron. These have been carefully compiled from the 
latest tables of the Pencoyd Works, which were selected on account of 
the large range of sizes made by those works, and the completeness of 
the data obtainable in regard to them. On p 524 (304) are illustrations 
of recent bridges of I beams in use on the Penna R R. The article on 
Shearing of (or vertical strains in) Beams, p 532 (642), is entirely re¬ 
written and re-illustrated. A new article on Riveted Girders is given, pp 
537 to 546 (214 to 217), in which much fuller rules, used in the best prac¬ 
tice of to-day, are substituted for those of Fairbairn ; and illustrations 
from the present standards of the Penna R R are given in place of 
those from the Charing Cross Railway of a quarter of a century ago. 

In Trusses, the remarks on Counter-bracing, pp564 to 570 (275 to 281), 
the treatment of the Fink Roof Truss, pp 574, &c (264, &c), and that 
of the Braced Arch, p 592 (274), have been corrected and simplified. 
Increased dimensions have been given in the tables of Wooden Howe 


xiv PREFACE TO THE TWENTY-SECOND THOUSAND. 


and Pratt Trusses, pp 595 and 596 (284 and 285), to provide for the modern 
weights of such loads as would probably be placed upon such structures. 

Considerable changes are made in Suspension Bridges, pp 615 to 625 
(588 to 597). The formula? for the strains in the main chains have been 
re-arranged, and those for the strains in the back-stays and on the piers 
have been much extended and illustrated, and made to cover the ground 
more fully. The description of the Pierce Well-borer, p 626 (636), has 
been corrected, other boring tools described and illustrated, and a new 
article on Artesian Well-Drilling added. The Nasmyth Steam-Ham¬ 
mer Pile-Driver is described on p 642. The articles on Stonework, 
pp 651 to 668 (310 to 313) have been greatly enlarged by the addi¬ 
tion of new articles on Machine Rock-Drills, Air Compressors, and 
Modern Explosives. In the first named, sectional views and descrip¬ 
tions of several of the more prominent modern drills are given. 

Under Railroads, pp 722 to 818 (409 to 428, Ac), will be found col¬ 
lected most, or all, of the former matter of the book, relating exclu¬ 
sively to that subject, and much that is new. A Table of Curves, with 
Radii, Ac, in metres , has been added, on p 728, for the convenience of 
those who have occasion to work by the metric system. Under Earth¬ 
work, pp 747, Ac (418 to 428, 435 to 441), original rules and tables 
have been added for estimating the cost of moving earth by means of 
the modern wheeled and drag scrapers, and that of moving earth and 
rock by cars and locomotives. In this connection, also, the operation 
of the modern steam excavators, or land dredges, is described, and data 
given concerning their capacity and their cost of operation. 

The article on Trestles, p 755, Ac (307, Ac), is greatly enlarged by 
descriptions of the new Portage, Kinzua, and other recent iron tres¬ 
tles. The former articles (390 to 408) on Rail-joints and Turnouts 
have been replaced by new, enlarged, and re-illustrated articles, pp 
763 to 789, based upon present practice. Under Rail-joints the prin¬ 
cipal place has been given to the fish and angle plates now so gener¬ 
ally used, after which follow the modern Fisher and Gibbon joints. 
The older forms, including most of those illustrated in former editions, 
are briefly referred to for purposes of comparison. Under Turnouts, 
the lengthy details of the stub-switch stand, Ac, are omitted, and the 
point and Wharton switches are given greater prominence. Several 
important improvements having been made in the Wharton since our j 
last edition (in which a new description of that switch was given), a 
further revision became necessary, and the new article is accompanied 
by a new cut, showing the switch in its latest form. Under Frogs, 
prominence is given to the Steel-Rail Frog, which has now supplanted 
the older forms on first-class lines; the Mansfield Elastic Frog is de-, , 
scribed and illustrated ; and an explanation of the working of the 
Spring-Rail Frog is added, with original cuts. The directions for laying <• 
out turnouts from curves have been simplified. 



PREFACE TO THE TWENTY-SECOND THOUSAND. XV 


Under Turntables, the description of the Sellers table has been re¬ 
written and brought into line with recent improvements; and several 
modern forms of plate and cast-iron turntables are described and illus¬ 
trated. The remarks on Track-Tanks, p 802 (434), have been extended, 
and a cut added, showing a cross-section of the Penna R R standard 
tank. 

The scattered data-(411 to 413) on Locomotives and Cars, and the 
Railroad Statistics formerly given (409, 410) have made room for tabu¬ 
lated modern data, pp 805 to 818, on these subjects, much fuller than 
those which they have replaced, and covering both standard and narrow- 
gauge roads. 

The following are among the more important of the minor changes. 


Compare 

with 


Vail at Base 


New page 

55 Exterior Angles. 

56 Definition of Complement and Supplement 
110 Table of Polygons 
112 Trigonometry. Case 2 
146 to 148 Circular Segments 
149 Ac Ellipse 
159 Cylindric Ungulas 
220 Records of Rain-falls . 

232 Rankine’s Formula for Thickness of 
246 Remarks Introductory to Table 2 

248 Art 3. 

293 Weights of Cast-Iron Pipes 
332 Polygon of Forces 
348 Center of Gravity (foot) 

380 Specific Gravity. Introduction . 

381 Ac Heading of Table . 

384 Foot-Note. 

425 Creosote, Ac. 

433 Paper. 

436 Crushing Loads of Masonry and of Ice (foot-notes) 

438, 439 Iron Pillars .... 

504 to 506 Deflections, Ac, of Beams . 

572 First paragraph .... 

573 Foot-Note. 

710 Brick Arches, Penna R R, Phila. 

803 Track Scales.409 


Old pago 
. 62 
. 62 
. 15 
. 39 
22, 24, 25 
25, 26, 630 
31, 32, 630 
. 518 
. 531 
. 541 
. 542 
. 364 
. 470 
. 442 
. 383 
384 Ac 


359 to 362 
. 151, 152 
. 175 
222, 223, 234 
196 to 198 
. 262 
. . 263 


A number of minor additions have been made to the glossary. 

Most of the new matter is in nonpareil, the larger of the two types 
heretofore used. Boldfaced type has been freely used ; but only for 
the purpose of guiding the reader rapidly to a desired division of a 
subject. For emphasis , italics have been employed. 

Illustrations which were lacking in clearness or neatness have been 
re-touched and re-lettered, or replaced with new and better cuts. The 
new matter is very freely illustrated. 

New rules have been put in the shape of formulae, and many of the 










XVi PREFACE TO THE TWENTY-SECOND THOUSAND. 


3 


old rules have been re-cast into the same form. In doing this, the 
terms of the formulse have, except where this would be exceedingly 
cumbersome, been written out in full, as in former editions, so that the 
reader is not compelled to look back over a number of pages to find 
the meaning of arbitrary symbols, and to tax his mind with remember¬ 
ing them when found. It is believed that the formul® will be found at 
least as easy to use as the rules. Take, for example, the second formula 
in Art 10, p 262 (556). In its former shape it read : 

“ To find the time reqd to fill the reservoir, to, from any level, e, above the top of 
the opening, to any upper level, d.” (This should have read “to the level a of the 
upper reservoir.”) 

“ Rule. First find the area in sq ft of a hor section of the reservoir to, which if 
supposed to be of uniform section throughout its depth. Mult together this area 
the constant number 2, and the sq rt of the vert height, a c, in ft. Call the prod p 
Mult together the area of the opening o, in sq ft; the coeff of contraction (usuallj 
about .62, whether the disch be into the air or under water) ; and the constant 8.02 j 
Call the prod y. Divp by y. The quot will be the reqd time in secs.” 


It is safe to say that any one, capable of using the above rule, can at 
readily use the following formula, in which it now appears 


Seconds required 

to raise level in m from 
c to a 


V height ac v hor area of v 0 
in ft * m in sq ft X " 

area of opening v ~ k 

o in sq ft X X b ' UJ 


and, moreover, the formula has the great advantages of showing tin 
whole operation at a glance, of making its principle more apparent 
and of being much more convenient for reference. The formula migh 
have been slightly condensed by giving the product of .62 X 8.03, in 
stead of the factors; but it was preferred to give the latter because the} 
show the principle of the formula. The coefficient .62 had already beer 
explained. The article referred to has also been otherwise improved. 

The addition of new matter, and a number of blank spaces necessaril} 
left in making the re-arrangement, have increased the number of page* 
about one-fifth. 

The new index is in stricter alphabetical order than that of formei 
editions, and contains more than twice as many entries, although mucl I 
repetition has been avoided by the free use of cross-references, withou i 
which this part of the work might have been indefinitely extended. 

The selection of articles of manufacture or merchandise for illustra 
tion, has been guided by no other consideration than their fitness for th< 
purpose, and the courtesy of the parties representing them, in supply 
ing information. 

The writer gratefully acknowledges the kindness of those who havo 
assisted in furnishing and arranging data. J. C. T., Jr. 

Philadelphia, January, 1885. 





PREFACE TO THE TWENTY-FIFTH THOUSAND. 


For this edition, the articles on Flow of Water in Channels, pp 
268 etc., Friction, pp 370 etc., and Timber Preservation, pp 425 etc., 
have been re-written. In the first named, Kutter’s formula is given, 
with tables to facilitate its use, and instructions for preparing a dia¬ 
gram from which its results may be taken by inspection. Under 
Friction are given the results of recent researches, including the now 
famous experiments of Capt. Douglas Galton on brake friction. The 
new article on Timber Preservation embodies, besides other matter, 
results recently published by a Committee of the American Society 
of Civil Engineers. 

Prices and descriptions of manufactured articles etc., have been re¬ 
vised to date. 

A number of changes have been made in the articles on Force in 
Rigid Bodies, and Trusses, in order to further simplify the treatment 
of those subjects. These changes include the enlargement, and the 
re-arrangement in more convenient form, of the articles on Gravity 
and Falling Bodies, Descent on Inclined Planes, Angular Velocity, 
Moment of Inertia, Radius of Gyration and Centrifugal Force. 

On p 230 a, the subject of Distribution of Pressure in Plane Sur¬ 
faces is explained more fully than in earlier editions. The empir¬ 
ical formula, p 243, for flow of water in pipes, has been so modified 
as to give more closely approximate results. An article on the Fa¬ 
tigue of Materials has been added, p 435. The rules for Strengths 
of Pillars, pp 439, etc., and those for Limited Deflections of Beams, 
pp 510 etc., have been simplified. On p 678 is given a summary of 
the results of Mr. Eliot C. Clarke’s recent experiments on the 
Strength etc. of Cements. Many other minor improvements have 
been made. 

Advantage has been taken of these changes to still further extend 
the substitution of the larger for the smaller type. 

J. C. T. Jr. 

Philadelphia, April, 1886. 

xvii 




PREFACE TO THE TWENTY-SEVENTH THOUSAND. 


The present edition contains revised formulae for thicknesses of cyl¬ 
inders under internal pressure, p. 232, a revised table of thicknesses 

c 

for cast-iron pipe, p. 233, additional tables of coefficient “c” by 
Rutter’s formula, pp. 275 etc., and a new and fuller table of values 
of foreign coins, p. 386. Mr. Pegram’s suggested uniform loading 
for railroad bridges, to be substituted for the usual wheel loads in 
specifications, is given on p. 546. The formulae for approximate 
weights of truss bridges, p. 605, have been revised, and the table- 
of dimensions and weights of large locomotives, p. 807, has been 
made to include some still heavier engines now in use. For minor 
changes see pp. 154, 155, 156, 159, 192, 358, 368, 397, 432 etc. 
About thirty pages more of the smaller type have been replaced 
by the larger. 

J. C. T., Jr. 

Philadelphia, March, 1887. 
xviii 



CONTENTS 


In many cases, a snbject may be found more qnickly by means of this table of 
contents, than by the index, page 833. See also Glossary, page 819. 


MATHEMATICS. 

Arithmetic. pace 


Fractions. 

Addition; subtraction; multiplica¬ 
tion ; division ; greatest common 
divisor; to reduce to lowest 

terms. 33 

Conversion into decimals. 34 

Decimals. 

Addition; subtraction; multiplica¬ 
tion; division.-. 35 

Duodecimals. 35 

Simple Proportion. 35 

Compound Proportion. Ratio. 35 

Arithmetical Progression. 36 

Geometrical Progression. 36 

Permutation... 36 

Combination. 36 

Alligation. 36 

Equation of Payments. 37 

Simple Interest. 37 

Compound Interest.. 37 

Discount. 37 

Commission, Brokerage. 37 

Insurance. 37 

Fellowship, Partnership. 37 

Logarithms. 

Table of.-. 38 

To find roots by. 39 

Hyperbolic or Naperian. 39 

Roots and Powers. Square and Cube. 

Tables of. 40 

Powers of large numbers. To find. 48 

Roots of large numbers. To find.. 52 

Roots of decimals. To find. 53 


Geometry, Mensnration, 
and Trigonometry. 

Lines, Figures, and Solids defined.... 54 


Linen. 

Lines. To divide. 


54 


Angles. 


PACE 


Angles. 

Defined; Different kinds of.. 54 

Interior and exterior. 55 

Right angles. To draw. 55 

Parallel lines. To draw. 56 

Angles. To draw; To bisect. 56 

Angles in a circle. 66 

Complement and Supplement.. 56 

Angles in a parallelogram. 57 

Minutes and Seconds in Decimals 
of a Degree. 

Table of. 57 

Angles. To measure by a 2-fit rule, 

Ac. 58 

Sines, tangents, Ac. 

Defined. 59 

Table of. 60 

Chords. Table of. 105 


Surfaces. 

Polygons. 

Regular. Tables, Ac, of. •••••••••••«•• 110 

Triangles. 

Different kinds of; properties of... 110 

Area of. To find. 110,111 

Side of. To find; having the area 

and angles.-.... Ill 

Right-angled. Properties of... 111,112 

Angles and sides of. To find. 112 

Trigonometrical Problems. 112 

Parallelograms. 

Properties of; To find area of, Ac... 119 
To draw a square on a given line.. 119 

Trapezoids; Trapeziums... 120 

Hexagons; Octagons, Ac. To draw.. 121 
Polygons. 

Regular. To draw. 121 

To reduce to a triangle of equal 

area.. 121 

To reduce a large figure to a smaller 

one. 122 

To reduce the scale of a map. 122 

Irregular figures. To measure..... 122 


xxiii 






















































XXIV 


CONTENTS, 


PAGE 


Vertical or “Shearing” Strains in 

Beams. 532 

Riveted Girders. 

Flanges. Strains in. 537 

Areas of; Effective area. 537 

Widths of. 538 

Thicknesses of. 538 

Rivets. Formulae for strains and 

numbers of.. 539 

Web. 

Its office. 539 

Stiffeners. Distance apart of; 

Areas of.. 539 

Shearing and buckling strains... 540 
Single and double webs com¬ 
pared. 540 

Strength. 

Formulae for ultimate loads. 540 

Factors of safety. 540 

Deflection. Formulae for. 541 

Results of experiment with a 

box girder. 541 

Weight. To find. 541 

Usual dimensions of parts. 541 

Methods of attaching stiffeners. 542 

Plate Girders for railroad bridges. 

Distance between girders. 542 

Lateral bracing. 542 

Dimensions, &c, of girders in ac¬ 
tual use. 543 

Diagram of moving loads. 546 

Trusses. 

Introduction ; Definitions. 547 

Strains in Trusses. Graphical 
methods of finding: 

in King and Queen trusses. 551 

in Warren and other trusses. 557 

Counterbracing. 564 

in roof-trusses, &c. 570 

in roof-trusses of the Fink sys¬ 
tem . 573 

Comparison between King, 
Queen, and Fink roof-trusses. 578 

Details of iron roof-trusses. 582 

in Fink bridge-trusses. 584 

in Bollmau bridge-trusses. 586 

in Bowstring and Crescent 

trusses. 588 

in the Braced Arch. 592 

in Cantilevers and Swing- 

bridges..... 593 

General arrangement of Sundry 
forms of truss. 

Howe bridge-truss. 594 

Pratt bridge-truss. 595 

Lattice bridge-truss. 596 

Bowstring bridge- and roof- 

trusses .. 597 

Examples. 599 

Moseley bridge-truss. 600 

Burr bridge-truss. 600 

Bollman bridge-truss. 603 

Fink bridge-truss. 603 

Weights of truss bridges. 605 

Loads on truss bridges. 606 

Factors of safety. 607 

Camber. 607 


PAGH 


Falseworks. 608 

Braces against overturning. 609 

Distance apart of trusses. 609 

Headway. 609 

Details. 

Floor girders. 610 

Transverse horizontal bracing... 610 

Chord splices. 610 

Eye Bars and Pins. 612 

Joints. 612 

Expansion rollers, &c. 614 

Suspension Bridges. 

Table of data for calculation of..... 615 
General remarks. 615 


Formulae for dimensions, &c, of, 
and strains in chains and pil¬ 


lars. 616-619 

Piers and Anchorages. 620 


Descriptions of actual bridges.. 622, &• 


CONSTRUCTION. 

Well Boring-. 

Tools for boring common wells. 626 

Tools for boring Artesian wells. 627 


Dredging. 

Cost of dredging. 631 

Dredging by bag-scoop. 632 

Weight of dredged material. 632 


Foundations. 


General remarks. 

Pierre perdue, or Rip-rap. 

w-ith piles. 

Cribs. 

Caissons... 

Coffer-dams. 

Wooden Piles. 

Sheet piles. 

Bearing piles. 

Grillage.. 

Pile-Driving Machines. 

Ordinary steam-drum machines. 

Cost of. 

Shaw’s gunpowder pile driver.... 
Nasmyth steam-hammer pile- 

driver. 

Bearing power of piles. 

Rules for; Examples of... 

Factors of safety. 

Splicing of piles. 

Blunt-ended piles. 

Friction of piles, and of cast-ivon 

cylinders. 

Penetrability aud elastic reaction 

of soils. 

Protection of feet and heads of 

piles. 

Driving below water; Follower. 

Withdrawing piles. 

Adherence of ice to piles... 


633 

634 

635 

635 

636 

637 

641 

641 

641 


641 

641 

642 

643 

644 
644 
644 


644 

644 

644 


645 


645 






























































































CONTENTS. 


XXV 


PAGE 


Forms for Level Note-books. 204 

Hand Level. Description ; Adjust¬ 
ment. 205 

Builder’s Plumb Level. Adjust¬ 
ment. 206 

Clinometer or Slope Instrument. 

Adjustment. 206 

Leveling by the barometer; by the 
boiling point. 207 


NATURAL PHENOMENA, 
FORCES AND SUBSTANCES. 

Sound. 

Sound. Velocity of; Distance trav¬ 
eled by. 211 


Ileat. 

Heat. Expansion of Solids by ; Es¬ 
timated heat of fires in common 

use. 212 

Thermometers. 

Conversion of readings by the three 
principal scales. Rules and tables 
for. 213 


Air and Water. 

Air. 

Extent; Composition; Weight; 

Quantity breathed. 215 

Temperature; Conduction of heat; 

Expansion by heat. 215 

Pressure in diving bells &c; Dew¬ 
point. 215 

Greatest recorded heat and cold.... 215 

Wind. Velocity and force of.. 216 

Water. 

I Composition; Weight at different 

temperatures. 217 

Weight of Sea Water. 217 

Ice. 217 

Compressibility of Water. 217 

Effects of fresh and salt water on 

metals, &c. 217 

Tides. 219 

Rain. 

Annual fall; Greatest recorded 
falls; Falls in different climates. 220 
One inch of depth. Equivalents 

of.. 221 

Snow. Weight of, &c. 221 

Evaporation, Filtration, Leakage. 222 




Hydrostatics. 

Hydrostatics defined. 222 

Pressure of water. 

In general. 222 

Against surfaces under varying 
conditions.•. 223 


PAG* 


To divide surfaces into portions 

sustaining equal pressures. 227 

Transmission of.. 227 

Center of. 227 

To find, under various conditions. 228 

Walls for resisting. 229 

Distribution of pressure in plane 

surfaces. 231a 

Cylinders for resisting. 

Reuleaux’s formula for thick¬ 
ness. 232 

Practical considerations. 233 

Thickness for riveted iron cylin¬ 
ders. 233 

Thickness for cast-iron pipes. 233 

Thickness for lead pipes. 234 

Valves must be closed slowly. 234 

Buoyancy, Flotation, Metacenter. 

General laws. 234 

Equilibrium of floating bodies. 235 

Stability of structures diminished 

by upward pressure of water. 236 

Draught of vessels. 236 

Compressibility of liquids...... 236 


Hydraulics. 


Hydraulics defined. 236 

Flow in pipes. 

Practical limitations to accuracy... 2.36 

Head defined. 237 

Total head. Divisions of. 237 

Pressures in pipes; Piezometer; 

Hydraulic grade line. 239 

Siphon. 240 

Approximate formulas for velocity 
in, and discharge through, 

pipes. 243 

Kutter’s formula. 244 

To find the diameter and slope 
required for a given velocity... 245 
Table of weight of water in pipes 

1 ft long. 246 

Table of areas of cross-section, con¬ 
tents of pipes, and square roots 

of diameters. 247 

To find heads, diameters, &c, for 
given velocities and dis¬ 
charges... 248 

Table of velocities, discharges, and 

heads in cast-iron pipes. 249 

Table of 5th roots and 5th powers.. 251 
Table of square roots of 5th powers. 253 
Table of square roots of 5th powers 

in ft. 253 

Discharge through a compound 

pipe of different diameters. 254 

Resistance of curved bends and of 

knees. 255 

Friction in pumping mains. 257 

Flow through openingsand adjutages 257 
Theoretical and actual velocities; 

vena contracta. 258 

Table of, and formula for, theoret¬ 
ical velocities. 258 































































XXVI 


CONTENTS. 


PAGE 


Foundations. 707 

Drains. 707 

Drainage of roadway. 708 

Table of contents of piers..*.... 708 

Brick Arches. 709 

Centers for Arches. 

General principles. 711 

Arrangements for striking. 711 

Settlement; Proper time for 

striking. 713 

Pressure of Arch-stones against 

centers. 713 

Designs for Centers. 714 


RAILROADS. 

Railroad Construction. 

Table of acres required per mile, &c, 


for different widths. 722 

Tables of grades. 723-725 

Curves. 

Table of Radii, Ac, of Curves, 

in feet. 726 

in metres. 728 

Table of long chords. 729 

Table of ordinates 5 ft apart. 730 

Earthwork. 

To prepare a table of level-cuttings. 732 

Tables of level-cuttings, Ac. 733 

Shrinkage of embankment. 741 

Cost of earthwork. 742 

Tunnels. 754 

Trestles. Wooden and iron. 755 

Roadway. 

Ballast; Ties. 759 

Rails.^60 

Table of middle ordinates for 

bending. 761 

Spikes. 762 

Rail-joints. 763 

Creeping of rails; Even and 

broken joints. 764 

Beveled joints; Fish and angle 

plates. 764 

Fisher joint. 766 

Gibbon joint. 767 

Old forms of joints. 767 

Turnouts. 

Switches. 770 

Stub switch . 771 

Switch levers and stands. 772 

Point switches. 774 

Stands for. 775 

Lorenz; and De Vout’s stand 

for.. 775, 776 

Lengths of switch rails. 776 

Prices. 776 

Wharton switch... 778 

Frogs. 

Cast-iron. 780 

Guide-rails, Ac. 781 

Rail frogs. 782 

Mansfield’s elastic frog. 783 

Spring rail frogs. 784 

Laying-out of Tnrnouts. 785 


Railroad Equipment. 


PAGE 

Turntables. 790 

General features; Minimum length. 791 

Sellers’ cast-iron turntable. 792 

Edge Moor wrouglit-iron turn¬ 
table. 793 

Fritzsche’s wrought-iron turn¬ 
table. 794 

Greenleaf’s and other turn¬ 
tables. 795,796 

Wooden turntable. 797 

Stops for turntables. 799 

Turntables with pivot at one end.. 799 

Engine-houses. Cost of. 799 

Shops. Cost of.. 799 

Water Stations. 

Dimensions, capacities, Ac. 800 

Burnham’s frost-proof tanks. 801 

Water supply; Capacities, Ac, of 

pumps. 801 

Reservoirs. 801 

Track tanks. 802 

Evaporation by locomotives. 803 

Thicknesses of tanks. Costs. 803 

Track-scales. 803 

Fences; Barbed-wire fences. 803 

Stations. Cost of.. 803 

Approximate estimate of cost of con¬ 
struction and equipment.. 804 


Railroad Rolling-Stock and 
Operation. 


Locomotives. 

Dimensions, Weights, Ac. 805 

Performance. 

Loads hauled; Fuel consump¬ 
tion; Expense of running. 808 

Cars. 

Dimensions, Capacities, W T eights, 

Cost, Ac. 811 

Resistance to motion. 812 

Wheels. Dimensions, weights, Ac. 

Cast-iron. 812 

Paper. 812 

Axles. Standard dimensions, 
weight, prices.... 813 


Railroad Statistics. 


For the United States. 

Plant and Operation. 814 

Items of annual expenses. 815 


Earnings and expenses of several 

lines. 816 

Statistics of several narrow-gauge 


roads... 818 

Miles of railroad in the world. 818 


Glossary of Technical 
Terms. 


819 



























































































CONTENTS. XXV11 


PAGE 

Strain. Further remarks on. 311 

Opposing forces. 313 

Point of application of force. 314 

Angle between line of force and sur¬ 
face acted upon. 314 

Quantity and rate of work. 316 

Vis viva. 317 

Energy. Kinetic and potential. 318 

Composition and resolution of forces. 319 

Forces in one plane. 319 

Parallelogram of forces. 320 

Polygon of forces. 329 

To ascertain resultants of forces 

by means of co-ordinates. 331 

Forces in different plaues, but 

tending to one point. 332 

Parallelopiped ot forces. 333 

Forces in different planes, and 

tending to different points. 334 

To ascertain resultants and com¬ 
ponents of forces by means of 

the angles between them. 334 

Moments; Leverage... 335 

Equilibrium of Moments. 338 

Virtual Velocities. 339 

Beam. Principle of the lever ap¬ 
plied to,. 339 

Arch. Principle of the lever ap¬ 
plied to. 342 

Compound levers; Gearing. 342 

Pulleys. 342 

The Cord or Funicular Machine.... 344 
Parallel forces. Remarks on; Re¬ 
sultant of. 347 

Center of Gravity 

of various figures. To find. 348 

of various solids. To find. 349 

of surfaces of solids. To find.... 349 

of hollow bodies. To find. 349 

Center of Pressure. 350 

Couples. 351 

Inclined Plane. 352 

Resolution of force upon. 353 

Stability on. 355,356 

Moment of Stability. 357 

Stability in arches, Ac. 359 

Gravity. Falling bodies. 362 

Descent on inclined planes. 363 

Pendulums. 364 

Center of Oscillation. 365 

Center of Percussion. 365 

Angular Velocity. 365 

Moment of Inertia. 365 

Table of Radii of Gyration. 366 

Centrifugal Force. 368 

Friction. 370 

Nature of. 370 

Static and kinetic. 370 

How estimated. 370 

Coefficient of.. 371 

Morin’s laws. 372 

Table of.. 373 

More recent experiments. 374 

“ Adhesion ” of locomotives. 3745 

Friction under great pressures. 3745 

Rolling friction. 3745 

Friction of liquids. 374c 


PAGE 

Of lubricated surfaces. 374c 

Launching friction. 374d 

Journal friction. 374d 

Friction rollers. 374e 

Friction of rolling stock. 374e 

Work of overcoming friction. 374/ 

Traction on roads, canals, and rail¬ 
roads... 375 

Animal Power. 377 


MATERIALS AND THEIR 
PROPERTIES. 

Specific Qravity, Weight, *c, 

Specific Gravity. 

Defined; Standards of. 380 

To find specific gravities of solids 

and liquids. 380,381 

Table of specific gravities and 
weights of various substances... 381 


Weights and Measures. 


American and British Standards. 385 

Troy Weight. 386 

Foreign Coins. Table of Values of, 

in U S Currency. 386 

Gold and Silver Coins, Ac. Weights, 

values, Ac, of. 387 

Apothecaries’ Weight. 387 

Avoirdupois Weight. 387 

Long Measure. 387 

Lengths of various conventional 

measures. 387 

Latitude and Longitude. Lengths 

of degrees of. 387, 388 

Inches reduced to Decimals of a 

foot. Table of. 388 

Square or Land Measure. 389 

Areas of various conventional 

measures. 389 

Cubic or Solid Measure. 389 

Contents of various conventional 

measures. 389 

Cubic foot, inch, and yard. Equiv¬ 
alents of. 389 

Spheres, 1 foot and 1 inch diam. 

Contents of.. 389 

Cylinders, 1 ft high. 1 ft and 1 in 

diam. Contents of. 390 

Liquid Measure. 390 

Basis of. The gallon and its equiv¬ 
alents. 390 

Contents of various conventional 

measures. 390 

Contents of cylinders in gallons, 

Ac. 390 

British measures. To reduce U S 

measures to, and vice versa. 390 

Dry Measure. 390 

Basis of. The bushel and its equiv¬ 
alents. 390 






























































































XXV111 


CONTENTS, 


i 

PAGE 


Contents of various conventional 

measures. 390 

British measures. To reduce U S 

measures to, and vice versa. 390 

British Imperial Measure. Liquid 

and Dr^. 391 

To obtain the size of Commercial 
Measures by means of the weight 

of water. 391 

Metric System. 

The metre. U S Standard. 391 

Metric Units of Length, U S and 
British Standards of Area and 

Solidity. 392 

Metric Weights. Avoirdupois 

equivalents. 393 

Old French Measures and Weights. 
Systeme Usuel. U S Equivalents.. 393 
Systeme Ancien. U S Equivalents. 393 

Russian Measures and Weights. 394 

Spanish Measures and Weights. 394 

Time. 

Civil or Clock. 395 

To regulate a watch by a star. 395 

Standard Railway Time. 396 

Dialling. 397 


Weights, Dimensions, Prop¬ 
erties, Prices, Ac, of various 
Substances an«l Articles. 


Iron. 

Cast. Table, &c, of weight of. 

Weight of patterns. 398 

Cast-iron pipes. Table of weight of 399 
Wrouglit-iron and steel. Tables 

, of weight of bars, &c . 400 

Wrought-iron and steel. Prices 

of. 402 

Rolled Star iron. Standard sizes 

of. 402 

Sheet iron; black, galvanized, and 
corrugated, Dimensions, weights, 
prices, methods of use, painting, 

strength. 403 

Wrought-iron pipes. 

Dimensions, weights, and prices. 405 

Fittings for. 405 

Wrought-iron boiler tubes. 

Dimensions, weights, and prices. 405 
‘ Bolts, nuts, and washers. 

Standard sizes, weights, and 

prices. 406 

Lock-nut washers. 408 

Table of weights and strengths 

of bolts. 408,409 

Buckled plates for bridge floors. 409 

Wire Gauges. 

Table of Birmingham, British 

(new), and American gauges. 410 

Birmingham gauge for brass, sil¬ 
ver, &c. 411 

Rolling mill gauge for sheet 

iron. 411 

Wire. 412 




PAGE 

Wire Rope. Iron and Steel. 413 

Rope. 414 

Chains. 414 

Lead, Copper, and Brass. 415 

Roof Copper. 416 

Sheet Lead. 416 

Balls of Lead, &c . 416 

Lead Pipe. 

Table of dimensions and weights. 416 

Remarks on. 416 

Tubes. Brass and Copper. 417 

Tin and Zinc. 

Method of using tinned and terne 
plates; Kinds in use ; Properties. 418 

Prices. 418 

Tinned and Terne Plates. Table of. 419 

Zinc sheets; Zinc vessels for water. 419 

Timber. 

Table of board measure. 420 

Durability and preservation of. 425 

Prices of lumber. 4256 

Nails; Holding power of; Prices 

and sizes. 4256 

Plastering. 

Materials and Methods. 426 

A day's work at. 426 

Cost of. 427 

Laths. 427 

Slating. 

Materials and Methods. 427 

Weights of Slate roofs. 428 

Shingles. 429 

Painting. 429 

Glass and Glazing. 431 

Draughting Materials. 

Paper, colors, lead pencils. 433 




I 


Strength of Materials. 


Strength of Materials in general. 434 

Modulus of Elasticity defined. 434 

Table of Moduli of Elasticity, Com¬ 
pression or Stretch under various 
loads, and Elastic Limits, of va¬ 
rious substances. 434 

Fatigue of Materials. 435 


Compressive Strength. 

Compressive Strength 

of timber. 436 

of stones, bricks, masonry, con¬ 
crete, Ac. 437 

of metals. 438 

Strength of Pillars. 

Law of. Gordon’s formula. 439 

Table of least radii of gyration. 440 

Tables of breaking loads. 442-456 

for wrought iron pillars of vari¬ 
ous shapes. 442, 443 

for iron pillars, per square inch 

of cross section. . 444 

for hollow cylindrical cast-iron 

pillars. 445 

for hollow cylindrical wrought- 
iron pillars .. 447 




































































CONTENTS. 


XXIX 


PAGE 


for solid cylindrical cast-iron 

pillars. 450 

for solid cylindrical wrought- 

iron pillars. 451 

for solid square cast-iron 

pillars. 452 

for solid square wrought-iron 

pillars... 453 

for Rolled I beam pillars. 454 

for channel-iron pillars. 456 

Usual practice ; Cautions; Shapes 

of Capitals. 457 

Shapes of pillars; Pillars placed 
obliquely to the line of 

pressure..... 457 

Steel pillars. 458 

Wooden pillars. 

Strength affected by seasoning; 

Proper factor of safety. 458 

Formula for strength ; Pine and 

Cast-Iron compared. 458 

Tables of breaking loads. 459 

Remarks. 460-462 

Tensile Strength. 

Tensile Strength 

of Timber. Table of.. 463 

of Metals. Table of. 464 

of Various Materials. Table of..... 466 
Diam of a round rod to bear a given 

pull. To find. 466 

Effect of cold on iron. 466 

Riveted joints. 468 

Shearing Strength. 

Shearing Strength. 476 

Torsional Strength. 

Torsional Strength. 476 

General Formulae; Constants for 

Metals and Woods. 477 

Angle of Torsion. 477 

To find Torsional Strength, having 

Shearing Strength. 477 

Torsional Strength in shafts of 

different shapes. 477 

Shafting, Wrought-iron. Strength 
of. 477 

Transverse Strength. 

Theory of. 478 

Moment of Rupture. 479 

of concentrated loads. 479 

of distributed loads. 480 

General rules for. 481 

Rules for special cases ; Diagrams. 482 

Theory of Resistance in Closed 

Beams. 485 

Moment of Resistance defined... 485 

Neutral axis. 485 

Coefficient of Resistance. 485 

Moment of Resistance. Formulas 

f„r. 486,488 

Moment of Inertia defined... 486, 487 


PAGE 


Moment of Inertia. To find. 486 

Neutral axis. To find. 487 

Breaking Load by the above 

theory.... 488 

Caution. Comparison between 

models and actual structures... 490 
Practical methods for finding 

strengths of beams. 491 

Constants for center breaking 

loads. To find. 491 

Constants for center breaking 

loads. Table of. 493 

Factors for different arrange¬ 
ments of beam and load. 494 

Breaking loads of beams of va¬ 


rious forms of cross-section. 494,495 
Allowable modifications in the 
longitudinal form of beams. 495 
Loads applied elsewhere than at 

the center of the span.. 496 

Inclined beams. 496 

Triangular beams. 496 

Dimensions required in beams. Tofind.497 
Tables of safe loads and deflections 

of wooden beams. 499 

Coefficients for iron beams. 500 

Stone Beams. Table of safe loads ... 501 
Cast-Iron Beams. Rectangular and 
Cylindrical. Tables of break¬ 
ing loads. 502, 503 

Elastic Limit in beams. 

Constants for. To find. 504 

Deflections of beams. 

General laws. 505 

Deflection within elastic limit. 

Constant for. To find. 506 

Constants for timber and metals... 507 

Amount of deflection. To find. 507 

Load to produce a given deflection. 

To find. 508 

Dimensions for a given deflection. 

To find. 508,609 

Deflection not to exceed a given frac¬ 
tion of the span. 510 

Center load. To find. 510 

Dimensions. To find. 510 

Table of loads for pine beams. 512 

Table of loads for cast-iron beams. 513 
Wooden beams for short railroad 

bridges. 514 

Continuous Beams. 515 

Hollow Beams. Results of experi¬ 
ments. 516 

Ilodgkinson Beams. 518 

Other shapes of Cast-iron Beams. 519 

Rolled I and Channel Beams. 520 

Formulae and Tables. 521 

Rolled I Beams for short railroad 

bridges. 524 

Angle- and T-iron. 

Formulae and Tables. 525 

Beams with thin webs. 

General principles. 528 

Flange strains. Formula for. 529 

Breaking loads. Formula for. 529 

Web members. Office of. 529 

Oblique and curved flanges. 530 











































































XXX 


CONTENTS. 


PAGE 


Vertical or “Shearing:” Strains in 

Beams. 532 

Riveted Girders. 

flanges. Strains in. 537 

Areas of; Effective area. 537 

Widths of. 638 

Thicknesses of. 638 

Rivets. Formulas for strains and 

d umhers of. 539 

Web. 

Its office. 639 

Stiffeners. Distance apart of; 

Areas of. 539 

Shearing and buckling strains... 540 
Single and double webs com¬ 
pared. 540 

Strength. 

Formulas for ultimate loads. 540 

Factors of safety. 540 

Deflection. Formulae for. 541 

Results of experiment with a 

box girder. 541 

Weight. To find. 641 

Usual dimensions of parts. 541 

Methods of attaching stiffeners. 542 

Plate Girders for railroad bridges. 

Distauce between girders. 542 

Lateral bracing. 542 

Dimensions, &c, of girders in ac¬ 
tual use........ 543 

Diagram of moving loads. 646 

Trusses. 

Introduction ; Definitions. 547 

Strains in Trusses. Graphical 
methods of finding: 

in King and Queen trusses. 551 

in Warren and other trusses. 557 

Counterbracing. 564 

in roof-trusses, &c. 570 

iu roof-trusses of the Fink sys- 

r tern. 573 

Comparison between King, 
Queen, and Fink roof-trusses. 578 

Details of iron roof-trusses. 682 

in Fink bridge-trusses. 684 

in Bollmau bridge-trusses. 686 

in Bowstring and Crescent 

trusses. 588 

in the Braced Arch. 692 

in Cantilevers and Swing- 

bridges. 693 

General arrangement of Sundry 
forms of truss. 

Howe bridge-truss. 594 

Pratt bridge-truss. 595 

Lattice bridge-truss. 596 

Bowstring bridge- and roof- 

trusses . 597 

Examples. 599 

Moseley bridge-truss. 600 

Burr bridge-truss.. 600 

Bollmau bridge-truss. 603 

Fink bridge-truss. 603 

Weights of truss bridges.. 605 

Loads on truss bridges. 606 

Factors of safety. 607 

Camber. 607 


PAGE 


Falseworks. 608 

Braces against overturning. 609 

Distance apart of trusses. 609 

Headway. 609 

Details. 

Floor girders. 610 

Transverse horizontal bracing... 610 

Chord splices. 610 

Eye Bars and Pins. 612 

Joints. 612 

Expansion rollers, &c. 614 

Suspension Bridges. 

Table of data for calculation of..... 615 

General remarks. 615 

Formulae for dimensions, &c, of, 
and strains in chains and pit- 

1.. - . fflft 


Piers and Anchorages. 620 


Descriptions of actual bridges.. 622, &c 

CONSTRUCTION. 


Well Boring. 

Tools for boring common wells.. 626 

Tools for boring Artesian wells. 627 

Dredging. 

Cost of dredging. 631 

Dredging by bag-scoop. 632 

Weight of dredged material. 632 

Foundations. 

General remarks. 633 

Pierre perdue, or Rip-rap. 634 

with piles. 635 

Cribs. 635 

Caissons. 636 

Coffer-dams. 637 

Wooden Piles. 

Sheet piles. 641 

Bearing piles. 641 

Grillage. 641 

Pile-Driving Machines. 

Ordinary steam-drum machines. 

Cost of. 641 

Shaw’s gunpowder pile driver.... 641 
Nasmyth steam-hammer pile- 

driver. 642 

Bearing power of piles. 

Rules for; Examples of. 643 

Factors of safety. 644 

Splicing of piles. 644 

Blunt-ended piles. 644 

Friction of piles, and of cast-iron 

cylinders. 644 

Penetrability and elastic reaction 

of soils. 644 

Protection of feet and heads of 

piles. 644 

Driving below water; Follower. 645 

Withdrawing piles. 645 

Adherence of ice to piles..... 645 


























































































CONTENTS. 


XXXI 


PAGE 


Iron Piles and Cylinders. 

Brunei’s process. 645 

Screw piles. 645 

Effect of salt water on iron. 645 

Jets. 646 

Iron piles driven by percussion. 647 

Potts’ vacuum process. 647 

Plenum process. 648 

Sinking Brickwork Cylinders. 650 

Sand-pump. 650 

Fascines. 650 

Sand-piles. 650 

Sundry Methods of forming Founda¬ 
tions . 651 

Diving-dress. Cost, &c. 651 


Stonework. 

Drilling. 

By hand ; jumper and churn drill.. 651 
By machinery. 

Diamond drills. Description, Ca- 


paci ty, Cost, Ac. 652 

Percussion drills. Description, 

Capacity, Cost, Ac. 653 

Principal points of difference in 

the more prominent makes. 656 

Hand-drilling machiues. 658 

Channeling. 658 

Air Compressors and Receivers. 
Dimensions, Capacities, Costs, 

Ac. 658 


Blasting. 

By powder. 

Explosive force, weight, cost, Ac. 660 
By the modern high explosives. 
Composition,properties, methods 


of use, cost, Ac. 

of Nitro-Glycerine. 661 

of Dynamite. 662 

of foreign explosives. 664 

Methods of firing the charge. 

By common fuse. 665 

By electricity... 665 

Simultaneous firing. 665 

Cost of quarrying stone. 667 

Cost of dressing stone. 667 

Cost of buildings per cubic foot. 668 


Mortar, Bricks, Cements, 
Concrete, «S:c. 

Mortar. 

Proportions; Quantity required,Ac. 669 


Lime. 669 

Grout... 670 

Strength. 670 

Effect upon wood. 670 

Adhesion to bricks. 670 

Bricks. 

Size, weight, absorption. 671 

Bricklaying and paving. 671 

Crushing strength. 671 

Enameled bricks. 671 

Tensile strength. 672 

Frozen mortar. 672 

To render brickwork impervious 

to water. 672 

White efflorescence on walls. 673 


PAGE 

Hydraulic Cements. 673 

Properties. Effect of moisture on. 673 

Restoration by re-burning. 674 

Rough-casting. 674 

Pointing mortar... 674 

Tests. Rapidity of setting Ac. 674, 678 

Effect of cold. 675 

Strength of cements. 675, 678 

Cement Mortar. 

Effect of sand in. 676 

Strengths of.. 676-678 

Adhesion to bricks and stone. 677 

Voids in sand and broken stone. 677,678 

White efflorescence on walls. 678 

Cement Concrete or Beton. 

Composition, Strength, Ac. 678, 679 

Admixture of lime with. Effect of. 680 

Ramming of Concrete. 680 

Stone Crushers. 680 

Methods of using. 680 

Mixing of; Cost of.. 681 

Coignet’s Beton. 681 

Concrete beams. Transverse 
strength of.. 682 

Retaining; Willis. 

Practical Rules for proportioning, Ac. 
When the earth is level with the 

top of the wall. 683 

When the wall is surcharged. 685 

Theory of Retaining walls. 686 

Thickness of walls with battered 

faces. 690 

Wharf walls. 691 

Transformation of profile. 691 

Buttresses. 692 

Protection against sliding. 692 

Counterforts; Land-ties; Curved pro¬ 
files. 692 

Revetment, counterscarp, and talus. 

Defined. 692 

Pressure as affected by width of 

backing. 692 

Table of contents per foot run of 
walls. 692 

Stone Bridges. 

Definitions. 693 

Keystone, To find depth of. 693 

Pressure sustained by arch-stones.... 694 
Table of dimensions of existing 

arches. 695 

Use of cement in arches. 696 

Keystones of elliptic arches. 696 

Table of depths of Keystones. 697 

Abutments. To proportion. 697 

Abutment-Piers. 699 

Inclination of Courses below the 

springs. 700 

Line of Resistance, Ac. 700 

To find the length of a culvert. 702 

Tables of quantities of masonry in 

arches. 703 

Tables of quantities of masonry in 

wing-walls. 704 

Tables of quantities of masonry in 
complete bridges... 706 

























































































CONTENTS, 


XXX11 


PAGE 

Foundations. 707 

Draius. 707 

Drainage of roadway. 708 

Table of contents of piers. 708 

Brick Arches. 709 

Centers for Arches. 

General principles. 711 

Arrangements for striking. 711 

Settlement; Proper time for 

striking. 713 

Pressure of Arch-stones against 

centers. 713 

Designs for Centers. 714 


RAILROADS. 

Railroad Construction. 

Table of acres required per mile, &c, 


for different widths. 722 

Tables of grades. 723-725 

Curves. 

Table of Radii, &c, of Curves, 

in feet. 726 

in metres. 728 

Table of long chords. 729 

Table of ordinates 5 ft apart. 730 

Earthwork. 

To prepare a table of level-cuttings. 732 

Tables of level-cuttings, &c. 733 

Shrinkage of embankment. 741 

Cost of earthwork. 742 

Tunnels. 754 

Trestles. Wooden and iron. 755 

Roadway. 

Ballast; Ties. 759 

Rails. 760 

Table of middle ordinates for 

bending. 761 

Spikes. 762 

Rail-joints. 763 

Creeping of rails; Even and 

broken joints. 764 

Beveled joints; Fish and angle 

plates. 764 

Fisher joint . 766 

Gibbon joint. 767 

Old forms of joints. 767 

Turnouts. 

Switches. 770 

Stub switch . 771 

Switch levers and stands. 772 

Point switches. 774 

Stands for. 775 

Lorenz; and De Vout’s stand 

for. 775, 776 

Lengths of switch rails. 776 

Prices. 776 

Wharton switch. 778 

Frogs. 

Cast-iron. 780 

Guide-rails, &c. 781 

Rail frogs. 782 

Mansfield’s elastic frog. 783 

Spring rail frogs. 784 

Laying-out of Turnouts. 785 


Railroad Equipment. 


PAGB 

Turntables. 790 

General features; Minimum length. 791 

Sellers’ cast-iron turntable. 792 

Edge Moor wrought-iron turn¬ 
table. 793 

Fritzsche’s wrought-iron turn¬ 
table. 794 

Greenleaf’s and other turn¬ 
tables. 795,796 

Wooden turntable. 797 

Stops for turntables. 799 

Turntables with pivot at one end.. 799 

Engine-houses. Cost of. 799 

Shops. Cost of. 799 

Water Stations. 

Dimensions, capacities, &c. 800 

Burnham’8 frost-proof tanks. 801 

Water supply; Capacities, &c, of 

pumps. 801 

Reservoirs. 801 

Track tanks. 802 * 

Evaporation by locomotives. 803 * 

Thicknesses of tanks. 803 

Track-scales. 803 

Fences; Barbed-wire fences. 803 

Stations. Cost of.. 803 

Approximate estimate of cost of con¬ 
struction and equipment. 804 


Railroad Rolling-Stock and 


Operation. 

Locomotives, 

Dimensions, Weights, &c. 805 

Performance. 

Loads hauled; Fuel consump¬ 
tion; Expense of running. 808 

Cars. 

Dimensions, Capacities, Weights, 

Cost, &c . 811 

Resistance to motion. 812 

Wheels. Dimensions, weights, &c. 

Cast-iron. 812 

Paper. 812 

Axles. Standard dimensions, 
weight, prices... 813 


Railroad Statistics. 


For the United States. 

Plant and Operation. 814 

Items of annual expenses. 815 

Earnings and expenses of several 

lines. 816 

Statistics of several narrow-gauge 

roads. 818 

Miles of railroad in the world. 818 

Glossary of Technical 
Terms. 819 
























































































THE 


CIVIL ENGINEER’S POCKET-BOOK. 


ARITHMETIC. 


On this subject we shall merely give a few examples for refreshing the memory of those who for 
want of constant practice cannot always recall the processes at the moment. 


Subtraction of Vulgar Fractions. 


1 l _ n 

2 ~ 2 — °- 


+ 7 ~ 2 ~ 


3 

4 ~ 


1 _ 
¥ - 


2 _ 
¥~ 


3 4 _. i 2 — 

S 7 1 3 ~ 


1 

7' 

2 5 

7 


2 _ 4 _ 1 
8—8 — 2' 


3 

¥“ 


_ 2 7 
“36 


2 0 _ 
3 6 — 


7_ 
3 6' 


_ 5 — 7.5. _ 

3 2 1 


35 - 4 0 — 119 
— 71 — 


7T 


2T‘ 


Addition of Vulgar Fractions. 


3.1 — 4 
¥ + ¥ _ ¥ 


— 1 6. 2 — s_ 

— 8 “ 8 “ « 


1 — 1 

8 


3.5—27 
¥ + “ AA 


-l 2 0 - 4 7 - ill 
*^36 36 *3 A* 


o4 i i 2. — 25j. 5 - 15 4. 35 - 1 1 0 — = 5 
°Y ‘ 1 3 ~ 7 ' 3 _ 2 1 ' 21 ~ 21 — °2T' 

2 . 3 . 8 — 40 i. 36 i. 120— 196 — «16 — »4 
3 + A + ¥~A0+A0 + 60“ 60 ~ 6 6 0 “ d TA" 

Multiplication of Vulgar Fractions. 


l — i 


X A = 


¥• 


3 v 1 — 3_ 6 v 2 —12 — 3_ 
¥ * ¥ ~ 1 6 * 8 * 8 — 64 — 16 * 


3 v 5 — 15 — 5 

4 A 9 - 36 - T2* 


31 X If- = 


2 — 25 


5—125 


X f = 


2T 


8—48 


X O_ ± 

a a 


6 0 - A' 


3 of 1 

¥ ° r 2 


of|of|=|xIx|xi- = 


7 — 105 — 21— 7 
A - T7 0 ~ 7¥ ~ S’ 


Division of Vulgar Fractions. 


lii-2-. 3^1—12 

2 ‘ 2 2' 4 * 4 — ¥ 


3 — q 6i2 — 41—24—12 — 3 — o 3i5-27-i 7_ 
T- d ’ 8 * A “ 1 6 ~ A ~ ¥ — T —¥ * 9~ 2A-*70' 


3 4i . 2 - 25 - 5 — 75 — 15—.,1 r - 7 — 5. c. 7 — 40 — s 5 

6 7 ’ x 3 - 7 * 3 - AA - 7 ~ l 7’ 0 * 8 - T ’ 8 - 7 - °7 > 

To find the greatest common divisor of a Vulgar Fraction. 


Ex. 1 . or T W 


70'175(2 

140 


Ex. 2. Of 


8 4 
2 O' 


20)84(4 

80 


35)70(2 

70 Ans 35. 


4)20(5 

20 Ans 4. 


To reduce a Vulgar Fraction to its lowest terms. 

First find the greatest common divisor: then divide both the numerator and denominator by it. 

H 0 2 8 4 2 1 

Thus, in the preceding example y'yj = tj- Ans. And 20 — A ^ ns * 


33 




34 


ARITHMETIC 


To reduce a Vulgar Fraction to a decimal form. 

Divide the numerator by the denominator. Thus, 

I =2)1.0(0.5 Ans. L 3 = 4)13(3.25 Ans. f-l = 40*32.0(0.8 Ans. 

10 4 12 40 32 0 

/ To 

8 


20 


Reduce 3 inches to the decimal of a foot. There are 12 ins in a foot; therefore, the question is 

to reduce yV to a decimal. Therefore, 12)3.0(0.25 of afoot. Ans. 

24 

~60 

60 

Reduce 2 ft 3 ins to the decimal of a yard. There are 36 ins in a yard; and 27 ins in 2 ft 3 ins; 

2 7 

therefore, -jg- of a yard = 36)27.0(0.75 of a yd. Ans. 

25 2 

180 

180 

How many feet and ins are there in .75 of a yard 7 Here 

.75 

3 ft in a yd. 

Ft 2).25 

12 ins in a ft. 


Ins 3.00 Ans 2 ft 3 ins. 

How many feet and ins are there in .0G25 of a yard 7 

.0625 

3 feet in a yd. 


No feet, .1875 

12 ins in a ft. 

- ft. ine. 

Ins 2.2500 Ans. 0 2.25. 


How many cnbic feet are there in .314 of r cub yard ? And cnb ins in .46 of a cub ft 7 
.314 .46 

27 cub ft in a yd. 1728 cub ins a cub ft. 

2198 368 

628 92 

- 322 

8.478 cub ft. Ans. 46 

794.88 cub ins. Ans. 

Fractions reduced to exact decimals. 

6t 

66 

3 

6T 

T6 

.015025 

.03125 

.040875 

.0025 

M- 

9 

3? 

19 

6 4 

1% 

.205025 

.28125 

.290875 

.3125 

33 

64 

u 

M 

9 

T6 

.515025 

.53125 

.540875 

.5625 

49 

6T 

2 5 
36 

5 1 
6¥ 

1 3 
T6 

.765025 : 
.78125 
.796875 
.8125 

th 

66 

6 ? ¥ 

1 

"S' 

.078125 

.09375 

.109375 

.125 

n 

n 

2 3 
6¥ 

3 

8 

.328125 

.34375 

.359375 

.375 

6¥ 

n 

36 

64 

5 

8 

.578125 

.59375 

.609375 

.025 

H 

27 

ST 

5 5 
64 

7 

6 

.828125 , 
.84375 
.859375 ‘ 
.875 

3 5 2 

M 

T6 

.140025 

.15025 

.171875 

.1875 

25 

64 

1 3 
66 
27 

6 4 

T6 

.390025 

.40025 

.421875 

.4375 

41 

64 

2 1 
62 

4 3 
64 

11 
T6 

.640625 

.05625 

.071875 

.0875 

57 

6 4 
29 
36 
59 

6 4 

1 5 
16 

.890025 

.90625 

.921875 

.9375 

1 3 
6¥ 

7 

66 

» 

.203125 

.21875 

.234375 

.25 

29 

6T 
15 
66 

3 1 
64 

1 

6 

.453125 

.40875 

.484375 

5 

45 

64 

2 3 

3 2 

41 

64 

3 

4 

.703125 

.71875 

.734375 

.75 

61 

64 

31 

36 

63 

64 

1 

.953125 

.90875 

.984375 

1. 

















































ARITHMETIC 


35 


Decimals. 

Addition. Add together .25 and .75; also .006, 1.3472, and 43. 

.25 

.75 

1.00 Ans. 


.006 

1.3472 

43. 


44.3532 

Subtraction. Subtract .25 from .75; also .0001 from 1 ; also 6.30 from 9.01. 


.75 1. 9.01 

.25 .0001 6.30 


Ans. 


.50 Ans. 


.9999 Ans. 


2.71 Ans. 


Multiplication. Mult 3 X .3; also .3 X .3 ; also .3 X -03; also 4.326 X .003. 

3 .3 .3 4.326 

.3 .3 .03 .003 


.9 Ans. .09 Ans. .009 Ans. .012978 Ans. 

Division. Divide 3. by .3 ; also .3 by .3 ; also .3 by .03 ; also 4.326 by .0003. 


.3j3.0(10. aus. 
3 

0 

Divide 62 by 87.042. 


Divide .006 by 20. 


.3j.3(l. Ans. 
3 


.03).30(10. Ans. 
3 


87.042)62.0000(0.712, «fcc. Ans. 
60.9294 

~ 107060~ 

87042 


200180 

20.000).0060000(0.0003 Ans. 
60000 


.0003)4.3260(14420. Ana. 
3 

13 

12 

12 

12 

6 

6 


Duodeci mals. 


Duodecimals refer to square feet of 144 sq ins; to twelfths of a square or duodecimal foot; each 
| such twelfth being called an inch; and being equal to 12 square inches; and to twelfths , each equal 
! to the 12th of a duodecimal inch, or to one square inch. The dimensions of the thing to be measd 
are supposed to be taken in common feet, ins, and 12ths of an inch ; but as ordinary measuring 
rules are divided into 8ths of an inch, it is usually guess-work to some extent. Duodecimals are 
very properly going out of use, in favor of decimals; we shall therefore give no rule for them. By 
means of our table of “ Inches reduced to Decimals of a Foot," p. 388, all dimensions in feet, ins, 
aud 8ths, &c, can be at once taken out in ft and decimals of a foot. 


Single Rule of Three; or. Simple Proportion. 

If 3 men lav 10000 bricks in a certain time, how many could 6 men lay in the same time? They 
will evidently lay more; therefore, the second term of the proportion must be greater than the first. 

3 : 6:: 10000 : 20000 Ans. 

6 

3)60000 
20000 Ans. 

If 3 men require 10 hours to lay a certain number of bricks, how many hours would 6 men 
require? They will evidently require less time; therefore, the second term of the proportion must 
be less than the first. 

6 : 3 : : 10 : 5 Ans. 

3 


Double Rule of Throe: or. Compound Proportion. 

If three men can lay 4000 bricks in 2 days, how many men can lay 12000 in 3 days? Here we see 
(hat 4000 bricks require 3 X 2 — 6 days' work; therefore 12000 will require, 

4000 : 12000 :: 6 : 18 days’ work. 

1 A 

But there are only 3 days to do the 18 days work in ; therefore the number of men must be y = 6 
nen. Ans. 

A moment's reflection will suffice to reduce any case of double rule of three to this simple form. 


Ratio. 
he ratio of 5 to 10 


Ratio. 

Simple ratio is a number denoting how often one quantity is contained in another. 


and the ratio of 10 to 5 is , or 2. 


Thus, 


uc ^ y—, or ^; and tne ratio oi iu io o is or t. When, of four numbers, two 

ave to each other thesame ratio that the other two have, the numbers are said to be tn proportion 
i each other. Thus, 6 has the same ratio (2) to 3. as 100 has to 50; therefore, 6, 3, 100. an- 50, are 
tid to be in proportion ; or, as 6 : 3 :: 100 : 50. In other words, an equality of ratios is called pro- 
ortion. Ratio and proportion are often confounded with one another; but the error is one oi no 
nportance. Duplicate ratio is that of the squares of numbers. 










36 


ARITHMETIC 


Arithmetical Progression, 


In a series of numbers, is a progressive increase or decrease in each successive number, by the add! 
tion or subtraction of the same amount at each step; as in 1, 2. 3. 4. 5, <fcc., in which 1 is added a( 
each step ; or 10, 8, 6, 4, &c., in which 2 is subtracted at each step; or *■£, %, 1, 1)4. &c. In any 

such series the numbers are called its terms; and the equal increase or decrease at each step its com 
man difference. 

To find the com diff, knowing the first and last terms; and the number of terms. Find the did 
between the first and last terms. Front the number of terms subtract I. Div the diff just found, by 
the rem. 

To find the last term, knowing the first term; the com diff; and the number of terms. From thi 
number of terms take 1. Mult the rem by the com diff. To the prod add the first term. 

To find the number of terms, having the first and last ones ; and the com diff. Take the ditl 
between the first and last terms. Div this diff by the com diff. To the quot add 1. 

To find the sum of all the terms, having the first and last ones; and the number of terms. Ad< 
together the first and last terms. Div their sum by 2. Mult the quot by the number of terms. 


Geometrical Progression, 


In a series of numbers, is a progressive increase or decrease in each successive number, by the sam- 
multiplier or divisor at each step ; as 3, 9, 27. 81, &c, where each succeeding term is increased by nml 
the preceding oue by 3. Or 48, 24. 12, 6, &c, or 27. 13Jsj, 6%, %%, &c. where each succeeding term i 
found by dividing the preceding one by 2. The multiplier or divisor is called the common ratio of th 
series, or progression. 

To find the last term, knowing the first one; the ratio; and the number of terms. Raise the rati 
to a power 1 less thau the number of terms. Mult this power by the first term. 

Fix. First term 10; ratio 3; number of terms 8; what is the last term? Here the number of term 
being 8, the ratio 3 must be raised to the 7th power; thus; 

3X3X3X3X3X3X3 = 2187, = 7th power. And 2187 X 10 = 21870 last term. Ans. 

A man agreed to buy 8 fine horses : paying $10 for the first; $30 for the second; $90 for the thirc 
&c: how much will the last oue cost him? Aus, $21870. as before. 

To find the sum of all the terms, knowing the first oue; the ratio: and the number of terms. Rais 
the ratio to a power equal to the whole number of terms. From this power subtract 1. Div the rei 
by 1 less than the ratio. Mult the quot by the first term. 

Ex.- As before. What is the sum of all the term'? Here the ratio must be raised to the 8l 
power; thus, 3X3X3X3X3X3X3X3= 6561 ~ 8th poxv. And 6560 div by 1 less than the rati 
6560 

3, ~ —— — 3280. And 3280 X 10 (or number of terms) — 32800 = sum. Ans. 


In the foregoing case, the 8 horses would cost $32800. 


Permutation 


Shows in how many positions any number of things can be arranged in a row. To do this, mu 
together all the uumbers used in couutiug the thiugs. Thus, in how many positions in a row can 
things be placed ? Here, 


1X2X3X4X5X6X"X8 X 9 = 362880 positions. Ans. 


I'omhiimtion 


Shows how manv combinations of a few things can be made out of a greater number of things, q 
do this, first set down that number which indicates the greater number of things; and after it a serit 
of numbers, diminishing by l, until there are in all as many as the number of the few things th: 
are to form each combination. Then beginning under the last one, set down said number of fe 
things ; and going backward, set down another series, also diminishing by 1. until arriving under tl 
first of the upper numbers. Mult together all the upper numbers to form one prod; and all the low* 
ones to form another. Div the upper prod by the lower one. 

Ex. How many combinations of 4 figures each, can be made from the 9 figs 1, 2, 3, 4, 5, 6, 7, 8, 5 
or from 9 any things? 

9 X 8 X 7X6 3024 

= 126 combinations. Ails. 


1 X 2 X 3 X 4 


24 




Alligation 

Shows the value of a mixture of different ingredients, when the quantity and value of each of thes 
last is known. 

Ex. What is the value of a pound of a mixture of 20 lbs of sugar worth 15 cts per lb; with 30 11 I 
worth 25 cts per lb ? 

lbs. cts. cts. 

20 X 15 — 300 
30 X 25 = 750 

50 lbs. 1050 cts. 


1050 

Therefore, —- =: 21 cts. Ans. 
oO 


\ 








ARITHMETIC 


37 


Equation of Payments. 


I 


' 

i 

. 




A owes B $1.00; of which $-100 are to be paid in 3 months; $500 in 4 months; and $300 in (t 
months ; all bearing interest until paid ; but it has been agreed to pay all at once. Now, at what tinn 
must this payment be made so that neither party shall lose any interest? 

$ months. 

400 X 3 = 1200 5000 

500 X 4 = 2000 Therefore, -= 4.16, &c, months. Ans. 

300 X 6 — 1800 1200 

1200 5000 


A owes B $1000 to be paid in 12 days; and $500 to be paid in 3 months. What would be the tw 
tor paying all at once ? 

$ days. 

1000 X 12 = 12000 57000 

500 X 90 = 45000 Therefore, —- = 38 days. Ans. 


1500 57000 


Simple Interest. 

What is the simple interest on $865.32 for one year, at 6 per ct per annum ? 

Principal. Interest. Principal. Interest. 

$100 : $6 : ; $865.32 : $51.0192 

6 

- $ cts. 

100)5101.92(51.9192 Ans. =r 51.01 -^2 

What is the interest on $865.32 for 1 year, 3 months, and 10 days, at 7 per cent per annum? 
First calculate the interest for 1 year only ; thus: 

Priu. Int. Prin. Int. 

$100 : $7 :: $865.32 : $60.5724 

7 


100)6057.24(60.5724 

Then say, If 1 year or 365 days give $60.5724 int, what will 465 days give? or 

Days. Iut. Days. Int. 

365 : $60.5724 :: 465 : $77.16, &c. Ans. 

At 5 per ct simple interest, money doubles itself in 20 years; at 6 per ct, in 16% years; and at 7 
per ct, in 14^ years. Simple Interest is Single Kule of Three. 

Compound Interest. 

When money is borrowed for more than a year at compound interest, find the simple interest at the 
end of the first year, and add it to the principal, for a second principal. Find the simple intere.-t oil 
this second enlarged principal for the next year, and add it to the enlarged principal for a third prin¬ 
cipal ; and so on for each successive year. 1 

At 5 per ct compound interest, money doubles itself in about 14^ years ; at 6 per ct, in about 11.9 
years; and at 7 per ct, in about 10% years. 

Discount 


Is a deduction of a part of the interest, when money at interest is paid before it is due. Or it is a 
deduction of the whole of the interest iu advance, at the time the money is lent. In the first case, if 
I borrow $100 for I year at 8 per ct. I must at the end of the year pay back $108; butif I pay at the 
eud of 3 months, I must add only $2, or the interest for those 3 months, paying back $102; and the 
' diff of $6 is the discount. Therefore, to find the discount in such cases, first find the interest for 
T the full time; then that for the short time; and take the diff. 

In the second case, if I borrow $100 from a bank for one year, at6 perct. I receive but 100 — 6— $91; 
but at the end of the year I must pay back $100. By discounting iu this manner, the bank actually 
gains more than 6 per ct; for it gains $6 for the use of $94 for 1 year. In the Uuited States, the banks 
deduct discount for 3 days more than the time stipulated iu the uote; these are called‘‘days of 
grace." The borrower is not obliged to pay before the last of these 3 days. 

Commission, or Brokerage, 


Is a percentage 'or so much per each $100) paid to commission merchants for selling our goods; or 
to brokers, or other kinds of agents, for transacting business for us. It is Single Rule of Three. 

Ex. If a broker makes purchases for me to the amount of $9362, at 2 per ct, what is his brokerage? 
Say, as 

Purchase. Brokerage. Purchase. Brokerage. 

$100 : $2 : : $9362 : $187.24 


Insnranee 

Is a percentage (called a premium) paid to a company for insuring onr property against fire, Ac. 
The company, or insurers, (called also underwriters,) deliver to the person insured, a paper bearing 
their seal, &c, and called the Policy of Insurance : which contains the conditions of the transaction. 
Insurance is calculated like Commissions, &c.; being merely Single Rule of Three. 

Fellowship. 

A puts $6000 into a business in partnership with B, who puts in $9000. At the end of a year they 
have made $2400; how much is eachone’s share? Here, $6000 -f- $9000 — $15000joint capital; then say, 
Joint cap. Total gain. A's cap. A’s share. 

$15000 : $2400 : : $6000 : $960 


B’s cap. 
$9000 


B's share. 
: $1440 


And 


$15000 


$2400 










38 


TABLE OF LOGARITHMS 




LApiritluns of Numbers, from O to 1000.* 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Prop. 

0 

0 

00000 

30103 

47712(60206 

69897 

77S15 

84510 

90309 

95424 


10 

00000 

00432 

00860 

01283 01703 

02118 

02530 

02938 

03342 

03742 

415 

11 

04139 

04532 

04921 

05307 

05690 

06069 

06445 

06818 

07188 

07554 

379 

12 

07918 

08278 

08636 

08990 09342 

09691 

10037 

10380 

10721 

11059 

349 

M 

11394 

11727 

12057 

12385 12710 

13033 

13353 

13672 

13987 

14301 

323 

14 

14613 

14921 

15228 

15533| 15836 

16136 

16435 

16731 

17026 

17318 

300 

15 

17609 

17897 

18184 

18469 

18752 

19033 

19312 

19590 

19866 

20139 

281 

16 

20412 

20682 

20951 

21218 

21484 

21748 

22010 

22271 

22530 

22788 

264 

17 

23045 

23299 

23552 

23804 

24054 

24303 

24551 

24797 

25042 

25285 

249 

18 

25527 

25767 

26007 

26245 

26481 

26717 

26951 

27184 

27415 

27646 

236 

19 

27S75 

28103 

28330 

28555 

28780 

29003 

29225 

29446 

29666 

29885 

223 

20 

30103 

30319 

30535 

30749 

30963 

31175 

31386 

31597 

31806 

32014 

212 

21 

32222 

32428 

32633 

32838 

33041 

33243 

33445 

33646 

33845 

34044 

202 

22 

34242 

3 4439 

346:35 

34830 

35024 

35218 

35410 

35602 

35793 

35983 

194 

23 

36173 

36361 

36548 

36735 

36921 

37106 

37291 

37474 

37657 

37839 

185 

24 

38021 

38201 

38381 

38560 

38739 

38916 

39093 

39269 

39445 

39619 

177 

25 

39794 

39J67 

40140 

40312 

40483 

40654 

40S24 

40993 

41162 

41330 

171 

26 

41497 

41664 

41830 

41995 

42160 

42324 

4248' 

42651 

42813 

42975 

164 

27 

43136 

43296 

43456 

43616 

43775 

43933 

44090 

44248 

44404 

44560 

158 

28 

44716 

44S70 

45024 

45178 

45331 

45484 

45 436 

45788 

45939 

46089 

153 

29 

46240 

46389 

46538 

46686 

4683 4 

46982 

47129 

47275 

47421 

47567 

148 

30 

47712 

47856 

48000 

48144 

48287 

48430 

48572 

48713 

48855 

48995 

143 

31 

49136 

49276 

49415 

49554 

49693 

49831 

49968 

50105 

50242 

50379 

138 

32 

50515 

50650 

50785 

50920 

51054 

51188 

51321 

51454 

51587 

51719 

134 

33 

51851 

51982 

52113 

52244 

52374 

52504 

52633 

52763 

52891 

53020 

130 

34 

53148 

53275 

53402 

53529 

53655 

53781 

53907 

54033 

54157 

54282 

126 

35 

54407 

54530 

5 4654 

54777 

54900 

55022 

55145 

55266 

55388 

55509 

122 

36 

55630 

55750 

55870 

55990 

56110 

56229 

56348 

56466 

56584 

56702 

119 

37 

56'20 

56937 

57054 

57170 

57287 

57403 

57518 

57634 

57749 

57863 

116 

38 

57978 

58092 

58206 

58319 

58433 

58546 

58658 

58771 

58883 

58995 

113 

39 

59106 

59217 

59328 

59439 

595 49 

59659 

59769 

59879 

59988 

60097 

110 

40 

60206 

60314 

60422 

60530 

60638 

60745 

60852 

60959 

61066 

61172 

107 

41 

61278 

61384 

61489 

61595 

61700 

61804 

61909 

62013 

62117 

62221 

104 

42 

62325 

62428 

62531 

62634 

62736 

62838 

62941 

63042 

63144 

63245 

102 

4; 

63347 

63447 

63548 

636 48 

63749 

63848 

639 IS 

64048 

64147 

64246 

99 

41 

61345 

64443 

64542 

64640 

64733 

64836 

64933 

65030 

65127 

65224 

98 

45 

65321 

65 417 

65513 

65609 

65705 

65S01 

65896 

65991 

660'6 

66181 

96 

46 

6627 6 

66370 

66464 

66558 

66651 

66745 

66838 

66931 

670.'4 

67117 

94 

47 

67210 

67302 

67394 

67486 

67577 

67669 

67760 

67851 

67942 

68033 

92 

48 

68121 

68214 

68304 

68394 

6S484 

68574 

68663 

68752 

68842 

68930 

90 

49 

69020 

69108 

69196 

092S4 

69372 

69460 

69548 

69635 

69722 

69810 

88 

50 

69897 

69983 

70070 

70156 

70243 

70329 

70415 

70500 

70586 

70671 

86 

51 

70757 

70842 

70927 

71011 

71096 

71180 

71265 

71349 

71433 

71516 

84 

52 

71600 

71683 

71767 

71850 

71933 

72015 

7209S 

72181 

72263 

72345 

82 

53 

72428 

72509 

72591 

72672 

72754 

72835 

72916 

72997 

73078 

73158 

81 

54 

73239 

73319 

73399 

73480 

73559 

73639 

73719 

73798 

73878 

73957 

80 

55 

74036 

74115 

74193 

74272 

74351 

74429 

74507 

74585 

74663 

74741 

78 

56 

74813 

74896 

74973 

75050 

75127 

75204 

75281 

75358 

75434 

75511 

77 

57 

75587 

75663 

75739 

75815 

75891 

75966 

76042 

76117 

76192 

76267 

75 

58 

76342 

76417 

76492 

76566 

76641 

76715 

76789 

76863 

76937 

77011 

74 

59 

77085 

77158 

77232 

77305 

77378 

77451 

77524 

77597 

77670 

77742 

73 

60 

77815 

77887 

77959 

78031 

78103 

78175 

78247 

78318 

78390 

78461 

72 

61 

78533 

78604 

78675 

78746 

78816 

78887 

78958 

79028 

79098 

79169 

71 

62 

79239 

79309 

79379 

79448 

79518 

79588 

79657 

79726 

79796 

79865 

70 

63 

79934 

8O002 

80071 

80140 

80208 

80277 

80345 

80413 

80482 

80550 

69 

64 

80618 

80685 

80753 

80821 

80888 

80956 

81023 

81090 

81157 

81224 

68 

65 

81291 

81358 

81424 

81491 

81557 

81624 

81690 

1 81756 

81822 

81888 

67 


♦Each log is supposed to have the decimal sign . before it. 


























































TABLE OF LOGARITHMS, 


39 


Logarithms of Numbers, from 0 to 1000*— (Continued.) 


No. 

1 

O 

1 

o 

3 

4 

5 

6 

7 

8 

9 

Prop 

66 

81954 

82020 

82085 

82151 

82216 

82282 

82347 

82412 

82477 

82542 

66 

67 

82607 

82672 

82736 

82801 

82866 

82930 

82994 

8305S 

83123 

83187 

65 

68 

83250 

83314 

83378 

83442 

83505 

83569 

83632 

83695 

83758 

83821 

64 

69 

83884 

83947 

84010 

84073 

84136 

84198 

84260 

84323 

84385 

84447 

63 

70 

84509 

84571 

84633 

84695 

84757 

84818 

84880 

84941 

85003 

85064 

62 

71 

85125 

85187 

85248 

85309 

85369 

85430 

85491 

85551 

85612 

85672 

61 

72 

85733 

85793 

85853 

85913 

85973 

86033 

86093 

86153 

86213 

86272 

60 

73 

86332 

86391 

86451 

86510 

86569 

86628 

86687 

86746 

86805 

86864 

59 

74 

86923 

86981 

87040 

87098 

87157 

87215 

87273 

87332 

87390 

87448 

58 

75 

87506 

87564 

87621 

87679 

87737 

87794 

87852 

87909 

87966 

88 024 

57 

76 

88081 

88138 

88195 

88252 

88309 

88366 

88422 

88479 

88536 

88592 

56 

77 

88649 

88705 

88761 

88818 

88874 

88930 

88986 

89042 

89098 

89153 

56 

78 

89209 

89265 

89320 

89376 

89431 

89487 

89542 

89597 

89652 

89707 

55 

79 

89762 

89817 

89872 

89927 

89982 

90036 

90091 

90145 

90200 

90254 

54 

80 

90309 

90363 

90417 

90471 

90525 

90579 

90633 

90687 

90741 

90794 

54 

81 

90848 

90902 

90955 

91009 

91062 

91115 

91169 

91222 

91275 

91328 

53 

82 

91381 

91434 

91487 

91540 

91592 

91645 

91698 

91750 

91803 

91855 

53 

83 

91907 

91960 

92012 

92064 

92116 

92168 

92220 

92272 

92324 

92376 

52 

84 

92427 

92479 

92531 

92582 

92634 

92685 

92737 

92788 

92839 

92S90 

51 

85 

92941 

92993 

93044 

93095 

93146 

93196 

93247 

93298 

93348 

93399 

51 

86 

93449 

93500 

93550 

93601 

93651 

93701 

93751 

93802 

93852 

93902 

50 

87 

93951 

94001 

9405L 

94101 

94151 

94200 

94250 

94300 

94349 

94398 

49 

88 

94448 

94497 

945-16 

94596 

94645 

94694 

94743 

94792 

94841 

94890 

49 

89 

94939 

94987 

95036 

95085 

95133 

95182 

95230 

95279 

95327 

95376 

48 

90 

95424 

95472 

95520 

95568 

95616 

95664 

95712 

95760 

95808 

95856 

48 

91 

95904 

95951 

95999 

96047 

96094 

96142 

96189 

96236 

96284 

96331 

48 

92 

96378 

98426 

96473 

96520 

96567 

96614 

96661 

96708 

96754 

96801 

47 

9.3 

96848 

96895 

96941 

96988 

97034 

97081 

97127 

97174 

97220 

97266 

47 

94 . 

97312 

97359 

97405 

97451 

97497 

97543 

97589 

97635 

97680 

97726 

46 

95 

97772 

97818 

97863 

97909 

97954 

98000 

98045 

98091 

98136 

98181 

46 

96 

98227 

98272 

98317 

98362 

98407 

98452 

98497 

98542 

985S7 

98632 

45 

97 

98677 

98721 

98766 

98811 

98855 

98900 

98945 

98989 

99033 

99078 

45 

98 

99122 

99166 

99211 

99255 

99299 

99343 

99387 

99431 

99475 

99519 

44 

99 

99563 

99607 

99651 

99694 

99738 

99782 

99825 

99869 

99913 

99956 

44 


♦ Each lo^ is supposed to have the decimal sign . before it. 


The log of 2870 is 3.45788 
“ “ “ 287 is 2.45788 

“ “ “ 28.7 is 1.45788 

“ “ “ 2.S7 is 0.45788 


The log of .287 is — 1.45788 

“ “ “ .028 is — 2.44716 

“ “ “ .002 is — 3.30103 

“ “ “ .0002 is — 4.30103 


What is the log of 2873? 

Here, log of 2870 = 3.45788 
And prop 153 X 8 = 459 

3.458339 

To find roots divide the log (with its index) of the given number, by that 

number which expresses the kiud of root. The quotient will be the log of the required root. 
Example. What is the cube root of 2870? 

3.45788 

Here, the log of 2870, with its index, is 3.45788. And-- - = 1.15263. Hence the cube root is 14.2, 

The Hyperbolic, or Napierian logarithm is the common log of 
the table multiplied by 2.3025851. 


















































40 


SQUARE AND CUBE ROOTS 


Square Roots ami Cube Roots of Numbers from .1 to 2S. 

No errors 


No . 

Square . 

Cube . 

Sq . Rt . 

C . Rt . 

No . 

Sq . Rt . 

C . Rt . 

No . 

Sq . Rt . 

C . Rt . 

.1 

.01 

.001 

.316 

.464 

.7 

2.387 

1.786 

.4 

3. f >61 

2.375 

.15 

.0225 

.0034 

,387 

.531 

.8 

2.408 

1.797 

.6 

3.688 

2.387 

.2 

.04 

.008 

.447 

.585 

.9 

2.429 

1.807 

.8 

3.715 

2.399 

.25 

.0625 

.0156 

.500 

.630 

6. 

2.449 

1.817 

14. 

3.742 

2.410 

.3 

.09 

.027 

.548 

.669 

.1 

2.470 

1.827 

.2 

3.768 

2.422 

.35 

.1225 

.0429 

.592 

.705 

.2 

2.490 

1.837 

.4 

3.795 

2.433 

.4 

.16 

.064 

.633 

.737 

.3 

2.510 

1.847 

.6 

3.821 

2.444 

45 

.2025 

.0911 

.671 

.766 

.4 

2.530 

1.857 

.8 

3.847 

2.455 

.5 

.25 

.125 

.707 

.794 

.5 

2.550 

1.866 

15. 

3.873 

2.466 

.55 

.3025 

.1664 

.742 

.819 

.6 

2.569 

1.876 

.2 

3.899 

2.477 

.6 

.36 

.216 

.775 

.843 

.7 

2.588 

1.885 

.4 

3.924 

2.488 

.65 

.4225 

.2746 

.806 

.866 

.8 

2.608 

1.895 

.6 

3.950 

2.499 

.7 

.49 

.343 

837 

.888 

.9 

2.627 

1.904 

.8 

3.975 

2 509 

.75 

.5625 

.4219 

.866 

.909 

7. 

2.646 

1.913 

16. 

4. 

2.520 

.8 

.64 

.512 

.894 

.928 

,i 

2.665 

1.922 

.2 

4.025 

2.530 

.85 

.7225 

.6141 

.922 

.947 

.2 

2.683 

1.931 

• 4 

4.050 

2.541 

.9 

.81 

.729 

.949 

.965 

.3 

2.702 

1.940 

.6 

4.074 

2.551 

.95 

.9025 

.8574 

.975 

.983 

.4 

2.720 

1.949 

.8 

4.099 

2.561 

1. 

1.000 

1.000 

1.000 

1.000 

.5 

2.739 

1.957 

17. 

4.123 

2.571 

.05 

1.103 

1.158 

1.025 

1.016 

.6 

2.757 

1.966 

.2 

4.147 

2.581 

i.i 

1.210 

1.331 

1.049 

1.032 

.7 

2.775 

1 975 

.4 

4.171 

2.591 

.15 

1.323 

1.521 

1.072 

1.048 

.8 

2.793 

1.983 

.6 

4.195 

2.601 

1.2 

1.440 

1.728 

1.095 

1.063 

.9 

2.811 

1.992 

.8 

4.219 

2.611 

.25 

1.563 

1.953 

1.118 

1.077 

8. 

2.8281 

2.000 

18. 

4.243 

2.621 

1.3 

1.690 

2.197 

1.140 

1.091 

.1 

2.846' 

2.008 

.2 

4.266 

2.630 

•35 

1.823 

2.460 

1.162 

1.105 

.2 

2.864 

2.017 

.4 

4.290 

2 640 

1.4 

1.960 

2.744 

1.183 

1.119 

.3 

2.881 

2.025 

.6 

4.313 

2.650 

.45 

2.103 

3.049 

1.204 

1.132 

.4 

2.898 

2.033 

.8 

4.336 

2.659 

1.5 

2.250 

3.375 

1.225 

1.145 

.5 

2.915 

2.041 

19. 

4.359 

2.668 

.55 

2.403 

3.724 

1.245 

1.157 

.6 

2.933 

2.049 

.2 

4.382 

2.678 

1.6 

2.560 

4.096 

1.265 

1.170 

.7 

2.950 

2.057 

.4 

4.405 

2.687 

.65 

2.723 

4.492 

1.285 

1.182 

.8 

2.966 

2.065 

.6 

4.427 

2.696 

17 

2.890 

4.913 

1.304 

1.193 

.9 

2.983 

2.072 

.8 

4.450 

2.705 

.75 

3.063 

5.359 

1.323 

1.205 

9. 

3. 

2.080 

20. 

4.472 

2.714 

1.8 

3.240 

5.832 

1.342 

1.216 

.1 

3.017 

2.088 

.2 

4.494 

2.723 

.85 

3.423 

6.332 

1.360 

1.228 

.2 

3.033 

2.095 

.4 

4.517 

2.732 

1.9 

3.610 

6.859 

1.378 

1.239 

.3 

3.050 

2.103 

.6 

4.539 

2.741 

.95 

3.803 

7.415 

1.396 

1.249 

.4 

3.066 

2.110 

.8 

4.561 

2.750 

a. 

4.000 

8.000 

1.414 

1.260 

.5 

3.082 

2.118 

21. 

4.583 

2.759 

.i 

4.410 

9.261 

1.449 

1.281 

.6 

3.098 

2.125 

.2 

4.604 

2.768 

.2 

4.840 

10.65 

1.483 

1.301 

.7 

3.114 

2.133 

.4 

4.626 

2.776 

.3 

5.290 

12.17 

1.517 

1.320 

.8 

3.130 

2.140 

.6 

4.648 

2.785 

.4 

5.760 

13.82 

1.549 

1.339 

.9 

3.146 

2.147 

.8 

4.669 

2.794 

.5 

6.250 

15.63 

1.581 

1.357 

10. 

3.162 

2.154 

22. 

4.690 

2.802 

.6 

6.760 

17.58 

1.612 

1.375 

.1 

3.178 

2.162 

.2 

4.712 

2.810 

.7 

7.290 

19.68 

1.643 

1.392 

.2 

3.194 

2.169 

.4 

4.733 

2.819 

.8 

7.840 

21.95 

1.673 

1.409 

.3 

3.209 

2.176 

.6 

4.754 

2.827 

.9 

8.410 

24.39 

1.703 

1.426 

.4 

3.225 

2.183 

.8 

4.775 

2.836 

8. 

9. 

27. 

1.732 

1.442 

.5 

3.240 

2.190 

23. 

4.796 

2.844 

.1 

9.61 

29.79 

1.761 

1.458 

.6 

3.256 

2.197 

.2 

4.817 

2.852 

.2 

10.24 

32.77 

1.789 

1.474 

.7 

3.271 

2.204 

.4 

4.837 

2.860 

.3 

10.89 

35.94 

1.811 

1.489 

.8 

3.286 

2.210 

.6 

4.858 

2.868 

.4 

11.56 

39.30 

1.844 

1.504 

.9 

3.302 

2.217 

.8 

4.879 

2.876 

.5 

12.25 

42.88 

1.871 

1.518 

11. 

3.317 

2.224 

24. 

4.899 

2.884 

.6 

12.96 

46.66 

1.897 

1.533 

.1 

3.332 

2.231 

.2 

4.919 

2.892 

.7 

13.69 

50.65 

1.924 

1.547 

.2 

3.347 

2.237 

.4 

4.940 

2.900 

.8 

14 44 

54.87 

1.949 

1.560 

.3 

3.362 

2.244 

.6 

4.960 

2.908 

.9 

15.21 

59.32 

1.975 

1.574 

.4 

3 376 

2.251 

.8 

4.980 

2.916 

4. 

16. 

64. 

2. 

1.587 

.5 

3.391 

2.257 

25. 

5. 

2.924 

.1 

16.81 

68.92 

2.025 

1.601 

.6 

3.406 

2.264 

.2 

5.020 

2.932 

.2 

17.64 

74.09 

2.049 

1 613 

.7 

3.421 

2.270 

.4 

5.040 

2.940 

.3 

18.49 

79.51 

2.074 

1.626 

.8 

3.435 

2.277 

.6 

5.060 

2.947 

.4 

19.36 

85.18 

2.098 

1.639 

.9 

3.450 

2.283 

.8 

5.079 

2.955 

.5 

20.25 

91.13 

2.121 

1.651 

12. 

3.464 

2 289 

26. 

5 099 

2.962 

.6 

21.16 

97.34 

2.145 

1.663 

.1 

3.479 

2.296 

.2 

5.119 

2.970 

.7 

22.09 

103.8 

2.168 

1.675 

.2 

3.493 

2.302 

.4 

5.138 

2.978 

.8 

23.04 

110.6 

2.191 

1.687 

.3 

3.507 

2.308 

.6 

5.158 

2.985 

.9 

24.01 

117.6 

2.214 

1.698 

.4 

3.521 

2.315 

.8 

5.177 

2.993 

5. 

25. 

125. 

2.236 

1.710 

.5 

3.536 

2.321 

27. 

5.196 

3 000 

.1 

26.01 

132.7 

2.258 

1.721 

.6 

3.550 

2.327 

.2 

5.215 

3 007 

.2 

27.04 

140.6 

2.280 

1.732 

.7 

3.564 

2.333 

.4 

5.235 

3.015 

.3 

28.09 

148.9 

2.302 

1.744 

.8 

3.578 

2 339 

.6 

5.254 

3.022 

.4 

29.16 

157.5 

2.324 

1.754 

.9 

3.592 

2.345 

.8 

5.273 

3.029 

.5 

3 C .25 

166.4 

2.345 

1.765 

13. 

3.606 

2.351 

28. 

5.292 

3.037 

.6 

| 31.36 

175.6 

2.366 

| 1.776 

.2 

3.633 

2.363 

.2 

5.310 

3.044 








































41 


SQUARES, CUBES, AND ROOTS. 


TABEE of Squares, Cubes, Square Roots, and Cube Roots, 
of Numbers from 1 to 1000. 

Remark on the following Table. Wherever the effect of a fifth decimal in the roots would be to 
add 1 to the fourth and final decimal in the table, the addition has been made. No errors. 


No . 


2 

3 

4 

5 

6 

7 

8 
9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

21 

25 

26 

27 

28 

29 

30 

31 


34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 


Square . 

Cube . 

Sq . Rt . 

C . Rt . 

No . 

Square . 

Cube . 

Sq . Rt . 

C . Rt . 

1 

1 

1.0000 

1.0000 

61 

3721 

226981 

7.8102 

3.9365 

4 

8 

1.4142 

1.2599 

62 

3844 

238328 

7.8740 

3.9579 

9 

27 

1.7321 

1.4422 

63 

3969 

250047 

7.9373 

3.9791 

16 

64 

2.0000 

1.5874 

61 

4096 

262144 

8.0000 

4 . 

25 

125 

2.2361 

1.7100 

65 

4225 

274625 

8.0623 

4.0207 

36 

216 

2.4495 

1.8171 

66 

4356 

287496 

8.1240 

4.0412 

49 

343 

2.6458 

1.9129 

67 

4489 

300763 

8.1854 

4 . 0 G 15 

64 

512 

2.8284 

2.0000 

68 

4624 

314432 

8.2462 

4.0817 

81 

729 

3.0000 

2.0801 

69 

4761 

328509 

8.3066 

4.1016 

100 

1000 

3.1623 

2.1544 

70 

4900 

343000 

8.3666 

4.1213 

121 

1331 

3.3166 

2.2240 

71 

5041 

357911 

8.4261 

4.1408 

144 

1728 

3.4641 

2.2894 

72 

5184 

373248 

8.4853 

4.1602 

169 

2197 

3.6056 

2.3513 

73 

5329 

389017 

8.5440 

4.1793 

196 

2744 

3.7417 

2.4101 

74 

5476 

405224 

8.6023 

4.1983 

225 

3375 

3.8730 

2.4662 

75 

5625 

421875 

8.6603 

4.2172 

256 

4096 

4.0000 

2.5198 

76 

5776 

438976 

8.7178 

4.2358 

289 

4913 

4.1231 

2.5713 

77 

5929 

456533 

8.7750 

4.2543 

324 

5832 

4.2426 

2.6207 

78 

6084 

474552 

8.8318 

4.2727 

361 

6859 

4.3589 

2.6684 

79 

6241 

493039 

8.8882 

4.2908 

400 

8000 

4.4721 

2.7144 

80 

6400 

512000 

8.9443 

4.3089 

441 

9261 

4.5826 

2.7589 

81 

6561 

531441 

9 . 

4.3267 

484 

10618 

4.6904 

2.8020 

82 

6724 

551368 

9.0554 

4.3445 

529 

12167 

4.7958 

2.8439 

83 

6889 

571787 

9.1104 

4.3621 

576 

13824 

4.8990 

2.8845 

84 

7056 

592704 

9.1652 

4.3795 

625 

15625 

5.0000 

2.9240 

85 

7225 

614125 

9.2195 

4.3968 

676 

17576 

5.0990 

2.9625 

86 

7396 

636056 

9.2736 

4.4140 

729 

19683 

5.1962 

3.0000 

87 

7569 

658503 

9.3274 

4.4310 

784 

21952 

5.2915 

3.0366 

88 

7744 

681472 

9.3808 

4.4480 

841 • 

24389 

5.3852 

3.0723 

89 

7921 

704939 

9.1340 

4.1647 

900 

27000 

5.4772 

3.1072 

90 

8100 

729000 

9.4868 

4.4814 

961 

29791 

5.5678 

3.1414 

91 

8281 

753571 

9.5394 

4.4979 

1024 

32768 

5.6569 

3.1748 

92 

8464 

778688 

9.5917 

4.5144 

1089 

35937 

5.7446 

3.2075 

93 

8649 

804357 

9.6437 

4.5307 

1156 

39304 

5.8310 

3.2396 

94 

8836 

830584 

9.6954 

4.5468 

1225 

42875 

5.9161 

3.2711 

95 

9025 

857375 

9.7468 

4.5629 

1296 

46656 

6.0000 

3.3019 

96 

9216 

884736 

9.7980 

4.5789 

1369 

50653 

6.0828 

3.3322 

97 

9409 

912673 

9.8489 

4.5917 

1444 

54872 

6.1644 

3.3620 

98 

9604 

941192 

9.8995 

4.6104 

1521 

59319 

6.2450 

3.3912 

99 

9801 

970299 

9.9499 

4.6231 

1600 

64000 

6.3246 

. 3.4200 

100 

10000 

1000000 

10 . 

4.6416 

1681 

68921 

6.4031 

3.4482 

101 

10201 

1030301 

10.0499 

4.6570 

1764 

74088 

6.4807 

3.4760 

102 

10104 

1061208 

10.0995 

4 . 67*23 

, 1849 

79507 

6.5574 

3.5034 

103 

10609 

1092727 

10.1489 

4.6875 

1936 

85184 

6.6332 

3.5303 

104 

10816 

1124864 

10.1980 

4.7027 

2025 

91125 

6.7082 

3.5569 

105 

11025 

1157625 

10.2470 

4.7177 

2116 

97336 

6.7823 

3.5830 

106 

11236 

1191016 

10.2956 

4.7326 

2209 

103823 

6.8557 

3.6088 

107 

11449 

1225043 

10.3141 

4.7475 

2304 

110592 

6.9282 

3.6342 

108 

11664 

1259712 

10.3923 

4.7622 

2401 

117619 

7.0000 

3.6593 

109 

11881 

1295029 

10.4403 

4.7769 

2500 

125000 

7.0711 

3.6840 

110 

12100 

1331000 

10.4881 

4.7014 

2601 

132651 

7.1414 

3.7084 

111 

12321 

1367631 

10.5357 

4.8059 

2704 

140608 

7.2111 

3.7325 

112 

12544 

1404928 

10.5830 

4.8203 

2809 

148877 

7.2801 

3.7563 

113 

12769 

1442897 

10.6301 

4.8346 

2916 

157464 

7.3485 

3.7798 

114 

12996 

1481544 

10.6771 

4.8488 

3025 

166375 

7.4162 

3.8030 

115 

13225 

1520875 

10 . 7*238 

4 . 86*29 

3136 

175616 

7.4833 

3.8259 

116 

13456 

1560896 

10.7703 

4.8770 

3249 

185193 

7.5498 

3.8485 

117 

13689 

1601613 

10.8167 

4.8910 

3304 

195112 

7.6158 

3 . 870 !) 

118 

13924 

1643032 

10.8628 

4.9049 

3481 

205379 

7.6811 

3.8930 

119 

14161 

1685159 

10.9087 

4 . 918 " 

3600 

216000 

7.7460 

3.9149 

120 

14400 

1728000 

10.9515 

4.9324 





































42 


SQUARES, CUBES, AND ROOTS 


TABLE of Squares, Cubes, Square Roots, an«l Cube Roots, 
of Numbers from 1 to 100(> — (Continued.) 


No . 

Square . 

Cube . 

Sq . Rt . 

C . Rt . 

No . 

Square . 

Cube . 

Sq . Rt . 

C . Rt . 


121 

14641 

1771561 

11 . 

4.9461 

186 

34596 

6434856 

13.6382 

5.7083 


122 

14884 

1815848 

11.0454 

4.9597 

187 

34969 

6539203 

113.(>748 

5.7185 


123 

15123 

1860867 

11.0905 

4.9732 

188 

35344 

6644672 

13.7113 

5.7287 


124 

15376 

1906624 

11.1355 

4.9866 

189 

35721 

6751269 

13.7477 

5.7388 


125 

15625 

1953125 

11.1803 

5 . 

190 

36100 

6859000 

13.7840 

5.7489 

P 

126 

15876 

2000376 

11.2250 

5.0133 

191 

36481 

6967871 

13.8203 

5.7590 


127 

16129 

2048383 

11.2694 

5.0265 

192 

36864 

7077888 

13.8564 

5.7690 


128 

16384 

2097152 

11.3137 

5.0397 

193 

37249 

7189057 

13.8924 

5 . 7790 . 


12 !) 

16641 

2146689 

11.3578 

5.0528 

194 

37636 

7301384 

13.9284 

5.7890 


130 

16900 

2197000 

11.4018 

5.0658 

195 

38025 

7414875 

13.9642 

5.7989 


131 

17161 

2248091 

11.4455 

5.0788 

196 

38416 

7529536 

14 . 

5.8088 


132 

17424 

2299968 

11.4891 

5.0916 

197 

38809 

7()45373 

14.0357 

5.8186 


133 

17689 

2352637 

11.5326 

5.1045 

198 

39204 

7762392 

14.0712 

5.8285 


134 

17956 

2406104 

11.5758 

5.1172 

199 

39601 

7880599 

14.1067 

5.8383 


135 

18225 

2460375 

11.6190 

5.1299 

200 

40000 

8000000 

14.1421 

5.8480 


136 

18496 

2515456 

11.6619 

5.1426 

201 

40401 

8120601 

14.1774 

5.8578 


137 

18769 

2571353 

11.7047 

5.1551 

202 

40804 

8242408 

14.2127 

5.8675 


138 

19044 

2628972 

11.7473 

5.1676 

203 

41209 

8365427 

14.2478 

5.8771 


13 !) 

19321 

2685619 

11.7898 

5.1801 

204 

41616 

8489664 

14.2829 

5.8868 


140 

19600 

2744000 

11.8322 

5.1925 

205 

42025 

8615125 

14.3178 

5.8964 


141 

19881 

2803221 

11.8743 

5.2048 

206 

42436 

8741816 

14.3527 

5.9059 


142 

20164 

2863288 

11.9164 

5.2171 

207 

42849 

8869743 

14.3875 

5.9155 


143 

20449 

2924207 

11.9583 

5.2293 

208 

43264 

8998912 

14.4222 

5.9250 


144 

20736 

2985984 

12 . 

5.2415 

209 

43681 

9129329 

14.4568 

5.9345 


145 

21025 

3048625 

12.0416 

5.2536 

210 

44100 

9261000 

14.4914 

5.9439 


146 

21316 

3112136 

12.0830 

5.2656 

211 

44521 

9393931 

14.5258 

5.9533 


147 

21609 

3176523 

12.1244 

5.2776 

212 

44944 

9528128 

14.5602 

5.9627 


148 

21904 

3241792 

12.1655 

5.2896 

213 

45369 

9663597 

14.5945 

5.9721 


14 !) 

22201 

3307949 

12.2036 

5.3015 

214 

45796 

9800344 

14.6287 

5.9814 


150 

22500 

3375000 

12.2474 

5.3133 

215 

46225 

9938375 

14.6629 

5.9907 

1 

151 

22801 

3442951 

12.2882 

5 3251 

216 

46656 

10077696 

14.6969 

6 . 


152 

23104 

3511808 

12.3288 

5.3368 

217 

47089 

10218!113 

14.7309 

6.0092 


153 

23403 

3581577 

12.3693 

5.3485 

218 

47524 

10360232 

14.7648 

6.0185 


154 

23716 

3652264 

12.4097 

5.3601 

219 

47961 

10503459 

14.7986 

6.0277 


155 

24025 

3723875 

12.4499 

5.3717 

220 

48400 

10648000 

14.8324 

6.0368 


156 

24336 

3796416 

12.4900 

5.3832 

221 

48841 

10793861 

14.8661 

6.0459 


157 

24649 

3869893 

12.5300 

5.3947 

222 

49284 

10941048 

14.8997 

6.0550 


158 

24964 

3944312 

12.5698 

5.4061 

223 

49729 

11089567 

14.9332 

6.0641 


15 !) 

25281 

4019679 

12.6095 

5.4175 

224 

50176 

11239424 

14.9666 

6.0732 


160 

25600 

4096000 

12.6491 

5.4288 

225 

50625 

11390625 

15 . 

6.0822 


161 

25921 

4173281 

12.6886 

5.4401 

226 

51076 

11543176 

15.0333 

6.0912 


162 

26244 

4251528 

12.7279 

5.4514 

227 

51529 

11697083 

15.0665 

6.1002 


163 

26569 

4330747 

12.7671 

5.4626 

228 

51984 

11852352 

15.0997 

6.1091 


164 

26896 

4410944 

12.8062 

5.4737 

229 

52141 

12008989 

15.1327 

6.1180 


165 

27225 

4492125 

12.8452 

5.4848 

230 

52900 

12167000 

15.1658 

6.1269 


106 

27556 

4574296 

12.8841 

5.4959 

231 

53361 

12326391 

15.1987 

6.1358 


107 

27889 

4657463 

12.9228 

5.5069 

232 

53824 

12487168 

15.2315 

6.1446 


168 

28224 

4741632 

12.9615 

5.5178 

233 

54289 

12649337 

15.2643 

45.1534 


163 

28561 

4826809 

13 . 

5.5288 

234 

54756 

12812904 

15.2971 

6.1622 


170 

28900 

4913000 

13.0384 

5.5397 

235 

55225 

12977875 

15.3297 

6.1710 


171 

29241 

5000211 

13.0767 

5.5505 

236 

55696 

13144256 

15.3623 

6.1797 


172 

29584 

5088448 

13.1149 

5.5613 

237 

56169 

13312053 

15.3948 

6.1885 


173 

29929 

5177717 

13.1529 

5.5721 

238 

56644 

13481272 

15.4272 

6.1972 


174 

30276 

5268024 

13.1909 

5.5828 

239 

57121 

13651919 

15.4596 

6.2058 


175 

30625 

5359375 

13.2288 

5.5934 

240 

57600 

13821000 

15.4919 

6.2145 


176 

30976 

5451776 

13.2665 

5.6041 

241 

58081 

13997521 

15.5242 

6.2231 


177 

31329 

5545233 

13.3041 

5.6147 

242 

58564 

14172488 

15.5563 

6.2317 


178 

31684 

5639752 

13.3417 

5.6252 

243 

59049 

14348907 

15.5885 

6.2403 


179 

32041 

5735339 

13.3791 

5.6357 

244 

59536 

14526784 

15.6205 

6.2488 


180 

32400 

5832000 

13.4164 

5.6462 

245 

60025 

14706125 

15.6525 

6.2573 


181 

32761 

5929741 

13.4536 

5.6567 

246 

60516 

14886936 

15.6844 

6.2658 


182 

33124 

6028568 

13.4907 

5.6671 

247 

61009 

15069223 

15.7162 

. 6.2743 


183 

33489 

6128487 

13.5277 

5.6774 

248 

61504 

15252992 

15.7480 

6.2828 


184 

33856 

6229504 

13.5647 

5.6877 

249 

62001 

15438249 

15.7797 

3.2912 


185 

34225 

6331625 

13.6015 

5.6980 

250 

62500 

15625000 

15.8114 

1 6.2936 



























SQUARES, CUBES, AND ROOTS 


43 


TABLE of Squares, Cubes, Square Roots, ami Cube Roots, 
of Numbers from 1 to 1000 — (Continued.) 


No . 

1 

Square . 

Cube . 

Sq . Rt . 

C . Rt 

No . 

Square . 

Cube . 

Sq . Rt . 

C . Rt . 

251 

63001 

15813251 

15.8430 

6.3030 

316 

99856 

31554496 

17.7764 

6.8113 

252 

63504 

16003008 

15.8745 

6.3164 

317 

100489 

31855013 

17.8045 

6.8185 

253 

64009 

16194277 

15.9060 

6.3247 

318 

101124 

32157432 

17.8326 

6.8256 

251 

64516 

16387064 

15.9374 

6.3330 

319 

101761 

32461759 

17.8606 

6.8328 

255 

65025 

16581375 

15.9687 

6.3413 

320 

102400 

32768000 

17.8885 

6.8399 

256 

65536 

16777216 

16 . 

6.3496 

321 

103041 

33076161 

17.9165 

6.8470 

257 

66049 

16974593 

16.0312 

6.3579 

322 

103684 

33386248 

17.9444 

6.8541 

258 

66564 

17173512 

16.0624 

6.3661 

323 

104329 

33698267 

17.9722 

6.8612 

259 

67081 

17373979 

16.0935 

6.3743 

324 

104976 

34012224 

18 . 

6.8683 

260 

67600 

17576000 

16.1245 

6.3825 

325 

105625 

34328125 

18.0278 

6.8753 

261 

68121 

17779581 

16.1555 

6.3907 

326 

106276 

34645976 

18.0555 

6.8824 

262 

68644 

17984728 

16.1864 

6.3988 

327 

106929 

34965783 

18.0831 

6.8894 

263 

69169 

18191447 

16.2173 

6.4070 

328 

107584 

35287552 

18.1108 

6.8964 

264 

, 69696 

18399744 

16.2481 

6.4151 

329 

108241 

35611289 

18.1384 

6.9034 

265 

70225 

18609625 

16.2788 

6.4232 

330 

108900 

35937000 

18.1659 

6.9104 

266 

70756 

18821096 

16.3095 

6.4312 

331 

109561 

36264691 

18.1934 

6.9174 

267 

71289 

19034163 

16.3401 

6.4393 

332 

110224 

36594368 

18.2209 

6.9244 

268 

71824 

19248832 

16.3707 

6.4473 

333 

110889 

36926037 

18.2483 

6.9313 

269 

72361 

19465109 

16.4012 

6.4553 

334 

111556 

37259704 

18.2757 

6.9382 

270 

72900 

19683000 

16.4317 

6.4633 

335 

112225 

37595375 

18.3030 

6.9451 

271 

73441 

19902511 

16.4621 

6.4713 

336 

112896 

37933056 

18.3303 

6.9521 

272 

73984 

20123648 

16.4924 

6.4792 

337 

113569 

38272753 

18.3576 

6.9589 

273 

74529 

20346417 

16.5227 

6.4872 

338 

114244 

38614472 

18.3848 

6.9658 

274 

75076 

20570824 

16.5529 

6.4951 

339 

114921 

38958219 

18.4120 

6.9727 

275 

75625 

20796875 

16.5831 

6.5030 

340 

115600 

39304000 

18.4391 

6.9795 

276 

76176 

21024576 

16.6132 

6.5108 

341 

116281 

39651821 

18.4662 

6.9864 

277 

76729 

21253933 

16.6433 

6.5187 

342 

116964 

40001688 

18.4932 

6.9932 

278 

77284 

21484952 

16.6733 

6.5205 

343 

117649 

40353607 

18.5203 

7 . 

279 

77841 

21717639 

16.7033 

6.5343 

344 

118336 

40707584 

18.5472 

7.0068 

280 

78400 

21952000 

16.7332 

6.5421 

345 

119025 

41063625 

18.5742 

7.0136 

281 

78961 

22188041 

16.7631 

6.5499 

346 

119716 

41421736 

18.6011 

7.0203 

282 

79524 

22425768 

16.7929 

6.5577 

347 

120409 

41781923 

13.6279 

7.0271 

283 

80089 

22665187 

16.8226 

6.5654 

348 

121104 

42144192 

18.6548 

7.0338 

284 

80656 

22906304 

16.8523 

6.5731 

349 

121801 

42508549 

18.6815 

7.0106 

285 

81225 

23149125 

16.8819 

6.5808 

350 

122500 

42875000 

18.7083 

7.0473 

286 

81796 

23393656 

16.9115 

6.5885 

351 

123201 

43243551 

18.7350 

7.0540 

287 

82369 

23639903 

16.9411 

6.5962 

352 

123904 

43614208 

18.7617 

7.0607 

288 

82944 

23887872 

16.9706 

6.6039 

353 

124609 

43986977 

18.7883 

7.0674 

289 

83521 

24137569 

17 . 

6.6115 

354 

125316 

44361864 

18.8149 

7.0740 

290 

84100 

24389000 

17.0294 

6.6191 

355 

126025 

44738875 

18.8414 

7.0807 

291 

84681 

24642171 

17.0587 

6.6267 

356 

126736 

45118016 

18.8680 

7.0873 

292 

85264 

24897088 

17.0880 

6.6343 

357 

127449 

45499293 

18.8944 

7.0940 

293 

85849 

25153757 

17.1172 

6.6419 

358 

128164 

45882712 

18.9209 

7.1006 

294 

86436 

25412184 

17.1464 

6.6494 

359 

128881 

46268279 

18.9473 

7.1072 

295 

87025 

25672375 

17.1756 

6.6569 

360 

129600 

46656000 

18.9737 

7.1138 

296 

87616 

25934336 

17.2047 

6.6644 

361 

130321 

47045881 

19 . 

7.1204 

297 

88209 

26198073 

17.2337 

6.6719 

362 

131044 

47437928 

19.0263 

7.1269 

298 

88804 

26463592 

17.2627 

6.6794 

363 

131769 

47832147 

19.0526 

7.1335 

299 

89401 

26730899 

17.2916 

6.6869 

364 

132496 

48228544 

19.0788 

7.1400 

300 

90000 

27000000 

17.3205 

♦ V 6943 

365 

133225 

48627125 

19.1050 

7.1466 

301 

90601 

27270901 

17.3494 

6.7018 

366 

133956 

49027896 

19.1311 

7.1531 

302 

91204 

27543608 

17.3781 

6.7092 

367 

134689 

49430863 

19.1572 

7.1596 

303 

91809 

27818127 

17.4069 

6.7166 

368 

135424 

49836032 

19.1833 

7.1661 

304 

92416 

28094464 . 

17.4356 

6.7240 

369 

136161 

50243409 

19.2094 

7.1726 

305 

93025 

28372625 

17.4642 

6.7313 

370 

136900 

50653000 

19.2354 

7.1791 

306 

93636 

28652616 

17.4929 

6.7387 

371 

137641 

51064811 

19.2614 

7.1855 

307 

94249 

28934443 

17.5214 

6.7460 

372 

138384 

51478848 

19.2873 

7.1920 

308 

94864 

29218112 

17.5499 

6.7533 

373 

139129 

51895117 

19.3132 

7.1984 

30 ) 

95481 

29503629 

17.5784 

6.7606 

374 

139876 

52313624 

19.3391 

7.2048 

310 

96100 

29791000 

17.6068 

6.7679 

375 

140625 

52 i 3437 o 

19.3649 

7.2112 

311 

96721 

30080231 

17.6352 

6.7752 

376 

141376 

53157376 

19.3907 

7.2177 

812 

97344 

30371328 

17.6635 

6.7824 

377 

142129 

53582633 

19.4165 

7.2240 

313 

97969 

30664297 

17.6918 

6.7897 

378 

142884 

54010152 

19.4422 

7.2304 

3 i 4 

98596 

30959144 

17.7200 

6.7969 

379 

143641 

54439939 

19.4679 

7.2368 

315 

99225 

31255875 

17.7482 

6.8041 

380 

14*400 

54872000 

19.4936 

7.2432 





































44 


SQUARES, CUBES, AND ROOTS 


9 


TARL.E of Squares, Cubes, Square Roots, and Cube Roots, 
of lumbers from 1 to 1000 — (Continued.) 


No. 

Square. 

Cube. 

Sq. Rt. 

C. Rt. 

No. 

Square. 

Cube. 

Sq. Rt. 

C. Rt. 

381 

145161 

55306341 

19.5192 

7.2495 

446 

198916 

88716536 

21.1187 

76103 

382 

145924 

55742968 

19.5448 

7.2558 

447 

199809 

89314623 

21.1424 

7.6460 

383 

146689 

56181887 

19.5704 

7.2622 

448 

200704 

89915392 

21.1660 

7.6517 

384 

147456 

56623104 

19.5959 

7.2685 

449 

201601 

90518849 

21.1896 

7.6574 

385 

148225 

57066625 

19.6214 

7.2748 

450 

202500 

91125000 

21.2132 

7.6631 

386 

148996 

57512456 

19.6469 

7.2811 

451 

203401 

91733851 

21.2368 

7.6688 

387 

149769 

57960603 

19.6723 

7.2874 

452 

204304 

92345408 

21.2603 

7.6744 

388 

150544 

58411072 

19.6977 

7.2936 

453 

205209 

92359677 

21.2838 

7.6801 

383 

151321 

58863869 

19.7231 

7.2999 

454 

206116 

93576664 

21.3073 

7.6857 

390 

152100 

59319000 

19.7484 

7.3061 

455 

207025 

94196375 

21.3307 

7.6914 

391 

152881 

59776471 

19.7737 

7.3124 

456 

207936 

94818816 

21.3542 

7.6970 

392 

153664 

60236288 

19.7990 

7.3186 

457 

208849 

95443993 

21.3776 

7.7026 

393 

154449 

60698457 

19.8242 

7.3248 

458 

209764 

93071912 

21.4009 

7.7082 

394 

155236 

61162984 

19.8494 

7.3310 

459 

210681 

96702579 

21.4243 

7.7138 

395 

156025 

61629875 

19.8746 

7.3372 

460 

211600 

97336000 

21.4476 

7.7194 

396 

156816 

62099136 

19.8997 

7.3434 

461 

212521 

97972181 

21.4709 

7.7250 

397 

157609 

62570773 

19.9249 

7.3496 

462 

213444 

98811128 

21.4942 

7.7306 

398 

158404 

63044792 

19.9499 

7.3558 

483 

214369 

99252847 

21.5174 

7.7362 

399 

159201 

63521199 

19.9750 

7.3619 

464 

215296 

99897344 

21.5407 

7.7418 

400 

160000 

64000000 

20. 

7.3681 

4(>5 

216225 

100544625 

21.5639 

7.7473 

401 

160801 

64481201 

20.0250 

7.3742 

466 

217156 

101194696 

21.5870 

7.7529 

402 

161604 

64964808 

20.0499 

7.3803 

467 

218089 

191847563 

21.6102 

7.7584 

403 

162409 

65450827 

20.0749 

7.38G4 

468 

219024 

102503232 

. 21.6333 

7.7639 

404 

163216 

65939264 

20.0996 

7.3925 

463 

219961 

103161709 

21.6564 

7.7695 

405 

164025 

66430125 

20.1246 

7.3986 

470 

220900 

103823000 

21.6795 

7.7750 

406 

164836 

66923416 

20.1494 

7.4047 

471 

221841 

104487111 

21.7025 

7.7805 

407 

165649 

67419143 

20.1742 

7.4108 

472 

222784 

105154048 

21.7256 

7.78(H) 

403 

166464 

67917312 

20.1990 

7.4169 

473 

223729 

105823817 

21.7486 

7.7915 

409 

167281 

63417923 

20.2237 

7.4229 

474 

224676 

108496121 

21.7715 

7.7970 

410 

168100 

68921000 

20.2485 

7.4290 

475 

225625 

107171875 

21.7945 

7.8025 

411 

168921 

69426531 

20.2731 

7.4350 

478 

226576 

107850176 

21.8174 

7.8079 

412 

169714 

6)934528 

20.2978 

7.4410 

477 

227529 

108531333 

21.8403 

7.8134 

413 

170569 

70444997 

20.3224 

7.4470 

478 

228484 

109215352 

21.8032 

7.8188 

414 

171396 

70957944 

20.3470 

7.4530 

479 

229441 

109902239 

21.8861 

7.8243 

415 

172225 

71473375 

20.3715 

7.4590 

480 

230400 

110592000 

21.9089 

7.8297 

416 

173056 

71991296 

20.3961 

7.4650 

481 

231361 

111284641 

21.9317 

7.8352 

417 

173889 

72511713 

20.4206 

7.4710 

482 

232324 

111380168 

21.9545 

7.8406 

418 

174724 

73034632 

20.4450 

7.4770 

483 

233289 

112678587 

21.9773 

7.8160 

419 

175561 

73560059 

20.4695 

7.4829 

484 

234256 

113379904 

• 22. 

7.8514 

420 

176400 

74088000 

20.4939 

7.4889 

485 

235225 

114084125 

22.0227 

7.8568 

421 

177241 

74618461 

20.5183 

7.4948 

486 

236196 

114791256 

22.0454 

7.8622 

422 

178084 

75151448 

20.5426 

7.5007 

487 

237169 

115501303 

22.0381 

7.8670 

423 

178929 

75686967 

20.5670 

7.5067 

488 

238144 

116214272 

22.0907 

7.873(0 

424 

179776 

76225024 

20.5913 

7.5126 

48.9 

233121 

116933169 

22.1133 

7.8784 

425 

180625 

76765625 

20.6155 

7.5185 

490 

240100 

117649000 

22.1359 

7.8837 

426 

181476 

77308776 

20.6398 

7.5244 

491 

241081 

118370771 

22.1585 

7.8891 

427 

182329 

77854483 

20.6640 

7.5302 

492 

242064 

119095488 

22.1811 

7.8944 

428 

183184 

78402752 

20.6882 

7.5361 

433 

243049 

119823157 

22.2036 

7.8998 

429 

184041 

78953589 

20.7123 

7.5420 

4)4 

244036 

120553784 

22.2261 

7.9051 

430 

184900 

79507000 

20.7364 

7.5478 

495 

245025 

121287375 

22.2486 

7.9105 

431 

185761 

80062991 

20.7605 

7.5537 

496 

246016 

122023936 

22.2711 

7.9158 

432 

186624 

80621568 

20.7846 

7.5595 

497 

247009 

122763473 

22.2935 

7.9211 

433 

187489 

81182737 

20.8087 

7.5654 

498 

248004 

123505902 

22.3159 

7.9264 

434 

188356 

81746504 

20.8327 

7.5712 

499 

249001 

124251499 

22.3383 

7.9317 

435 

189225 

82312875 

20.8567 

7.5770 

500 

250000 

125000000 

22.3607 

7.9370 

436 

190096 

82881856 

20.8806 

7.5828 

501 

251001 

125751501 

22.3830 

7.9423 

437 

190969 

83453453 

20.9045 

7.5886 

502 

252004 

126503008 

22.4054 

7.9476 

438 

191844 

84027672 

20.9284 

7.5944 

503 

253009 

127263527 

22.4277 

7.9528 

439 

192721 

84604519 

20.9523 

7.6001 

504 

254016 

128021061 

22.4499 

7.9581 

440 

193600 

85184000 

20.9762 

7.6059 

505 

255025 

128787625 

22.4722 

7.9634 

441 

194481 

85766121 

21. 

7.6117 

506 

256036 

129554216 

' 22.4944 

7.9686 

442 

195364 

86350888 

21.0238 

7.6174 

507 

2570 49 

130323843 

265167 

7.9739 

44!! 

196249 

86938307 

21.0476 

7.6232 

508 

258064 

131096512 

22.5389 

7.9791 

444 

197136 

87528384 

21.0713 

7.6289 

509 

259081 

131872229 

22.5610 

. 7.9843 

445 

198025 

88121125 

21.0950 

7.6346 

510 

260100 

132651000 

22.5832 

7.9896 
































SQUARES, CUBES, AND ROOTS 


45 


TABLE of Squares, Cubes, Square Roots, and Cube Roots, 
of lumbers from 1 to 1000 — (Continued.) 


No. 

Square 

Cube. 

Sq. Ht. 

C. Ht. 

No. 

Square. 

Cube. 

Sq. Ht. 

C. Ht. 

511 

261121 

133432831 

22.6053 

7.9948 

576 

331776 

191102976 

24. 

8.3203 

512 

262144 

134217728 

22.6274 

8. 

577 

332929 

192100033 

24.0208 

8.3251 

513 

263169 

135005697 

22.6495 

8.0052 

578 

334084 

193100552 

24.0416 

8.3300 

5U 

264196 

135796744 

22.6716 

8.0104 

579 

335241 

194104539 

24.0624 

8.3348 

515 

265225 

136590875 

22.6936 

8.0156 

680 

336400 

195112000 

24.0832 

8.3396 

516 

266256 

137388096 

22.7156 

8.0208 

581 

337561 

196122941 

24.1039 

8.3443 

517 

267289 

138188413 

22.7376 

8.0260 

582 

338724 

197137368 

24.1247 

8.3491 

518 

268324 

138991832 

22.7596 

8.0311 

583 

339889 

198155287 

24.1454 

8.3539 

519 

269361 

139798359 

22.7816 

8.0363 

584 

341056 

199176704 

24.1661 

8.3587 

520 

270400 

140608000 

22.8035 

8.0415 

585 

342225 

200201625 

24.1868 

8.3634 

521 

271441 

141420761 

22.8254 

8.0466 

586 

343396 

201230056 

24.2074 

8.3682 

522 

272484 

142236648 

22.8473 

8.0517 

587 

344569 

202262003 

24.2281 

8.3730 

523 

273529 

143055667 

22.8692 

8.0569 

588 

345744 

203297472 

24.2487 

8.3777 

521 

274576 

143877824 

22.8910 

8.0620 

589 

346921 

204336469 

24.2693 

8.3825 

525 

275625 

144703125 

22.9129 

8.0671 

590 

348100 

205379000 

24.2899 

8.3872 

526 

276676 

145531576 

22.9347 

8.0723 

591 

349281 

206425071 

24.3105 

8.3919 

527 

277729 

146363183 

22.9565 

8.0774 

592 

350464 

207474688 

24.3311 

8.3967 

528 

278784 

147197952 

22.9783 

8.0825 

593 

351649 

208527857 

24.3516 

8.4014 

529 

279841 

148035889 

23. 

8.0876 

594 

352836 

209584584 

24.3721 

8.4061 

530 

280900 

148877000 

23.0217 

8.0927 

595 

354025 

210644875 

24.3926 

8.4108 

531 

281961 

149721291 

23.0434 

8.0978 

596 

355216 

211708736 

24.4131 

8.4155 

532 

283024 

150568768 

23.0651 

8.1028 

597 

356409 

212776? 73 

24.4336 

8.4202 

533 

284089 

151419437 

23.0868 

8.1079 

598 

357604 

213847192 

24.4540 

8.4249 

534 

285156 

152273304 

2.3.1084 

8.1130 

599 

358801 

214921799 

24.4745 

8.4296 

535 

286225 

153130375 

23.1301 

8.1180 

600 

360000 

216000000 

24.4949 

8.4343 

536 

287296 

153990656 

23.1517 

8.1231 

601 

361201 

217081801 

24.5153 

8.4390 

537 

288369 

154854153 

23.1733 

8.1281 

602 

362404 

218167208 

24.5357 

8.4437 

538 

289444 

155720872 

23.1948 

8.1332 

603 

363609 

219256227 

24.5561 

8.4484 

539 

290521 

156590819 

23.2164 

8.1382 

604 

364816 

220348864 

24.5764 

8.4530 

540 

291600 

157464000 

23.2379 

8.1433 

605 

366025 

221445125 

24.5967 

8.4577 

541 

292681 

158340421 

23.2594 

8.1483 

606 

367236 

222545016 

24.6171 

8.4623 

542 

293764 

159220088 

23.2809 

8.1533 

607 

368449 

223648543 

24.6374 

8.4670 

543 

291849 

160103007 

23.3024 

8.1583 

608 

369664 

224755712 

24.6577 

8.4716 

544 

295936 

160989184 

23.3238 

8.1633 

609 

370881 

225866529 

24.6779 

8.4763 

545 

297025 

161878625 

23.3452 

8.1683 

610 

372100 

226981000 

24.6982 

8.4809 

546 

298116 

162771336 

23.3666 

8.1733 

611 

373321 

228099131 

24.7184 

8.4856 

547 

299209 

163667323 

23.3880 

8.1783 

612 

374544 

229220928 

24.7383 

8.4902 

548 

300304 

164566592 

23.4094 

8.1833 

613 

375769 

230346397 

24.7588 

8.4948 

549 

301401 

165469149 

23.4307 

8.1882 

614 

376996 

231475544 

24.7790 

8.4994 

550 

302500 

166375000 

23.4521 

8.1932 

615 

378225 

232608375 

24.7992 

8.5040 

551 

303601 

167284151 

23.4734 

8.1982 

616 

379456 

233744896 

24.8193 

8.5086 

552 

304704 

168196608 

23.4947 

8.2031 

617 

380689 

234885113 

24.8395 

8.5132 

553 

305809 

169112377 

23.5160 

8.2081 

618 

381924 

236029032 

24.8596 

8.5178 

554 

306916 

170031464 

23.5.372 

8.2130 

619 

383161 

237176659 

24.8797 

8.5224 

555 

308025 

170953875 

23.5584 

8.2180 

620 

384400 

238328000 

24.8998 

8.5270 

556 

309136 

171879616 

23.5797 

8.2229 

621 

385641 

239483061 

24.9199 

8.5316 

557 

310249 

172808693 

23.6008 

' 8.2278 

622 

386884 

240641848 

24.9399 

8.5362 

558 

311364 

173741112 

23.6220 

8.2327 

623 

388129 

241804367 

24.9600 

8.5408 

559 

312481 

174676879 

23.6432 

8.2377 

624 

389376 

242970624 

24.9800 

8.5453 

560 

313600 

175616000 

23.6643 

8.2426 

625 

390625 

244140625 

25. 

8.5499 

561 

314721 

176558481 

23.6854 

8.2475 

626 

391876 

245314376 

25.0200 

8.5544 

562 

315844 

177504328 

23.7065 

8.2524 

627 

393129 

246491883 

25.0400 

8.5590 

563 

316969 

178453547 

23.7276 

8.2573 

628 

394384 

247673152 

25.0599 

8.5635 

564 

318096 

179406144 

23.7487 

8 2621 

629 

395641 

248858189 

25.0799 

8.5681 

565 

319225 

180362125 

23.7697 

8.2670 

630 

396900 

250047000 

25.0998 

8.5726 

566 

320356 

181321496 

23.7908 

8.2719 

631 

398161 

251239591 

25.1197 

8.5772 

567 

321489 

182284263 

23.8118 

8.2768 

632 

399424 

252435968 

25.1396 

8.5817 

568 

322624 

183250432 

23.8328 

8.2816 

633 

400689 

253636137 

25.1595 

8.5862 

569 

323761 

184220009 

23.8537 

8.2865 

634 

401956 

254840104 

25.1.794 

8.5907 

570 

324900 

185193000 

23.8747 

8.2913 

635 

403225 

256047875 

25.1992 

8.5952 

571 

326041 

186169411 

23.8956 

8.2962 

636 

404496 

257259456 

25.2190 

8.5997 

572 

327184 

187149248 

23.9165 

8.3010 

637 

405769 

258474853 

25.2389 

8.6043 

573 

328329 

188132517 

23.9374 

8.3059 

638 

407044 

259694072 

25.2587 

8.6088 

571 

329 476 

189119224 

23.9583 

8.3107 

639 

408321 

260917119 

25.2784 

8.6132 

575 

330625 

190109375 

23.9792 

8.3155 

640 

409600 

262144000 

25.2982 

8.6177 












































46 


SQUARES, CUBES, AND ROOTS, 


TABLE of Squares, Cubes, Square Roots, ami Cube Roots, 
of Numbers from I to 100b — (Continued.) 


No. 

Square. 

Cube. 

Sq. Rt. 

C. Rt. 

No. 

Square. 

Cube. 

Sq. Rt. 

C. Rt. 

641 

410881 

263374721 

25.3180 

8.6222 

706 

498436 

351895816 

26.5707 

8.9043 

642 

412164 

264609288 

25.3377 

8.6267 

707 

499849 

353393243 

26.5895 

8.9085 

643 

413449 

265847707 

25.3574 

8.6312 

708 

501264 

354894912 

26.6083 

8.9127 

644 

414736 

267089984 

25.3772 

8.6357 

709 

502681 

356400829 

26.6271 

8.9169 

645 

416025 

268336125 

25.3969 

8.6401 

710 

504100 

357911000 

26.6458 

8.9211 

646 

417316 

269586136 

25.4165 

8.6446 

711 

505521 

359425431 

26.6646 

8.9253 

647 

418609 

270840023 

25.4362 

8.6190 

712 

506944 

360944128 

26.6833 

8.9295 

648 

419904 

272097792 

25.4558 

8.6535 

713 

508369 

362467097 

26.7021 

8.9337 

649 

421201 

273359449 

25.4755 

8.6579 

714 

509796 

363994344 

26.7208 

8.9378 

650 

422500 

274625000 

25.4951 

8.6624 

715 

511225 

365525875 

26.7395 

8.9420 

651 

423801 

275894451 

25.5147 

8.6668 

716 

512656 

367061696 

26.7582 

6.9462 

652 

425104 

277167808 

25.5343 

8.6713 

717 

514089 

368601813 

26.7769 

8.9503 

653 

426 409 

278445077 

25.5539 

8.6757 

718 

515524 

370146232 

26.7955 

8.9545 

654 

427716 

279726264 

25.5734 

8.6801 

719 

516961 

371694959 

26.8142 

8.9587 

655 

429025 

281011375 

25.5930 

8.6845 

720 

518400 

373248000 

26.8328 

8.9628 

656 

430336 

282300416 

25.6125 

8.6890 

721 

519841 

374805361 

26.8514 

8.9670 

657 

431649 

283593393 

25.6320 

8.6934 

722 

521284 

376367048 

26.8701 

8.9711 

658 

432964 

284890312 

25.6515 

8.6978 

723 

522729 

377933067 

26.8887 

8.9752 

659 

434281 

286191179 

25.6710 

8.7022 

724 

524176 

379503424 

26.9072 

8.9794 

660 

435600 

287496000 

25.6905 

8.70(»(J 

725 

525625 

381078125 

26.9258 

8.9835 

661 

436921 

288804781 

25.7099 

8.7110 

726 

527076 

382657176 

26.9444 

8.9876 

662 

438244 

290117528 

25.7294 

8.7154 

727 

528529 

384240583 

26.9629 

8.9918 

663 

439569 

291434247 

25.7488 

8.7198 

728 

529984 

385828352 

26.9815 

8.9959 

664 

440896 

292754944 

25.7682 

8.7241 

729 

531441 

387420489 

27. 

9. 

665 

442225 

294079625 

25.7876 

8.7285 

730 

532900 

389017000 

27.0185 

9.0041 

666 

443556 

295408296 

25.8070 

8.7329 

731 

534361 

390617891 

27.0370 

9.0082 

6(57 

444889 

296740963 

25.8263 

8.7373 

732 

535824 

392223168 

27.0555 

9.0123 

668 

446224 

298077632 

25.8457 

8.7413 

733 

537289 

393832837 

27.0740 

9.0164 

669 

447-61 

299418309 

25.8650 

8.7460 

734 

538756 

395446904 

27.0924 

9.0205 

670 

448900 

300763000 

25.8844 

8.7503 

735 

540225 

397065375 

27.1109 

9.0246 

671 

450241 

302111711 

25.9037 

8.7547 

736 

541696 

398688256 

27.1293 

9.0287 

672 

451584 

303464448 

25.9230 

8.7590 

737 

543169 

400315553 

27.1477 

9.0328 

673 

452929 

304821217 

25.9422 

8.7634 

738 

544644 

401947272 

27.1662 

9.0369 

674 

451276 

306182024 

25.9615 

8.7677 

739 

546121 

403583419 

27.1816 

9.0410 

675 

455625 

307546875 

25.9808 

8.7721 

740 

547600 

405224000 

27.2029 

9.0450 

676 

456976 

308915776 

26. 

8.7764 

741 

549081 

406869021 

27.2213 

9.0491 

677 

458329 

310288733 

26.0192 

8.7807 

742 

550564 

408518488 

27.2397 

9.0532 

678 

459684 

311665752 

26.0384 

8.7850 

743 

552049 

410172407 

27.2580 

9.0572 

679 

461041 

313046839 

26.0576 

8.7893 

744 

553536 

411830784 

27.2764 

9.0613 

680 

462400 

314432000 

26.0768 

8.7937 

745 

555025 

413493625 

27.2947 

9.0654 

681 

463761 

315821241 

26.0960 

8.7980 

746 

556516 

415160936 

27.3130 

9.0694 

682 

465124 

317214568 

26.1151 

8.8023 

747 

558009 

416832723 

27.3313 

9.0735 

683 

466189 

318611987 

26.1343 

8.8066 

748 

559504 

418508992 

27.3496 

9.0775 

684 

467856 

320013504 

26.1534 

8.8109 

749 

561001 

420189749 

27.3679 

9.0816 

685 

469225 

321419125 

26.1725 

8.8152 

750 

562500 

421875000 

27.3861 

9.0856 

686 

470596 

322828856 

26.1916 

8.8194 

751 

564001 

423564751 

27.4044 

9.0896 

687 

471969 

324242703 

26.2107 

8.8237 

752 

565504 

425259008 

27.4226 

9.0937 

688 

473344 

325660672 

26.2298 

8.8280 

753 

567009 

426957777 

27.4408 

9.0977 

689 

474721 

327082769 

26.2488 

8.8323 

754 

568516 

428661064 

27.4591 

9.1017 

690 

476100 

328509000 

26.2679 

8.8366 

755 

570025 

430368875 

27.4773 

9.1057 

691 

477481 

329939371 

26.2869 

8.8408 

756 

571536 

432081216 

27.4955 

9.1098 

692 

478864 

331373888 

26.3059 

8.8451 

757 

573049 

433798093 

27.5136 

9.1138 

693 

480249 

335.812557 

26.3249 

8.8493 

758 

574564 

435519512 

27.5318 

9.1178 

694 

481636 

334255384 

26.3439 

8.8536 

759 

576081 

437245479 

27.5500 

9.1218 

695 

483025 

535702375 

26.3629 

8.8578 

760 

577600 

438976000 

27.5681 

9.1258 

696 

484416 

337153536 

26.3818 

8.8621 

761 

579121 

440711081 

27.5862 

9.1298 

697 

485809 

338608873 

26.4008 

8.8663 

762 

580644 

442450728 

27.6043 

9.1338 

698 

487204 

340068392 

26.4197 

8.8706 

763 

582169 

444194947 

27.6225 

9.1378 

699 

488601 

341532099 

26.4386 

8.8748 

764 

583696 

445943744 

27.6105 

9.1418 

700 

490000 

343000000 

26.4575 

8.8790 

765 

585225 

447697125 

27.6586 

9.1458 

701 

491401 

344472101 

26.4761 

8.8833 

766 

586756 

449455096 

27.6767 

9.1498 

702 

492804 

345948408 

26.4953 

8.8875 

767 

588289 

451217663 

27.6948 

9.1537 

703 

494209 

347428927 

26.5141 

8 8917 

768 

589824 

452984832 

27.7128 

9.1577 

704 

495616 

348913664 

26.5330 

k 8.8959 

769 

591361 

454756609 

27.7308 

9.1617 

705 

497025 

350402625 

26.5518 

8.9001 

770 

592900 

456533000 

27.7489 

9.1657 




























SQUARES, CUBES, AND ROOTS 


47 


TABIiE of Squares, Cubes, Square Roots, and Cube Roots, 
of Numbers from 1 to 1000— (Continued.; 


No. 

Square 

771 

594441 

772 

595984 

773 

597529 

774 

599076 

775 

600H25 

776 

602176 

777 

603729 

778 

605284 

779 

606841 

780 

608400 

781 

609961 

782 

611524 

783 

613089 

784 

614656 

785 

616225 

786 

617796 

787 

619369 

788 

620944 

789 

622521 

790 

624100 

791 

625681 

792 

627264 

793 

628849 

794 

630436 

795 

632025 

796 

633616 

797 

635209 

798 

636804 

799 

638401 

800 

640000 

801 

641601 

802 

643204 

803 

644809 

804 

646416 

805 

648025 

806 

649636 

807 

6512 49 

808 

652864 

809 

654481 

810 

656100 

811 

657721 

812 

659344 

813 

660969 

814 

662596 

815 

664225 

816 

665856 

817 

667489 

818 

669124 

819 

670761 

820 

672400 

821 

674041 

822 

675684 

823 

677329 

82 4 

678976 

825 

680625 

826 

682276 

827 

683929 

828 

685584 

829 

687241 

830 

688900 

an 

690561 

832 

692224 

833 

693889 

834 1 

695556 

835 1 

697225 


Cube. 

Sq. Rt. 

a 

W 

rf 

No. 

Square. 

458314011 

27.7669 

9.1696 

836 

698896 

460099648 

27.7849 

9.1736 

837 

700569 

461889917 

27.8029 

9.1775 

838 

702244 

463684824 

27.8209 

9.1815 

839 

703921 

465484375 

27.8388 

9.1855 

840 

705600 

467288576 

27.8568 

9.1894 

841 

707281 

469097433 

27.8747 

9.1933 

842 

708964 

470910952 

27.8927 

9.1973 

843 

710649 

472729139 

27.9106 

9.2012 

844 

712336 

474552000 

27.9285 

9.2052 

845 

714025 

476379541 

27.9464 

9.2091 

846 

715716 

478211768 

27.9643 

9.2130 

847 

717409 

480048687 

27.9821 

9.2170 

848 

719104 

481890304 

28. 

9.2209 

849 

720801 

483736625 

28.0179 

9.2248 

850 

722500 

485587656 

28.0357 

9.2287 

851 

724201 

487443403 

28.0535 

9.2326 

852 

725904 

489303872 

28.0713 

9.2365 

853 

727609 

491169069 

28.0891 

9.2404 

854 

729316 

493039000 

28.1069 

9.2443 

855 

731025 

494913671 

28.1247 

9.2482 

856 

732736 

496793088 

28.1425 

9.2521 

857 

734449 

498677257 

28.1603 

9.2560 

858 

736164 

500566184 

28.1780 

9.2599 

859 

737881 

502459875 

28.1957 

9.2638 

860 

739600 

504358336 

28.2135 

9.2677 

861 

741321 

506261573 

28.2312 

9.2716 

862 

743044 

508169592 

28.2489 

9.2754 

863 

744769 

510082399 

28.2666 

9.2793 

864 

746496 

512000000 

28.2843 

9.2832 

865 

748225 

513922401 

28.3019 

9.2870 

866 

749956 

515849608 

28.3196 

9.2909 

867 

751689 

517781627 

28.3373 

9.2948 

868 

753424 

519718464 

28.3549 

9.2986 

869 

755161 

521660125 

28.3725 

9.3025 

870 

756900 

523606616 

28.3901 

9.3063 

871 

758641 

525557943 

28.4077 

9.3102 

872 

760384 

527514112 

28.4253 

9.3140 

873 

762129 

529475129 

28.4429 

9.3179 

874 

763876 

531441000 

28.4605 

9.3217 

875 

765625 

533411731 

28.4781 

9.3255 

876 

767376 

535387328 

28.4956 

9.3294 

877 

769129 

537367797 

28.5132 

9.3332 

878 

770884 

539353144 

28.5307 

9.3370 

879 

772641 

541343375 

28.5482 

9.3408 

880 

774400 

543338496 

28.5657 

9.3447 

881 

776161 

545338513 

28.5832 

9.3485 

882 

777924 

547343432 

28.6007 

9.3523 

883 

779689 

549353259 

28.6182 

9.3561 

884 

781456 

551368000 

28.6356 

9.3599 

885 

783225 

553387661 

28.6531 

9.3637 

886 

784996 

555412248 

28.6705 

9.3675 

887 

786769 

557441767 

28.6880 

9.3713 

888 

788544 

559476224 

28.7054 

9.3751 

889 

790321 

561515625 

28.7228 

9.3789 

890 

792100 

563559976 

28.7402 

9.3827 

891 

793881 

565609283 

28.7576 

9.3865 

892 

795664 

567668552 

28.7750 

’ 9.3902 

893 

797149 

569722789 

28.7924 

9.3940 

894 

799236 

571787000 

28.8097 

9.3978 

895 

801025 

573856191 

28.8271 

9.4016 

896 

802816 

575930368 

28.8444 

9.4053 

897 

804609 

578009537 

28.8617 1 

9.4091 

898 

806104 

580093704 

28.8791 

9.4129 

899 

808201 

582182875 

28.8964 1 

9.4166 

900 1 

810000 


Cube. 


584277056 

586376253 

588480472 

590589719 

592704000 

594823321 

596947688 

599077107 

601211584 

603351125 

605495736 

607645423 

609800192 

611960049 

614125000 


Sq. Rt. 


28.9137 

28.9310 

28.9482 

28.9655 

28.9828 

29 

29.0172 
29.0345 
29.051 1 
29.0689 

29.0861 

29.1033 

29.1204 

29.1376 

29.1548 


C. Rt.. 


9.4204 

9.4241 

9.4279 

9.4316 

9.4354 

9.4391 

9.4429 

9.4466 

9.4503 

9.4541 

9.4578 

9.4615 

9.4652 

9.4690 

9.4727 


616295051 

618470208 

620650477 

622835864 

625026375 


29.1719 
29 1890 
29.2062 
29.2233 
29.2404 


9.4764 
9.4801 
9 4838 
9.4875 
9.4912 


627222016 

629422793 

631628712 

633839779 

636056000 


29.2575 

29.2746 

29.2916 

29.3087 

29.3258 


9.4949 

9.4986 

9.5023 

9.5060 

9.5097 


638277381 

640503928 

642735647 

644972544 

647214625 


29.3428 

29.3598 

29.3769 

29.3939 

29.4109 


9.5134 

9.5171 

9.5207 

9.5244 

9.5281 


649461896 

651714363 

653972032 

656234909 

658503000 


29.4279 

29.4449 

29.4618 

29.4788 

29.4958 


9.5317 

9.5354 

9.5391 

9.5427 

9.5464 


660776311 

663054848 

665338617 

667627624 

669921875 


29.5127 

29.5296 

29.5466 

29.5635 

29.5804 


9.5501 

9.5537 

9.5574 

9.5610 

9.5647 


672221376 

674526133 

676836152 

679151439 

681472000 


29.5973 

29.6142 

29.6311 

29.6479 

29.6648 


9.5683 

9.5719 

9.5756 

9.5792 

9.5828 


683797841 

686128968 

688465387 

690807104 

693154125 


29.6816 

29.6985 

29.7153 

29.7321 

29.7489 


9.5865 

9.5901 

9.5937 

9.5973 

9.6010 


695506456 

697864103 

700227072 

702595369 

704969000 


29.7658 

29.7825 

29.7993 

29.8161 

29.8329 


9.6046 

9.6082 

9.6118 

9.6154 

9.6190 


707347971 

709732288 

712121957 

714516984 

716917375 


29.8496 

29.8664 

29.8831 

29.8998 

29.91(H) 


9.6226 

9.6262 

9.6298 

9.6334 

9.6370 


719323136 

721734273 

724150792 

726572699 

729000000 


29.9333 

29.9500 

29.9666 

29.9833 

30. 


9.6106 
9.6442 
9.0477 
9 6513 
9.6o49 








































48 


SQUARES, CUBES, AND ROOTS 


TABLE of Squares, Cubes, Square Roots, and Cube Roots, 
of Numbers from 1 to 1000— (Continued.) 


No. 

Square. 

Cube. 

Sq. Rt. 

C. Rt. 

No. 

Square. 

Cube. 

Sq. Rt. 

C. Rt. 

901 

811801 

731432701 

30.0167 

9.6585 

951 

904401 

860985351 

30.8383 

9.8339 

902 

813601 

733870808 

30.0333 

9.6620 

952 

906304 

862501408 

30.8545 

{J..8.S74 

903 

815109 

736314327 

30.0500 

9.6656 

953 

908209 

865523177 

30.8707 

9.8408 

904 

817216 

738763204 

30.0666 

9.6692 

954 

910116 

868250664 

30.8869 


905 

819025 

741217625 

30.0832 

9.6727 

955 

912025 

870983875 

30.9031 

9.S477 

906 

820836 

743677116 

30.0998 

9.6763 

956 

913936 

873722816 

30.9192 

9.8511 

907 

822649 

746142643 

30.1164 

9.6799 

957 

915849 

876407493 

30.9354 

9.8546 

908 

824464 

748613312 

30.1330 

9.6834 

958 

917764 

879217912 

30.9516 

9.85S0 

90) 

826281 

751089429 

30.1496 

9.6870 

959 

919681 

881974079 

30.9677 

9.8614 

910 

828100 

753571000 

30.1662 

9.6905 

960 

921600 

884736000 

30.9839 

y.8648 

911 

829921 

756058031 

30.1828 

9.6941 

961 

923521 

887503681 

31. 

9.8683 

912 

831714 

758550528 

30.1993 

9.6976 

962 

925444 

890277128 

31.0161 

9.8717 

913 

833569 

761048497 

30.2159 

9.7012 

963 

927369 

893056347 

31.0322 

9.C751 

914 

835396 

763551914 

30.2324 

9.7017 

964 

929296 

895841344 

31.0483 

9.8785 ; 

915 

837225 

7tifi060875 

30.2490 

9.7082 

965 

931225 

898632125 

31.0644 

9.8819 

916 

839056 

768575296 

30.2655 

9.7118 

966 

933156 

901428696 

31.0805 

9.8854 

917 

840889 

771095213 

30.2820 

9.7153 

967 

935089 

904231063 

31.0966 

9.8888 | 

918 

842724 

773620632 

30.2985 

9.7188 

968 

937024 

907039232 

31.1127 

9.8922 j 

919 

814561 

776151559 

30.3150 

9.7224 

969 

93S961 

909S53209 

31.1288 

9.8956 i 

920 

846400 

778688000 

30.3315 

9.7259 

970 

940900 

912673000 

31.1448 

9.8990 

921 

818241 

781229961 

30.3480 

9.7294 

971 

942841 

915498611 

31.1609 

9.9024 

922 

850084 

783777448 

80.3645 

9.7329 

972 

914784 

918330048 

31.1769 

9.9058 

923 

851929 

786330467 

30.3809 

9.7364 

973 

946729 

921167317 

31.1929 

9.9092 

924 

853776 

788889024 

30.3974 

9.7400 

974 

948676 

924010424 

31.2090 

9.9126 

925 

855625 

791453125 

30.4138 

9.7435 

975 

950625 

926859375 

31.2250 

9.9160 

926 

857476 

794022776 

30.4302 

9.7470 

976 

952576 

929714176 

31.2410 

9.9194 

927 

859329 

796597983 

30.4467 

9.7505 

977 

954529 

932574833 

31.2570 

9.9227 

928 

861184 

799178752 

30.4631 

9.7540 

978 

956484 

935441352 

31.2730 

9.9261 

929 

863011 

801765089 

30.4795 

9.7575 

979 

958441 

938313739 

31.2890 

9.9295 

930 

'864900 

804357000 

30.4959 

9.7610 

980 

960400 

941192000 

31.3050 

9.9329 

931 

866761 

806954491 

30.5123 

9.7645 

981 

962361 

944076141 

31.3209 

9.9363 

932 

868624 

809557568 

30.5287 

9.7680 

982 

964324 

946966168 

31.3369 

9.9396 

933 

870489 

812166237 

30.5450 

9.7715 

983 

<(66289 

949862087 

31.3528 

9.9430 

931 

872356 

S14780504 

30.5614 

9.7750 

984 

968256 

952763904 

31.3688 

9.9464 

935 

874225 

817400375 

30.5778 

9.7785 

985 

970225 

955671625 

31.3847 

9.9497 

936 

876096 

820025856 

30.5941 

9.7819 

986 

972196 

958585256 

31.4006 

9.9531 

937 

87796^ 

822653953 

30.6105 

9.7854 

987 

974169 

961504803 

31.4166 

9.9565 

938 

879814 

825293072 

30.6268 

9.7889 

988 

976144 

964430272 

31.4325 

9.9598 

939 

881721 

827936019 

30.6431 

9.7924 

989 

978121 

967361669 

31.4484 

9.9632 

940 

883600 

830584000 

30.6594 

9.7959 

990 

980100 

970299000 

31.4643 

9.9666 

911 

885481 

833237621 

30.6757 

9.7993 

991 

982081 

973242271 

31.4802 

9.9699 

912 

887364 

835896888 

30.6920 

9.8028 

992 

984064 

976191488 

31.4960 

9.9733 

943 

889219 

838561807 

30.7083 

9.8063 

993 

986049 

979146657 

31.5119 

9.9766 

914 

891136 

841232384 

30.7246 

9.8097 

994 

988036 

982107784 

31.5278 

9.9800 

945 

893025 

843908625 

30.7409 

9.8132 

995 

990025 

985074875 

31.5436 

9.9833 

916 

894916 

846590536 

30.7571 

9.8167 

996 

992016 

988047936 

31.5595 

9.9866 

917 

896809 

84927812.3 

30.7734 

9.8201 

997 

994009 

991026973 

31.5753 

9.9900 

948 

898701 

851971392 

30.7896 

9.8236 

998 

996004 

994011992 

31.5911 

9.9933 

919 

900601 

854670349 

30.8058 

9.8270 

999 

998001 

997002999 

31.6070 

9.9967 

950 

902500 

857375000 

30.8221 

9.8305 

1000 

1000000 

1000000000 

31.6228 

10. 


To find the square or cube of any whole number ending 
with ciphers. First, omit all the final ciphers. Take from the table th 

square or cube (as the case may be) of the rest of the number. To this square add twice as man 
ciphers as there were Huai ciphers in the original number. To the cube add three times as many a 
in the original number. Thus, for 905002; 90S 2 = Ml9025. Add twice 2 ciphers, obtaining 819025000( 
For 905003, 9053 — 741217625. Add 3 times 2 ciphers, obtaining 741217625000000. 

For tables of fifth roots, fifth powers, and square roots of fiftl 

pow ers, see pp 251 to 253. 
































SQUARE AND CUBE ROOTS, 


49 


Square Hoots and Cube Hoots of Numbers from 1000 to 10000. 

.—. ---- No errors. 


Num. 

Sq. Rt 

. Cu. Rt 

Num. 

Sq. Rt 

^Cu. Rt 

Num. 

Sq. Rt. 

Cu. Rt. 

Num. 

Sq. Rt. 

Cu. Rt 

1005 

31.70 

10.02 

1405 

37.48 

11.20 

1805 

42.49 

12.18 

2205 

46.96 

13.02 

loll) 

31.78 

10.0.1 

1410 

37.55 

11.21 

1810 

42.54 

12.19 

2210 

47.01 

13.03 

JOlo 


10.05 

1415 

37.62 

11.23 

1815 

42.60 

12.20 

2215 

47.06 

13.04 

10-0 

31.94 

10.07 

1420 

37.68 

11.24 

1820 

42.66 

12.21 

2220 

47.12 

13.05 

1015 

32.02 

10.08 

1425 

37.75 

11.25 

1825 

42.72 

12.22 

2225 

47.17 

13.05 

1030 

32.09 

10.10 

1430 

37.82 

11.27 

1830 

42.78 

12.23 

2230 

47.22 

13.06 

1055 

32.17 

10.12 

1435 

37.88 

11.28 

1835 

42.84 

12.24 

2235 

47.28 

13.07 

1040 

32.25 

10.13 

1440 

37.95 

11.29 

1840 

42.90 

12.25 

2240 

47.33 

13.08 

1015 

32.33 

10.15 

1445 

38.01 

11.31 

1845 

42.95 

12.26 

2245 

47.38 

13.09 

1050 

32.40 

10.16 

1450 

38.08 

11.32 

1850 

43.01 

12.28 

2250 

47.43 

13.10 

1055 

32.48 

10.18 

1455 

38.14 

11.33 

1855 

43.07 

12.29 

2255 

47.49 

13.11 

1000 

32.56 

10.20 

1460 

38.21 

11.34 

1860 

43.13 

12.30 

2260 

47.54 

13.12 

1005 

32.63 

10.21 

1465 

38.28 

11.36 

1865 

43.19 

12.31 

2265 

47.59 

13.13 

1070 

32.71 

10.23 

1470 

38.34 

11.37 

1870 

43.24 

12.32 

2270 

47.64 

13.14 

1075 

32.79 

10.24 

1475 

38.41 

11.38 

1875 

43.30 

12.33 

2275 

47.70 

13.15 

1080 

32.86 

10.26 

1480 

38.47 

11.40 

1880 

43.36 

12.34 

2280 

47.75 

13.16 

1085 

32.94 

10.28 

1485 

38.54 

11.41 

1885 

43.42 

12.35 

2285 

47.80 

13.17 

1090 

33.02 

10.29 

1490 

38.60 

11.42 

1890 

43.47 

12.36 

2290 

47.85 

13.18 

1095 

33.03 

10.31 

1495 

38.67 

11.43 

1895 

43.53 

12.37 

2295 

47.91 

13.19 

1100 

33.17 

10.32 

1500 

38.73 

11.45 

1900 

43.59 

12.39 

2300 

47.96 

13.20 

1105 

33.24 

10.34 

1505 

38.79 

11.46 

1905 

43.65 

12.40 

2305 

48.01 

13.21 

1110 

33.32 

10.35 

1510 

38.86 

11.47 

1910 

43.70 

12.41 

2310 

48.06 

13.22 

1115 

33.39 

10.37 

1515 

38.92 

11.49 

1915 

43.76 

12.42 

2315 

48.11 

13.23 


1120 

33.47 

10.38 

1520 

38.99 

11.50 

1920 

43.82 

12.43 

2320 

48.17 

13.24 


1125 

33.54 

10.40 

1525 

39.05 

11.51 

1925 

43.87 

12.44 

2325 

48.22 

13.25 


1130 

33.62 

10.42 

1530 

39.12 

11.52 

1930 

43.93 

12.45 

2330 

48.27 

13.26 


1135 

33.69 

10.43 

1535 

39.18 

11.54 

1935 

43.99 

12.46 

2335 

48.32 

13.27 


1140 

33.76 

10.45 

1540 

39.24 

11.55 

1940 

44.05 

12.47 

2340 

48-37 

13.28 


1145 

33.84 

10.46 

1545 

39.31 

11.56 

1945 

44.10 

12,48 

2345 

48.43 

13.29 


1150 

33.91 

10.48 

1550 

39.37 

11.57 

1950 

44.16 

12.49 

2350 

48.48 

13.30 


1155 

33.99 

10.49 

1555 

39.43 

11.C9 

1955 

44.22 

12.50 

2355 

48.53 

13.30 


1100 

34.06 

10.51 

1560 

39.50 

11.60 

I960 

44.27 

12.51 

2360 

48.58 

13.31 


1165 

34.13 

10.52 

1565 

39.56 

11.61 

1965 

44.33 

12.53 

2365 

48.63 

13.32 


1170 

34.21 

10.54 

1570 

39.62 

11.62 

1970 

44.38 

12.54 

2370 

48.68 

13.33 


1175 

34.28 

10.55 

1575 

39.69 

11.63 

1975 

44.44 

12.55 

2375 

48.73 

13.34 


1180 

34.35 

10.57 

1580 

39.75 

1 1 .65 

1980 

44.50 

12.56 

2380 

48.79 

13.35 


1185 

34.42 

10.58 

1585 

39.81 

11.66 

1985 

44.55 

12.57 

2385 

48.84 

13.36 


1190 

34.50 

10.60 

1590 

39.87 

11.67 

1990 

44.61 

12.58 

2390 

48.89 

13.37 


1195 

34.57 

10.61 

1595 

39.94 

11.68 

1995 

44.67 

12.59 

2395 

48.94 

13.38 


1200 

34.64 

10.63 

1600 

40.00 

11.70 

2000 

44.72 

12.60 

2400 

48.99 

13.39 


1205 

34.71 

10.64 

1605 

40.06 

11.71 

2005 

44.78 

12.61 

2405 

49.04 

13.40 


1210 

34.79 

10.66 

1610 

40.12 

11.72 

2010 

44.83 

12.62 

2410 

49.09 

13.41 


1215 

34.86 

10.67 

1615 

40.19 

11.73 

2015 

44.89 

12.63 

2415 

49.14 

13.42 


1220 

34.93 

10.69 

1620 

40.25 

11.74 

2020 

44.94 

12.64 

2420 

49.19 

13.43 


1225 

35.00 

10.70 

1625 

40.31 

11.76 

2025 

45.00 

12.65 

2425 

49.24 

13.43 


1230 

35.07 

10.71 

1630 

40.37 

11.77 

2030 

45.06 

12.66 

2430 

49.30 

13.44 

1 

1235 

35.14 

10.73 

1635 

40.44 

11.78 

2035 

45.11 

12.67 

2435 

49.35 

13.45 

1 

1240 

35.21 

10.74 

1640 

40.50 

11.79 

2040 

45.17 

12.68 

2440 

49.40 

13.46 


1215 

35.28 

10.76 

1645 

40.56 

11.80 

2045 

45.22 

12.69 

2445 

49.45 

13.47 

A 

1250 

35.36 

10.77 

1650 

40.62 

11.82 

2050 

45.28 

12.70 

2450 

49.50 

13.48 


1255 

35.43 

10.79 

1655 

40.68 

11.83 

2055 

45.33 

12.71 

2460 

49.60 

13.50 


1260 

35.50 

10.80 

1660 

40.74 

11.84 

2060 

45.39 

12.72 

2470 

49.70 

13.52 


1265 

35.57 

10.82 

1665 

40.80 

11.85 

2065 

45.44 

12.73 

2480 

49.80 

13.54 

i 

1270 

35.64 

10.83 

1670 

40.87 

11.88 

2070 

45.50 

12.74 

2490 

49.90 

13.55 


1275 

35.71 

10.84 

1675 

40.93 

11.88 

2075 

45.55 

12.75 

2500 

50.00 

13.57 


1280 

35.78 

10.86 

1680 

40.99 

11.89 

2080 

45 61 

12.77 

2510 

50.10 

13.59 


1285 

35.85 

10.87 

1685 

41.05 

11.90 

2085 

45.66 

12.78 

2520 

50.20 

13.61 


1290 

35.92 

10.89 

1690 

41.11 

11.91 

2090 

45.72 

12.79 

2530 

50.30 

13.63 


1295 

35.99 

10.90 

1695 

41.17 

11.92 

2095 

45.77 

12.80 

2540 

50.40 

13.64 


1300 

36.06 

10.91 

1700 

41.23 

11.93 

2100 

45.83 

12.81 

2550 

50 50 

13.66 


1305 

36.12 

10.93 

1705 

41.29 

11.95 

2105 

45.88 

12.82 

2560 

50.60 

13 68 


1310 

36.19 

10.94 

1710 

41.35 

11.96 

2110 

45.93 

12.83 

2570 

50.70 

13.70 


1315 

36.26 

10.96 

1715 

41.41 

11.97 

2115 

45.99 

12.84 

2580 

50.79 

13.72 

> 

1320 

86.33 

10.97 

1720 

41.47 

11.98 

2120 

46.04 

12.85 

2590 

50.89 

13.73 


1325 

36.40 

10.98 

1725 

41.53 

11.99 

2125 

46.10 

12.86 

2600 

50.99 i 

13.75 

S 

1330 

36.47 

11.00 

1730 

41.59 

12.00 

2130 

46.15 

12.87 

2610 

51.09 

13.77 

ip 

1335 

36.54 

11.01 

1735 

41.65 

12.02 

2135 

46.21 

12.88 

2620 

51.19 

13.79 

nT 

1340 

36.61 

11.02 

1740 

41.71 

12.03 

2140 

46.26 

12.89 

2630 

51.28 

13.80 

L 

1345 

36.67 

11.04 

1745 

41.77 

12.04 

2145 

46.31 

12.90 

2640 

51.38 

13.82 


1350 

36.74 

11.05 

1750 

41.83 

12.05 

2150 

46.37 

12.91 

2650 ; 

51.48 

13.84 


1355 

36.81 

11.07 

1755 

41.89 

12.06 

2155 

46.42 

12.92 

2660 

51.58 

13.86 


1300 

36.88 

11.08 

1760 

41.95 

12.07 

2160 

46.48 

12.93 

2670 

51.67 

13.87 

ti 

1365 

36.95 

11.09 

1765 

42.01 

12.09 

2165 

46.53 

12.94 

2680 

51.77 

13.89 


1370 

37.01 

11.11 

1770 

42.07 

12.10 

2170 

46.58 

12.95 

2690 

51.87 

13.91 



37.08 

11.12 

1775 

42.13 

12.11 

2175 

4f».fr4 

12.96 

2700 

51.96 

13.92 


1380 

37.15 

11.13 

1780 

42.19 

12.12 

2180 

46.69 

12.97 

2710 

52.06 

13.94 



37.22 

11.15 

1785 

42.25 

12.13 

2185 

46.74 

12.98 

2720 

52.15 

13.96 


1390 

37.28 

11.16 

1790 

42.31 

12.14 

2190 

46.80 

12.99 

2730 

52.25 

13.98 


1395 

87.35 

11.17 

1795 

42.37 

12.15 

2195 

46.85 

13.00 

2740 

52.35 

13.99 

1 

1400 

37.42 

11.19 

1800 

42.43 

12.16 

2200 

46.90 

13.01 

2750 

52.44 

14.01 


4 
















































50 


SQUARE AND CUBE ROOTS, 


Square Roofs and Cube Roots of Numbers from 1000 to 10000 

—(Continued.) 


Num. 

Sq. Rt. 

Cu. Rt. 

Num. 

Sq. Rt. 

Cu. Rt. 

Num. 

Sq. Rt. 

Cu. Rt. 

Num. 

Sq. Rt. 

Cu. Rt. 

2760 

52.54 

14.03 

3550 

59.58 

15.25 

4340 

65.88 

16.31 

5130 

71.62 

17.25 

2770 

52.63 

14.04 

3560 

59.67 

15.27 

4350 

65.95 

16.32 

5140 

71.69 

1726 

2780 

52.73 

14.06 

3570 

59.75 

15.28 

4360 

66.03 

16.34 

5150 

71.76 


2790 

52.82 

14.08 

3580 

59.83 

15.30 

4370 

66.11 

16.35 

5160 

71.83 

17.28 

2800 

52.92 

14.09 

3590 

59.92 

15.31 

4380 

66.18 

16.36 

5170 

71.90 

17.29 

2810 

53.01 

14.11 

3600 

60.00 

15.33 

4390 

66.26 

16.37 

5180 

71.97 

17 30 

2820 

53.10 

14.13 

3610 

60.08 

15.34 

4400 

66.33 

16.39 

5190 

72.04 

17.31 

2830 

53.20 

14.14 

3620 

60.17 

15.35 

4410 

66.41 

16.40 

5200 

72.11 

17.32 

2810 

53.29 

14.16 

3630 

60.25 

15.37 

4420 

66.48 

16.41 

5210 

72.18 

17.34 

2850 

53.39 

14.18 

3640 

60.33 

15.38 

4430 

66.56 

16.42 

5220 

72.25 

17.35 

2H(>0 

53.48 

14.19 

3650 

60.42 

15.40 

4440 

66.63 

16.44 

5230 

72.32 

17.36 

2870 

53.57 

14.21 

3660 

60.50 

15.41 

4450 

66.71 

16.45 

5240 

72.39 

17.37 

2880 

53.67 

14.23 

3670 

60.58 

15.42 

4460 

66.78 

16.46 

5250 

72.46 

17.38 

2890 

53.76 

14.24 

3680 

60.66 

15.44 

4470 

66.86 

16.47 

5260 

72.53 

17.39 

2900 

53.85 

14.26 

3690 

60.75 

15.45 

4480 

66.93 

16.49 

5270 

72.59 

1740 

2910 

53.94 

14.28 

3700 

CO.83 

15.47 

4490 

67.01 

1650 

5280 

72.66 

17.41 

2920 

54.04 

14.29 

3710 

60.91 

15.48 

4500 

67.08 

16.51 

5290 

72.73 

17.42 

29.50 

51.13 

14.31 

3720 

60.99 

15.49 

4510 

67.16 

16.52 

5300 

72.80 

17.44 

2910 

54.22 

14.33 

3730 

61.07 

15.51 

4520 

67.23 

16.53 

5310 

72.87 

17.45 

2950 

54.31 

14.34 

3740 

61.16 

15.52 

4530 

67.31 

16.55 

5320 

72.94 

17.46 

2960 

54.41 

14.36 

3750 

61.24 

15.54 

4540 

67.38 

16.56 

5330 

73.01 

17.47 

2970 

54.50 

14.37 

3760 

61.32 

15.55 

4550 

67.45 

16.57 

5340 

73.08 

17.48 

2980 

54.59 

14.39 

3770 

61.40 

15.56 

4560 

67.53 

16.58 

5350 

73.14 

17.49 

2990 

54.68 

14.41 

3780 

61.48 

15.58 

4570 

67.60 

16.59 

5360 

73.21 

17.50 

8000 

54.77 

14.42 

3790 

61.56 

15.59 

4580 

67.68 

16.61 

5370 

73.28 

17.51 

3010 

54.86 

14.44 

3800 

61 64 

15.60 

4590 

67.75 

16.62 

5380 

73.35 

17.52 

3020 

54.95 

14.45 

3810 

61.73 

15.62 

4600 

67.82 

16.63 

5390 

73.42 

17.53 

3030 

55.05 

14.47 

3820 

6} .81 

15.63 

4610 

67.90 

16.64 

5400 

73.48 

17.54 

3010 

55.14 

14.49 

3830 

61.89 

15.65 

4620 

67.97 

16.66 

5410 

73.55 

17.55 

3050 

do. 23 

14.50 

3840 

61.97 

15.66 

4630 

68.04 

16.67 

5420 

73.62 

17.57 

3000 

55.32 

14.52 

3850 

62.05 

15.67 

4640 

68.12 

16.68 

5430 

73.69 

17.58 

3070 

55.41 

14.53 

3860 

62.13 

15.69 

4650 

68.19 

16.69 

5440 

73.76 

17.59 

3080 

55.50 

14.55 

3870 

62.21 

15.70 

4660 

68.26 

16.70 

5450 

73.82 

17.60 

3090 

55.59 

14.57 

3,"SO 

62.29 

15.71 

4670 

68.34 

16.71 

5460 

73.89 

17.61 

3100 

55.68 

11.58 

3890 

62.37 

15.73 

4680 

68.41 

16.73 

5470 

73.96 

17.62 

3110 

55.77 

14.60 

3900 

62.45 

15.74 

4690 

68.48 

16.74 

5480 

74.03 

17.63 

3120 

55.86 

14.61 

3910 

62.53 

15.75 

4700 

68 56 

16.75 

5490 

74.09 

17.64 

3130 

55.95 

14.63 

3920 

62.61 

15.77 

4710 

68.63 

16.76 

5500 

74.16 

17.65 

3110 

56.04 

14.64 

3930 

62.69 

15.78 

4720 

68.70 

16.77 

5510 

74.23 

17.66 

3150 

56.12 

14.66 

3940 

62.77 

15.79 

4730 

68.77 

16.79 

5520 

74.30 

17.67 

3160 

56.21 

14.67 

3950 

62.85 

15.81 

4740 

68.85 

16.80 

5530 

74.36 

17.68 

3170 

56.30 

14.69 

3960 

62.93 

15.82 

4750 

68.92 

16.81 

5540 

74.43 

17.69 

3180 

56.39 

14.71 

3970 

63.01 

15.83 

4760 

68 99 

16.82 

5550 

74.50 

17.71 

3190 

56.48 

14.72 

3980 

63.09 

15.85 

4770 

69.07 

16.83 

5560 

74.57 

17.72 

3200 

56.57 

14.74 

3990 

63.17 

15.8(5 

4780 

69.14 

16.85 

5570 

74.63 

17.73 

3210 

56.66 

14.75 

4000 

63.25 

15.87 

4790 

69 21 

16.86 

5580 

74.70 

17.74 

3220 

56.75 

11.77 

4010 

63.32 

15.89 

4800 

69.28 

16.87 

5590 

74.77 

17.75 

•3230 

56.83 

14.78 

4020 

63.40 

15.90 

4810 

69.35 

16.88 

5600 

74.83 

17.76 

3210 

56.92 

14.80 

4030 

63.48 

15.91 

4820 

69.43 

16 89 

5610 

74.90 

17.77 

3250 

57.01 

14 81 

4040 

63.56 

15.93 

4830 

69.50 

16.90 

5620 

74.97 

17.78 

3260 

57.10 

14.83 

4050 

63.64 

15.94 

4840 

69.57 

16.92 

5630 

75.03 

17.79 

3270 

57.18 

14.84 

4060 

63.72 

15.95 

4850 

69.64 

16.93 

5640 

75.10 

17.80 

3280 

57.27 

14.86 

4070 

63 80 

15.97 

4860 

69.71 

16.94 

5650 

75.17 

17.81 

3290 

57.36 

14.87 

4080 

63.87 

15.98 

4870 

69.79 

16.95 

5660 

75.23 

17.82 

3300 

57.45 

14.89 

4090 

63.95 

15.99 

4880 

69.86 

16.96 

5670 

75.30 

17.83 

3310 

57.53 

14.90 

4100 

64.03 

16.01 

4890 

69.93 

16.97 

5680 

75.37 

17.84 

3320 

57.62 

14.92 

4110 

64.11 

16.02 

4900 

70.00 

16.98 

5690 

75.43 

17.85 

3380 

57.71 

14.93 

4120 

64.19 

16.03 

4910 

70.07 

17.00 

5700 

75.50 

17.86 

3310 

57.79 

14.95 

4130 

64.27 

16.04 

4920 

70.14 

17.01 

5710 

75.56 

17.87 

3350 

57.88 

14.96 

4140 

64.34 

16.06 

4930 

70.21 

17.02 

5720 

75.63 

17.88 

3360 

57.97 

14.98 

4150 

64.42 

16.07 

4940 

70.29 

17.03 

5730 

75.70 

17.89 

3370 

58.05 

14.99 

4160 

64.50 

16.08 

4950 

70.36 

17.04 

5740 

75.76 

17.90 

3380 

58.14 

15,01 

4170 

64.58 

16.10 

4960 

70.43 

17.05 

5750 

75.83 

17.92 1 

3390 

58.22 

15.02 

4180 

64.65 

16.11 

4970 

70.50 

17.07 

5760 

75.89 

17.93 

3100 

58.31 

15.04 

4190 

64.73 

16.12 

4980 

70.57 

17.08 

5770 

75.96 

17.94 ! 

3110 

58.40 

15.05 

4200 

64.81 

16.13 

4990 

70.64 

17.09 

5780 

76.03 

17.95 i 

3420 

58.48 

15.07 

4210 

64.88 

16.15 

5000 

70.71 

17.10 

5790 

76.09 

17.96 

3130 

58.57 

15.08 

4220 

64.96 

16.16 

5010 

70.78 

17.11 

5800 

76.16 

17.97 I 

3440 

58.65 

15.10 

4230 

65.04 

16.17 

5020 

70.85 

17.12 

5810 

76.22 

17.98 

3450 

58.74 

15.11 

4240 

65.12 

16.19 

5030 

70.92 

17.13 

5820 

76.29 

17.99 p 

3460 

58.82 

15.12 

4250 

65.19 

16.20 

5010 

70.99 

17.15 

5830 

76.35 

18.00 

3470 

58.91 

15.14 

4260 

65.27 

16.21 

5050 

71.06 

17.16 

5840 

76.42 

18.01 

3480 

58.99 

15.15 

4270 

65.35 

16.22 

5060 

71.13 

17.17 

5850 

76.49 

18.02' 

3190 

59.08 

15.17 

4280 

65.42 

16.24 

5070 

71.20 

17.18 

5860 

76.55 

18.03 

3500 

59.16 

15.18 

4290 

65.50 

16.25 

5080 

71.27 

17.19 

5870 

76.62 

18.04 

3510 

59.25 

15.20 

4300 

65 57 

16.26 

5090 

71.34 

17.20 

5880 

76.68 

18.05 

3520 

59.33 

15.21 

4310 

65.65 

16.27 

5100 

71.41 

17.21 

5890 

76.75 

18.06 

3530 

59.41 

15.23 

4320 

65.73 

16.29 

5110 

71.48 

17.22 

5900 

76.81 

18.07 

3540 

59.50 

15.24 

4330 

65.80 

16.30 

5120 

71.55 

17.24 

5910 

76.88 

18.08 













































SQUARE AND CUBE ROOTS, 


51 


Square Roots and Cube Roots of Numbers from 1000 to 10000 


— (Continued.) 


Kura. 

Sq. RtJcu. Rt 

Num. 

Sq. Rt. Cu. Rt 

Num. 

Sq. Rt 

Cu. Rt 

Num. 

Sq. Rt 

Cu. Rt 

5920 

76 94 

18.09 

6710 

81.91 

18.86 

7500 

86.60 

19.57 

8290 

91.05 

20.24 

593C 

77.01 

18.10 

6720 

81.98 

18.87 

7510 

86.66 

19.58 

8300 

91.10 

20.25 

5910 

77.07 

18.11 

6730 

82.04 

18.88 

7520 

86.72 

19.59 

8310 

91.16 

20.26 

5950 

77.14 

18.12 

6740 

82.10 

18.89 

7530 

86.78 

19.60 

8320 

91.21 

20.26 

6900 

77.20 

18.13 

6750 

82.16 

18.90 

7540 

86.83 

19.61 

8330 

91.27 

20.27 

697C 

77.27 

18.14 

6760 

82.22 

18.91 

7550 

86.89 

19.62 

8340 

91.32 

20.28 

5980 

77.33 

18.15 

6770 

82.28 

18.92 

7560 

86.95 

19.63 

8350 

91.38 

20.29 

5990 

77.40 

18.16 

6780 

82.34 

18.93 

7570 

87.01 

19.64 

8360 

91.43 

20.30 

6000 

77.46 

18.17 

6790 

82.40 

18.94 

7580 

87.06 

19.64 

8370 

91.49 

20.30 

6010 

77.52 

18.18 

6800 

82.46 

18.95 

7590 

87.12 

19.65 

8380 

91.54 

20.31 

6020 

77.59 

18.19 

6810 

• 82.52 

18.95 

7600 

87.18 

19.66 

8390 

91.60 

20.32 

6050 

77.65 

18.20 

6820 

82.58 

18.96 

7610 

87.24 

19.67 

8400 

91.65 

20.33 

6010 

77.72 

18.21 

6830 

82.64 

18.97 

7620 

87.29 

19.68 

8410 

91.71 

20.34 

6050 

77.78 

18.22 

6840 

82.70 

18.98 

7630 

87.35 

19.69 

8420 

91.76 

20.34 

6000 

77.85 

18.25 

6850 

82.76 

18.99 

7640 

87.41 

19.70 

8430 

91.82 

20.35 

6070 

77.91 

18.24 

6860 

82.83 

19.00 

7650 

87.46 

19.70 

8440 

91.87 

20.36 

6080 

77.97 

18.25 

6870 

82.89 

19.01 

7660 

87.52 

19.71 

8450 

91.92 

20.37 

6090 

78.04 

18.26 

6880 

82.95 

19.02 

7670 

87.58 

19.72 

8460 

91.98 

20.38 

6100 

78.10 

18.27 

6890 

83.01 

19.03 

7680 

87.64 

19.73 

8470 

92.03 

20.38 

6110 

78.17 

18.28 

6900 

83.07 

19.04 

7690 

87.69 

19.74 

8460 

92.09 

20.39 

6120 

78.23 

18.29 

6910 

83.13 

19.05 

7700 

87.75 

19.75 

8490 

92.14 

20.40 

6150 

78.29 

18.30 

6920 

83.19 

19.06 

7710 

87.81 

19.76 

8500 

92.20 

20.41 

6110 

78.36 

18.31 

6930 

83.25 

19.07 

7720 

87.8o 

19.76 

8510 

92.25 

20.42 

6150 

78.42 

18.32 

6940 

83.31 

19.07 

7730 

87.92 

19.77 

8520 

92.30 

20.42 

6160 

78.49 

18 33 

6950 

83.37 

19.08 

7740 

87.98 

19.78 

8530 

92.36 

20.43 

6170 

78.55 

18.34 

6960 

83.43 

19.09 

7750 

88.03 

19.79 

8540 

92.41 

20.44 

6180 

78.61 

18.35 

6970 

83.49 

19.10 

7760 

88.09 

19.80 

8550 

92.47 

20.45 

6190 

78.68 

18.36 

6980 

83.55 

19.11 

7770 

88.15 

19.61 

8560 

92.52 

20.46 

6200 

78.74 

18.37 

6990 

83.61 

19.12 

7780 

88.20 

19.81 

8570 

92.57 

20.46 

6210 

78.80 

18.38 

7000 

83.67 

19.13 

7790 

88.26 

19.82 

8580 

92.63 

20.47 

6220 

78.87 

18.39 

7010 

83.73 

19.14 

7800 

88.32 

19.83 

8590 

92.68 

20.48 

6250 

78.93 

18.40 

7020 

83.79 

19.15 

7810 

88.37 

19.84 

8600 

92.74 

26.49 

6210 

78.99 

18.41 

7030 

83.85 

19.16 

7820 

88.43 

19.85 

8610 

92.79 

20.50 

6250 

79.06 

18.42 

7040 

83.90 

19.17 

7830 

88.49 

19.86 

8620 

92.84 

20.50 

6260 

79.12 

18.43 

7050 

83. 

19.17 

7840 

88.54 

19.87 

8630 

92.90 

20.51 

6270 

79.18 

18.44 

7060 

84.02 

19.18 

7850 

88.60 

19.87 

8640 

92.95 

20.52 

6280 

79.25 

18.45 

7070 

84.08 

19.19 

7860 

88.66 

19.88 

8650 

93.01 

20.53 

6290 

79 31 

18.46 

7080 

84.14 

19.20 

7870 

88.71 

19.89 

8660 

93.06 

20.54 

6300 

79.37 

18.47 

7090 

84.20 

19.21 

7880 

88.77 

19.90 

8670 

93.11 

20.54 

6510 

79.44 

18.48 

7100 

84.26 

19.22 

7890 

88.83 

19.91 

8680 

93.17 

20.55 

6520 

79.50 

18.49 

7110 

84.32 

19.23 

7900 

88.88 

19.92 

8690 

93.22 

20.56 

6550 

79.56 

18.50 

7120 

84.38 

19.24 

7910 

88.94 

19.92 

8700 

93.27 

20.57 

6510 

79.62 

18.51 

7130 

84.44 

19.25 

7920 

88.99 

19.93 

8710 

93.33 

20.57 

6550 

79.69 

18.52 

7140 

84.50 

19.26 

7930 

89.05 

19.94 

8720 

93.38 

20.58 

6500 

79.75 

18.53 

7150 

84.56 

19.26 

7940 

89.11 

19.95 

8730 

93.43 

20.59 

6570 

79.81 

18.54 

7160 

84.62 

19.27 

7950 

89.16 

19.96 

8740 

93.49 

20.60 

6580 

79.87 

18.55 

7170 

84.68 

19.28 

7960 

89.22 

19.97 

8750 

93.54 

20.61 

6590 

79.94 

18.56 

7180 

84.73 

19.29 

7970 

89.27 

19.97 

8760 

93.59 

20.61 

6100 

80.00 

18.57 

7190 

84.79 

19.30 

7980 

89.33 

19.98 

8770 

93.65 

20.62 

6110 

80.06 

18.58 

7200 

84.85 

19.31 

7990 

89.39 

19.99 

8780 

93.70 

20.63 

6120 

80.12 

18.59 

7210 

84.91 

19.32 

8000 

89.44 

20.00 

8790 

93.75 

20.64 

6150 

80.19 

18.60 

7220 

84.97 

19.33 

8010 

89.50 

20.01 

8800 

93.81 

20.65 

6110 

80.25 

18.60 

7230 

85.03 

19.34 

8020 

89.55 

20.02 

8810 

93.86 

20.65 

6150 

80.31 

18.61 

7240 

85.09 

19.35 

8030 

89.61 

20.02 

8820 

93.91 

20.66 

6160 

80.37 

18.62 

7250 

85.15 

19.35 

8040 

89.67 

20.03 

8830 

93.97 

20.67 

6170 

80.44 

18.63 

7260 

85.21 

19.36 

8050 

89.72 

20.04 

8840 

94.02 

20.68 

6180 

80.50 

18.64 

7270 

85.26 

19.37 

8060 

89.78 

20.05 

8850 

94.07 

20.68 

6190 

80.56 

18.65 

7280 

85.32 

19.38 

8070 

89.83 

20.06 

8860 

94.13 

20.69 

6500 

80.62 

18.66 

7290 

85.38 

19.39 

8080 

89.89 

20.07 

8870 

94.18 

20.70 

6510 

80.68 

18.67 

7300 

85.44 

19.40 

8090 

89.94 

20.07 

8880 

94.23 

20.71 

6520 

80.75 

18.68 

7310 

85.50 

19.41 

8100 

90.00 

20.08 

8890 

94.29 

20.72 

6530 

80.81 

18.69 

7320 

85.56 

19.42 

8110 

90.06 

20.09 

8900 

94.34 

20.72 

6510 

80.87 

18.70 

7330 

85.62 

19.43 

8120 

90.41 

20.10 

8910 

94.39 

20.73 

6550 

80.93 

18.71 

7340 

85.67 

19.43 

8130 

90.17 

20.11 

8920 

94.45 

20.74 

6560 

80.99 

18.72 

7350 

85.73 

19.4 4 

8140 

90.22 

20.12 

8930 

94.50 

20.75 

6570 

81.06 

18.73 

7360 

85.79 

19.45 

8150 

90.28 

20.12 

8940 

94.55 

20.75 

6580 

81.12 

18.74 

7370 

85.85 

19.46 

8160 

90.33 

20.13 

8950 

94.60 

20.76 

6590 

81.18 

18.75 

7380 

85.91 

19.47 

8170 

90.39 

20.14 

8960 

94.66 

20.77 

6600 

81.24 

18.76 

7390 

85.97 

19.48 

8180 

90.44 

20.15 

8970 

94.71 

20.78 

6610 

81.30 

18.77 

7400 

86.02 

19.49 

8190 

90.50 

20.16 

8980 

94.76 

20.79 

6620 

81.36 

18.78 

7410 

86.08 

19.50 

8200 

90.55 

20.17 

8990 

94.82 

20.79 

6630 

81.42 

18.79 

7420 

86.14 

19.50 

8210 

90.61 

20.17 

9000 

94.87 

20.80 

6610 

81.49 

18.80 

7430 

86.20 

19.51 

8220 

90.66 

20.18 

9010 

94.92 

20.81 

6650 

81.55 

18.81 

7440 

86.26 

19.52 

8230 

90.72 

20.19 

911211 

94.97 

20.82 

6660 

81.61 

18.81 

7450 

86.31 

19.53 

8240 

90.77 

20.20 

9030 

95.03 

20.82 

6670 

81.67 

18.82 

7460 

86.37 

19.54 

8250 

90.83 

20.21 

9040 

95.08 

20.83 

6680 

81.73 

18.83 

7470 

86.43 

19.55 

8260 

90.88 

20.21 

9050 

95.13 

20.84 

6690 

81.79 

18.84 

7480 

86 49 

19.56 

8270 

90.94 

20.22 

9060 

95.18 

20.85 

6700 

81.85 

18.85 

7490 

86.54 

19.57 

8280 | 

90.99 

20.23 

9070 

95.24 | 

20.86 











































52 


SQUARE AND CUBE ROOTS 



Square Roots and Cube Roots of Numbers from 1000 to 10000 

— (Continued.) 


Num. 

Sq. Rt. 

Cu. Rt 

9080 

95.29 

20.86 

9090 

95.34 

20.87 

9100 

95.39 

20.88 

9110 

95.45 

20.89 

9120 

95.50 

20.89 

9130 

95.55 

20.90 

9140 

95.60 

20.91 

9150 

95.66 

20.92 

9160 

95.71 

20.92 

9170 

95.76 

20.93 

9180 

95.81 

20.94 

9190 

95.86 

20.95 

9200 

95.92 

20.95 

9210 

95.97 

20.96 

9220 

96.02 

20.97 

9230 

96.07 

20.98 

9240 

96.12 

20.98 

9250 

96 18 

20.99 

9260 

96.23 

21.00 

9270 

96.28 

21.01 

9280 

96.33 

21.01 

9290 

96.38 

21.02 

9300 

96.44 

21.03 

9310 

96.49 

21.04 


Num. 


9820 

9830 

9340 

9350 

9360 

9370 

9380 

9390 

9400 

9410 

9420 

9430 

9440 

9450 

9460 

9470 

9480 

9490 

9500 

9510 

9520 

9530 

9540 


Sq. Rt. 

Cu. Rt. 

Num. 

Sq. Rt. 

Cu. Rt. 

Num. 

Sq. Rt. 

Cu. Rt. 

96.54 

21.04 

9550 

97.72 

21.22 

9780 

98.89 

21.39 

96.59 

21.05 

9560 

97.78 

21.22 

9790 

98.94 

21.39 

96.64 

21.06 

9570 

97.83 

21.23 

9800 

98.99 

21.40 

96.70 

21.07 

9580 

97 88 

21.24 

9810 

99.05 

21.41 

96.75 

21.07 

9590 

97.93 

21.25 

9820 

99.10 

21.41 

96.80 

21.08 

9600 

97.98 

21.25 

9830 

99.15 

21.42 

96.85 

21.09 

9610 

98.03 

21.26 

9840 

99.20 

21.43 

96.90 

21.10 

9620 

98.08 

21.27 

9850 

99.25 

21 44 

96.95 

21.10 

9630 

98.13 

21.28 

9860 

99.30 

21.44 

97.01 

21.11 

9640 

98.18 

21.28 

9870 

99.35 

21.45 

97.06 

21.12 

9650 

98.23 

21.29 

9880 

99.40 

21.46 

97.11 

21.13 

9660 

98.29 

21.30 

9890 

99.45 

21.47 

97.16 

21.13 

9670 

98.34 

21.30 

9900 

99.50 

21.47 

97.21 

21.14 

9680 

98.39 

21.31 

9910 

99.55 

21 48 

97.26 

21.15 

9690 

98.44 

21.32 

9920 

99.60 

21.49 

97.31 

21.16 

9700 

98.49 

21.33 

9930 

99.65 

21.49 

9 (.37 

21.16 

9710 

98.54 

21.33 

9940 

99.70 

21.50 

97.42 

21.17 

9720 

98.59 

21.34 

9950 

99.75 

21.51 

97.47 

21.18 

9730 

98.64 

21.35 

9960 

99.80 

21.52 

97.52 

21.19 

9740 

98.69 

21.36 

9970 

99.85 

21.52 

97.57 

21.19 

9750 

98.74 

21.36 

9980 

99.90 

21.53 

97.62 

21.20 

9760 

98.79 

21.37 

9990 

99.95 

21.54 

97.67 

21.21 

9770 

98.84 

21.38 

10000 

100.00 

21.54 




To fiml Square or Cube Roots of larg-e numbers not com 
tained in tlie column of numbers of the table. 


Such roots may sometimes be taken at once from the table, by merely regarding the columns of 
powers as being columns of numbers; and those of numbers as being those of roots. Thus, if thf 
sq re of 25281 is reqd, first dud that number in the column of squares; and opposite to it, in thf 
column of numbers, is its sq rt 159. For the cube rt of 857375, find that number in the column of 
cubes; and opposite to it, in the col of numbers, is its cube rt 95. When the exact number is not con 
tained in the column of squares, or cubes, as the case may he, we may use instead the number neares 
to it, if no great accuracy is reqd. But when a considerable degree of accuracy is necessary, thf E 
following very correct methods may be used. 

For the square root. 

This rule applies both to whole numbers, and to those which are partly (not wholly) decimal. First 
in the foregoing manner, take out the tabular number, which is nearest to the given one; and also it 
tabular sq rt. Mult this tabular number by 3; to the prod add the given number. Call ttie sum A 
Then mult the given number by 3 ; to the prod add the tabular number. Call the sum B. Then 

A : B : : Tabular root : Reqd root. 


Ex. Bet the given number be 946.53. Here we find the nearest tabular number to be 947; and it 
tabular sq rt 30.7734. Hence, 

947 = tab num r 946.53 = given num. 

3 3 


2841 

946.53 = given num. 


3787.53 = A. 


and 




2839.59 

947 ~ tab num. 


(3786.59 — B. 


A. R. Tab root. Reqd root. 

Then 3787.53 : 3786.59 : : 30.7734 ; 30.7657 +. 

The root as found by actual mathematical process is also 30.7657 -(-. 

For the cube root. 

This rule applies both to whole numbers, and to those which are partly decimal. First take out tv 
tabular number which is nearest to the given one; and also its tabular cube rt. Mult this tahuk' 
number by 2; and to the prod add the given number. Call the sum A. Then mult the given numb 
by 2; and to the prod add the tabular number. Call the sum B. Then 

A : B : : Tabular root : Reqd root. 

Ex. Let the given number be 7368. Here we find the nearest tabular number (in the column i 
cu6e«) to be 68o9; and its tabular cube rt 19. Hence, ' 


B. 

3786.59 


Tab root. 
30.7734 




6859 = tab num. 
2 


13718 

7368 == given num. 


► and 


7368 = given num. 
2 


14736 

6859 — tab num. 


21595 = B. 


21086 rr a. 

A. R. Tab Root. 

Then, as 21086 ; 21595 ; : 19 : 

The root as found by correct mathematical process is 19.4588. 


Reqd Rt, 
19.4585 


The engineer rarely requires ev 
















































SQUARE AND CUBE ROOTS 


53 


this degree of accuracy; for his purposes, therefore, this process is greatly preferable to the ordinary 
laborious one. 

To find the square root of a number which is wholly 

decimal. 

Very simple, and correct to the third numeral figure inclusive. If the number does not contain at 
least five figures, counting from tile first numeral , anti including it, add one or more ciphers to make 
live. If, after that, the whole number is not separable into twos, add another cipher to make it so. 
Then beginning at the first numeral figure, and including it, assume the number to be a whole one. 
Ill the table find the uumber nearest to this assumed one ; take out its tabular sq rt; move the deci¬ 
mal point of this tabular root to the left, half as many places as the dually modified decimal uumber 
has figures. 

Ex. What is the sq rt of the decimal .002? Here, in order to have at least five decimal figures, 

, couuting from the first numeral (2). and including it, add ciphers thus, .00.20.00 0. But. as it is not 
' uow separable into twos, add another cipher, thus, .00,20,00,00. Theu beginning at the first numeral 
(2), assume this decimal to be the whole uumber 200000. The nearest to this in the table is 109809; 
and the sq rt of this is 147. Now, the decimal uumber as finally modified, namely. .00.20,00,00, has 
eight figures; one-half of which is 4; therefore, move the decimal point of the root 447, four places to 
the left; making it .0447. This is the reqd sq rt of .002, correct to the third numeral 7 included. 

To find the cube root of a number which is wholly decimal. 

Very simple, aud correct to the third numeral inclusive. 

1 If the number does not contain at least five figures, counting from the first numeral, and including 
it, add one or more ciphers to make five. If. after that, the number is not separable into threes, add 
one or more ciphers to make it so. Then beginning at the first numeral, and including it, assume 
the number to be a whole one. In the table find the number nearest to this assumed one, and take 
out its tabular cub rt. Move the decimal point of this rt to the left, one-third as many places as the 
tiually modified decimal number has figures. 

Ex. What is the cube rt of the decimal .002? Here, in order to have at least five figures, counting 
Hrom the first numeral (2), and including it, add ciphers thus, .002 000.0. But as it is not now separ¬ 
able into threes, add two more ciphers to make it so; thus, .002.000,000. Then beginning with the 
first numeral (2), assume the decimal to be the whole number 2000000. The nearest cube to this in 
the table in the column of cubes, is 2000276; and its tabular cube rt as found in the col of numbers, 
is 126. Now, the decimal number as finally modified, namely, .002 000 000, has nine figures; one-third 
of which is 3; therefore, move the decimal poiut, of the root 126, three places to the left, ntakiug it 
.126. This is the reqd cube rt of the decimal .002, correct to the third uumeral 6 included. 

To And roots by logarithms, see p 39. 

For tables of fifth roots, fiflli pow ers, and square roots of filth 

powers, see pp 251 to 253. 



54 


GEOMETRY. 


GEOMETRY. 



Lines. Figures. Solids, defined. Strictly speaking a geometrical line ,, 

is simply length, or distance. The lines we draw on paper have not only length, but breadth and 
thickness ; still they are the most convenient symbol we can employ for denoting a geometrical line. 

Straight lines are also called right lines. A vortical lineisonethat points 
toward the center of the earth; aud a horizontal one is at right .angles to a 
vert one. A plane figure is merely any flat surface or area entirely enclosed f_ 

by lines either straight or curved ; which are called its outline, boundary, circumf, or periphery. Wt 
often confound the outline with the fig itself as when we speak of drawing circles, squares, &c; for 
we actually draw only their outlines. Geometrically speaking, a fig lias length and breadth only; nc 
thickness. A solid is any body; it has length, breadth, and thickness. 

Geometrically similar figs or solids, are not necessarily of the same 
size; but only of precisely the same shape. Thus, any two squares are,scien- | 

tifically speaking, similar to each other ; so also any two circles, cubes, &c, no matter how difleren! 
thev may be in size. When they are not only of the same shape, but of the same size, they are said 
to be similar, and equal. 

The quantities of lines are to each other simply as their lengths; but i 
the quantities, or areas, or surfaces of similar fit'll res, are as, or in proportion 
to, the squares of any one of the corresponding lines or sides which enclose tht 
figures, or which may be drawn upon them ; and the quantities, or solidities of 
similar solids, are as the cubes of any of the corresponding lines which font 

their edges, or the figures by which they are enclosed. 

Rem. —Simple as the following operations appear, it is only by care, and good instruments, that 
they are made to give accurate results. Several of them can be much better performed by means of a 
metallic triangle having one perfectly accurate right angle. In the field, the tape-line, chain, or a 
lueasuriug-rod will take the place of the dividers aud ruler used indoors. 



To divide a given line, a b, into two equal parts. 


From its ends a and b as centers, and with any rad greater than one-half of a b 
describe the arcs c and rf, and join e/. '( the line a b is very long, first lav off 

equal dists a n and bg. each way from tne ends, so as to approach convenientlj 
near to each other ; and then proceed as if o g were the line to he divided. Oi 
measure a 6 by a scale, aud thus ascertain its center. 



To divide a given line, m it, into any 
given number of equal parts. 


From to and n draw any two parallel lines to o and » a 
to an indefinite dist; aud on them, from to and « step off th< 
reqd number of equal parts of any convenient length : final 
ly, join the corresponding points thus stepped off. Or 011 I 3 
one line, as too. may he drawn and stepped off. as to s 
then join « n ; and draw the other short lines parallel to it 


This is done on the same principle as the last: thus, let the proportion be as 1 to 3. First drav 
any me to 0 ; and with any convenient opening of the dividers make to z equal to one step ; aud x 
equal to three steps. Join sn; and parallel to it draw xc. Then to c is to cm as 1 is to 3^ 


ANGLES, 


I 


Angles. When two straight, or right lines meet each other at any inclina 
tion, the inclination is cadled an angle; and is measured by the degrees con 

tamed in the arc of a circle described from the point of meeting as a center. Since all circles whetbe 
large or small are supposed to be divided iuto 360 degrees, it follows that any number of degrees of 
small circle will measure the same degree of inclination as will the same number of a large one 

to the Cl na??nnT; a * 8 °*U “\°» i ?.“ oh a n,;lIlne . r cimation o n a is equt 

to the inclination o n 6, then the two lines are said to be 

perpendicular to each other; and the angles o n a and 
o n b , are called right angles ; and are each measd bv or 
are equal to. 110°, or one-fourth part of the circumf of a circle. Any angle 

as ced ,smaller than a right angle, is called acute orsharV’ 
and one c ef, larger than a right angle, is called obtuse or 

biunt. When one ime as in the first Fig on opposite’page, the two angles on the 



• i .. .., " > ,u urst on opposite page, the two angles ou the 

f n ifare d fdh!ce^t th ![ ,?,? are f allea conti S ,, ««s, or adjacent. Thus', vus and 

vn w are adjacent, also t u s and t uw;sut and s u v; w u laiidit; u v. The sum of two adjacent 
angles is always equal to two right angles; or to lhU°. Therefore, if we know the number of de- 
g.ues contained in one ol them, and subtract it from 180=, we obtain the other 













GEOMETRY 


55 



w 


When two straight lines cross each other, forming four 
angles, either pair of those angles which point in exactly 
opposite directions are called opposite, or vertical 
angles ; thus, the pair s u t, and v u w are opposite an¬ 
gles ; also the pair s u v and t n w. The opposite angles 
of any pair are always equal to each other. 

When a straight line a b crosses two parallel lines c d, 
ef, the alternate angles which form a kind of Z are 
equal to each other. Thus, the angles d o n, and o n /, are 
equal: as are also e o n, and one. Also the sum of the 
two internal angles on the same side of a b, is equal to two 
right angles, or 180°; thus, c o n o nf = 180°; also 
don -{-one — 180°. 

An interior angle. 

In any fig, is any angle formed inside of that fig, by the meet¬ 
ing of two of its sides, as the angles c a b, a b c, b c a, of this 
triangle. All the interior augles of any straight-lined figure of 
any number of sides whatever, are together equal to twice as 
many right angles minus four, as the figure has sides. Thus, a 
triangle has 3 sides; twice that number is 6 ; and 6 right augles, 
J or 6 X 90° =540°; from which take 4right angles, or 360°; and 
u there remain 180°, which is the number of degrees in every 
plane, or straight lined triangle. This principle furnishes an 
easy means of testing our measurements of the angles of any 
fig; for if the sum of all our measurements does not agree with 
the sum given by the rule, it is a proof that we have committed some error. 

An exterior angle 

Of any straight-lined figure, is any angle, as a b d. formed octsidk of that fig, 
by the meeting of any side, as a b, with the prolongation of an adjacent side, 
as c b ; so likewise the angles c a s, and 6 c w.* All the exterior angles of any 
straight-lined fig, no matter how many sides it may have, amount to 360° ; 
but if any of the angles are re-entering’, i e, pointing 
inwards, as g ij, the supplements, as g i x, corresponding to such, must be 
taken as negative or minus Thus abd-\-bcw-\-cas = 360°; and y hj -f- 
zji-gix + i g w — 360°. Angles, as a, b, c, g, A, and j, which poiut uiU- 
ward y are called salient. 


From any given point, p, on a line s t, 
to draw a perp, p a. 

From p, with any convenient opening of the dividers, step off the 
equals p o,p g. From o and g as centers, with any opening greater 
than half o g, describe the two short arcs b and c ; and join a p. 
Or still better, describe four arcs, and join a y. 

Or from p with any convenient scale describe two 

short arcs g and c either one of them with a radius 3, and the other 
with a rad 4. Then from g with rad 5 describe the arc b. Join p a. 




If the point p is at one end of the line, 
or very near it. 

Extend the- line, if possible, and proceed as above. But if this 
caunot be done, then from any convenient point, w, open the divid¬ 
ers to p, and describe the semicircle, sp o; through o w draw o w 
s; join p s. 

Or use tlie last foregoing process with 

rads 3, 4, and 5. 



From a given point, o, to let fall a 
perp o ff t to a given line, m n. 

From o, measure to the line m n, any two equal dists, o c, 
o e; and from c and e as centers, with any opening greater 
than half of c e, describe the two arcs a and b ; join o f. Or 
from nnv point, as d on the line, open the dividers to o, and 
describe the arc o g ; make i x equal tot a; and join o x. 


* An exterior angle a b d, or y A j, is the supplement (p 56) 
Of the interior angle a b c, or g hj, at the same point b, or A. 


















56 


GEOMETRY, 


If the line, a b, is on the ground. 

And a perp is reqd to be drawn from c, first measure off any two 
equal dists, c to, c n. At to and n, hold the ends of a piece of string, 
tape line, or chain, m s n; then tighten out the string, &c, as shown 
by to s n ; s being its ceuter. Then will s c be the reqd perp. Or if 
the perp x z is to be drawn from the end of the Hue w x, first measure x y 
upon the liDe, and equal to three feet; then holding the end of a tape- 
line at x, and its nine feet mark at y. hold the four feet mark at z, keep¬ 
ing z x and z y equally stretched. Then zx will be the reqd perp, because 

8 . 1, and 5, make the sides of a right-angled triangle. Instead of 3, 4, and 
6 , any multiples of those numbers may be used, such as 6, 8, and 10 ; or 

9, 12, 15, &c: also instead of feet, we may use yards, chains, &c. 




Through a jriven point, a, to draw a 
line, a c, parallel to another line, 

e f. 

With the perp dist, a e, from any point, n, in e /, describe 
an arc, t ; draw a c just touching the arc. 



At any point, «, in a line a b, 
to make an angle cn b, equal 
to a given angle, m w o. 

From wand a, with any convenient rad, describe 
She arcs st.de ; measure s {, and make t d equal 
to it; through a d draw a c. 




To bisect, or divide any angle, w x y, into 
two equal parts. 

From x set off any two equal dists, xr,xs. From r and s with any rad 
describe two arcs intersecting, as at o; and join ox. If the two sides of 
the angle do not meet, as c / and g h, either first extend them until they 
do meet; or else draw lines x w, and xy, parallel to them, and at equal 
dists from them, so as to meet; then proceed as before. 


All angles, as n a to, n o to. at the circumf of a semicircle, and stand¬ 
ing on its diam n to, are right angles; or, as it is usually expressed, 

all angles in a semicircle are right angles. 

An angle ns a: at the centre of a circle, is twice as great as an angle n 
m x at the circumf, when both stand upon the same arc n x. 



P 


All angles, as y d p, y e p, y g p. at the circumf of a circle, and standing 
upon the same arc. as y p, are equal to each other ; or, as usually expressed, 

all angles in the same segment of a circle are 
equal. 


The complement of an angle is what it lacks of 90°. Thus, the com¬ 
plement of 80° is 90° — 80° = 10°; and that of 210° is 90° — 210° — — 120°. 
The supplement of an angle is what it lacks of 180°. Thus, the supple¬ 
ment of 80° is 180° — 80° = 100 °; and that of 210 ° is 180° — 210 ° - — 30°. 
But ordinarily we may neglect the signs -|- and —, before complements and 
supplements, and call the complement of an angle its diff from 90°; and 
the supplement its diff from 180°. 
















ANGLES. 


57 


a 


Anglos In a Parallelogram. 

A parallelogram is any four-sided straight-lined fig- 
r ure whose opposite sides are equal, as abed ; or a 
square, Ac. Any line drawn across a parallelogram 
between 2 opposite angles, is called a diagonal, as a c, 
or b d. A diag divides a parallelogram into two equal 
parts; as does also any line rn n drawn through the 
center of either diag; and moreover, the line rn u 
itself is div into two equal parts by the diag. Two 
diags bisect each other; they also divide the parallel¬ 
ogram into four triangles of equal areas. The sum 
of the two angles at the ends of any one side is = 180° ; thus, dab+a bc = abc + 
b c d =■ 180° ; and the sum of the four angles, d a b, a b c, b c d, c d a — 3G0°. 

The sum of the squares of the four sides, is equal to the sum of the squares of the 
two diags. 



To reduce Minutes and Seconds to Degrees and decimals 

of a Degree, etc. 

In anj’ given angle— 

Number of degrees = Number of minutes 60. 

= Number of seconds 3600. 

Number of minutes = Number of degrees X 60. 

= Number of seconds - 4 - 60. 

Number of seconds = Number of degrees X 3600. 

= Number of minutes X 60. 


Table of Minutes and Seconds in decimals of a Degree. 


Min. 

Deg. 

Min. 

Deg. 

Min. 

Deg. 

Sec. 

Deg. 

Sec. 

Deg. 

Sec. 

Deg. 

1 

.016666 

21 

.350000 

41 

.683333 

1 

.000278J 

21 

.005833 

41 

.011389 

2 

.033333 

22 

.366666 

42 

.700000 

2 

.000556 

22 

.006111 

42 

.011667 

3 

.050000 

23 

.383333 

43 

.716666 

3- 

.000833 

23 

.006389 

43 

.011944 

4 

.066666 

24 

.400000 

44 

.733333 

4 

.001111 

24 

.006667 

44 

.012222 

5 

.083333 

25 

.416666 

45 

.750000 

5 

.001389 

25 

.006944 

45 

.012500 

6 

.100000 

26 

.433333 

46 

.766666 

6 

.001667 

26 

.007222 

46 

.012778 

7 

.116666 

27 

.450000 

47 

.783333 

7 

.001944 

27 

.007500 

47 

.013056 

8 

.133333 

28 

•466666 

48 

.800000 

8 

.002222 

28 

.007778 

48 

.013333 

9 

.150000 

29 

.483333 

49 

.816666 

9 

.002500 

29 

.008056 

49 

.013611 

10 

.166666 

30 

.500000 

50 

.833333 

10 

.002778 

30 

.008333 

50 

.013889 

11 

.183333 

31 

.516666 

51 

.850000 

11 

.003056 

31 

.008611 

51 

.014167 

12 

.200000 

32 

.533333 

52 

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12 

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32 

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52 

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13 

.216666 

33 

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53 

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13 

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33 

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53 

.014722 

14 

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34 

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54 

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14 

.003889 

34 

.009444 

54 

.015000 

15 

.250000 

35 

.583333 

55 

.916666 

15 

.00+167 

35 

.009722 

55 

.015278 

16 

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36 

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56 

.933333 

16 

.004444 

36 

.010000 

56 

.015556 

17 

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37 

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57 

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17 

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37 

.010278 

57 

.015833 

18 

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38 

.633333 

58 

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18 

.005000 

38 

.010556 

58 

.016111 

19 

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39 

.650000 

59 

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19 

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39 

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59 

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20 

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40 

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20 

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40 

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60 

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x .0002777778. 









































58 


ANGLES. 


To Measure Angles by a 2 Ft Rule, etc. 

The four finders of tl»e hand, held at right angles to the arm, and at 

arms-lengih from the eye, cover about 7 degree*. Aud 7 C corresponds to about 12.2 ft iu 100 ft; or to 
36.6 ft iu 100 yds; or to 615 ft in a mile ; or iu the same proportion as the distance. 

The following Table 

mav sometimes be found useful for the rough measurement of angles, either on a drawing, or be¬ 
tween distant objects in the field. If the inner edges or a common two-foot rule be opened to the ex¬ 
tent shown in the column of inches, its edges will be inclined at the angles shown in the columns of 
angles. Since an opening of % of an inch up to 19 inches or about 105°. corresponds to from about 
to 1 °, no great accuracy is to be expected; and beyond 105° still less; the liability to error in¬ 
creasing very rapidly as the opening becomes greater. Thus, the last X inch corresponds to about 12 °. 

As to the table itself, angles for openings intermediate of those therein given, may be calculated to 
the nearest minute or two, by simple proportion, up to 23 inches of openiug, or about 147°. 

Table of Angles corresponding to openings of a 2-foot rule. 

(Original.) 

D, degrees; M. minutes. Correct. 


Ins. 

D. 

M. 

Ius. 

D. 

M. 

lus. 

D. 

1 

M. 

Ius. 

D. 

M. 

Ins. 

D. 

M. 

Ins. 

D. 

M. 

x 

1 

12 

*x 

20 

24 


40 

13 

viy 

61 

23 

\GX 

85 

14 

•>0J4 

115 

5 


1 

48 

21 



40 

51 


62 

5 


86 

3 


116 

12 

x 

2 

24 

x 

21 

37 

X 

41 

29 

X 

62 

47 

X 

86 

52 

X 

117 

20 


3 

00 

22 

13 


42 

T 

63 

28 


87 

41 


118 

30 

x 

3 

36 

x 

22 

50 

X 

42 

46 

X 

64 

11 

X 

88 

31 

X 

119 

40 

4 

11 

23 

27 


43 

24 

64 

53 


89 

21 


120 

52 

i 

4 

47 

5 

24 

3 

9 

44 

3 

13 

65 

35 

17 

90 

12 

21 

122 

6 


5 

23 


24 

39 


44 

42 


66 

18 


91 

3 


123 

20 

X 

5 

58 

X 

25 

16 

X 

45 

21 

X 

67 

1 

X 

91 

54 

X 

124 

36 

6 

34 

25 

53 


45 

59 

67 

44 

92 

46 


125 

54 

X 

7 

10 

X 

26 

30 

14 

46 

38 

X 

68 

28 

X 

93 

38 

X 

127 

14 

7 

46 


27 

7 


47 

17 

69 

12 

94 

31 


128 

35 

X 

8 

22 

X 

27 

44 

X 

47 

56 

X 

69 

55 

X 

95 

24 

X 

129 

59 

8 

58 

28 

21 


48 

35 


70 

38 

96 

17 


J31 

25 

2 

9 

34 

6 

28 

58 

10 

49 

15 

14 

71 

22 

18 

97 

11 

22 

132 

53 


10 

10 


29 

35 


49 

54 


72 

6 


98 

5 


134 

24 


10 

46 

x 

30 

ii 

X 

50 

34 

X 

72 

51 

X 

99 

00 

X 

135 

58 

11 

22 

30 

49 

51 

13 

73 

36 

99 

55 

137 

35 

X 

11 

58 

x 

31 

26 

14 

51 

53 

X 

74 

21 

X 

100 

51 

X 

139 

16 

12 

34 

.32 

3 


52 

33. 

75 

6 

101 

48 


141 

1 

X 

13 

10 

hi 

32 

40 

X 

53 

13 

X 

75 

61 

X 

102 

45 

X 

142 

51 

13 

46 

33 

17 

53 

53 


76 

36 

103 

43 


144 

46 

3 

14 

22 

7 

33 

54 

u 

54 

34 

15 

77 

22 

19 

104 

41 

23 

146 

48 


14 

58 


34 

33 


55 

14 


78 

8 


105 

40 


148 

58 

x 

15 

34 

X 

35 

10 

X 

55 

55 

X 

78 

54 

X 

106 

39 

X 

151 

17 

16 

10 

35 

47 

56 

35 

79 

40 

107 

40 

153 

48 

x 

16 

46 

14 

36 

25 

X 

57 

16 

X 

80 

27 

X 

108 

41 

X 

156 

34 


17 

22 

37 

3 

57 

57 


81 

14 

109 

43 


159 

43 

x 

17 

59 

X 

37 

41 

X 

58 

38 

X 

82 

2 

X 

110 

46 

X 

163 

27 

18 

35 

38 

19 

59 

19 

82 

49 


Ill 

49 

168 

18 

4 

19 

12 

8 

38 

57 

12 

60 

00 

16 

83 

37 

20 

112 

53 

24 

180 

00 


1 19 

48 


39 

i 

35 


60 

41 


84 

26 


i 113 

58 





Or this table may be used thus. From any point measure 12 ft 

towards each object, and place marks. Measure the dist in ft between these 
marks. Suppose the first cols in the table to be ft instead of ins; then opposite 
the dist in ft will be the angle. Oue-eiglith of a ft is 1.5 ius. 

The following is a good way to measure an angle. Measure 

100 or any other number of ft towards each object, and place marks. Measure the 
dist between the marks. Then 

As dist measured . 1 . . Half the dist . nat sine of 
toward one object • 1 • • between marks • Half the angle. 

Find this nat sine in the table of nat sines, take out the corresponding angle, 
and multiply it by 2. See near foot of p 114. 























SINES, TANGENTS, ETC 


59 


Sines, Tangents, Ac. 

Sine, a s, of any angle, a c ft, or which is the same thing, the sine of any circular arc, a ft, 
which subtends or measures the angle, is a straight line drawn from one end, as a, of the arc, at right 
angles to, and terminating at, the rad c 6, drawn to the other end ft of the arc. It is, therefore, equal 
to half the chord an, of the arc aim, which is equal to twice the arc aft; or, the sine of an angle is 
always equal to half the chord of twice that angle; and vice versa, the chord of an angle is always 
equal to twice the sine of half the angle. 

The siue t c of an angle t, c 6, or of au arc 
t a ft. of 90°. is equal to the rad of the arc 
or of the circle ; and this sine of 90° is 
treater than that of any other angle. 

Vosine c s of an angle a c b, 

is that part of the rad w hich lies between 
the sine and the ceuter of the circle, it 
is always equal to the sine y a of the 
complement t c a of a c ft: or of what a 
eft wautsof being 90°. The prefix co be¬ 
fore sines, &c, means complement; thus, 
cosine meaus sine of the complement. 

Versed Mine s b of any angle 
a c 6 , is that part of the diam which lies 
between the sine, aud the outer end 6 . 

It is very common, but erroueous, when 
speaking of bridges, &c, to call the rise 
or height s 6 of a circular arch a ft n, its 
versed sine; while it is actually the versed 
sineofonly half the arch. Thisabsurdity 
should cease; for the word rise or height 
is not ouly more expressive,but is correct. 

Tan^eii tbworad, of any angle 

a c ft, is a liue drawn from, aud at right 
angles to, the end 6 or a of either rad c 6, 
or c a, which forms one of the legs of the 
angle ; and terminating as at tv, or d, in 
the prolongation of the rad which forms 
the other leg. This last rad thus pro¬ 
longed, that is, c vs, or c cJ.as the case may 

be, is the secant of the angle 

a c ft. The angle t c 6 being supposed 
to be equal to 90°, the angle tea becomes the complement of the angle a c 6, or what a c 6 wants 
of being 90°; and the sine y a of this complement; its versed sine t y : its tangent t o ; and its secant 
c o, are respectively the co sine, co-versed sine; co tangent; and co-secant, of the angle a c ft. Or, 
vice versa, the sine, &c, of a c 6, are the cosine, &c, of tea; because the angle a c ft is the comple¬ 
ment of the angle lea. When the rad c ft, c a, or c t, is assumed to be equal to unity, or 1, the cor¬ 
responding sines, tangents, &c, are called natural ones; and their several lengths for dilf angles, 
for said rad of unity, have been calculated: constituting the well-known tables of nat sines, &c. In 
any circle whose rad is either larger or smaller than 1, the sines, &c, of the angles will be in the 
same proportion larger or smaller than those in the tables, aud are consequently fouud by mult the 
sine, &c, of the table, by said larger or smaller rad. 

The following 1 table of natural sines. Ac. does not contain nat 
versed sines, co-versed sines, secants, nor cosecants, but these may be found thus; 
for any angle not exceeding 90 degrees. 

Versed Sine. From 1 take the nat cosine. 

Co-versed Sine. From 1 take the nat sine. 

Secant. Divide 1 by the nat cosine. 

Cosecant. Divide 1 by the nat sine. 

For angles exceeding 90°; to find the sine, cosine, tangent, cotang, secant or cosec, (but not 
the versed sine or co-versed sine), take the angle from 180°: if betweeu 180° and 270° take 180° from 
the angle ; if bet 270° and 360°, take the angle from 360°. Then in each case take from the table the 
sine, cosine, tang, or cotang of the remainder. Find its secant or cosec as directed above. For the 
versed sine; if between 900 an ,i 270°, add cosine to 1; if bet 270° and 360°, take cosine from 1. (The 
engineer seldom needs sines, &c, exceeding 180°. 

To find the nat sine, cosine, fang-, secant, versed sine, Ac, 
of an angfle containing’ seconds. First find that due to the given deg 

and min ; then the next greater one. Take their did. Then as 60 sec are to this ditf, so are the sec 

only of the given angle to a dec quantity to be added to the one first taken out 
if it is a sine, tang, secant, Ac ; or to be subtracted lrom it if it is a cosine, 

cotang, cosecant, Ac. . . . _ . 

The tangents in flic table are strict trigonometrical ones; that is, 
tangents to gTven angles; and which must extend to meet the secants of the angles 
to which they belong. Ordinary, or geometrical tangents, as those on 
p 124 may extend as far as we please. In the field practice of railroad 
curves, two trigonometrical tangents terminate where they meet each other. 
Each of these tangs is the tang of half the curve. It is usually, hut improperly, 
called “ the tang of the curve." “Apex dist of the curve,” as suggested by Mr 
Sbunk, would be better. 


W 










NATURAL SINES AND TANGENTS TO A RADIUS 1. 
0 Deg. 0 Deg. 0 Deg. 


60 


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2 Deg. 2 Deg. 2 Deg. 


62 



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TABLE OF CHORDS. 


105 


The table of chords, below, furnishes the means of laying down angles on 

paper more accurately than by an ordinary protractor. To do this, after having drawn 
and measured the first side (say a c) of the figure that is 
to be plotted; from its end c as a center, describe an arc 
n 1 / of a circle of sufficient extent to subtend the angle at 
that point. The rad cn with which the arc is described 
should be as great as convenience will permit; and it is to 
be assumed as unity or 1; and must be decimally divided, 
and subdivided, to be used as a scale for laying down the 
chords taken from the table, in which their lengths are 
given in parts of said rad 1. Having described the arc, find 
in the table the length of the chord nt corresponding to 
the angle act. Let us suppose this angle to be 45°; then 
we find that the tabular chord is .7654 of our rad 1. There¬ 
fore from n we lay off the chord nt , equal to .7G54 of our radius-scale; and the line 
cs drawn through the point t will form the reqd angle act of 45°. And so at each 
angle. The degree of accuracy attained will evidently depend on the length of the 
rad, and the neatness of the drafting. The method becomes preferable to the com¬ 
mon protractor in proportion as the lengths of the sides of the angles exceed the rad 
of the protractor. With a protractor of 4 to 6 ins rad, and with sides of angles not 
much exceeding the same limits, the protractor will usually be preferable. The di¬ 
viders in boxes of instruments are rarely* fit for accurate arcs of more than about 6 
ins diam. In practice it is not necessary to actually describe the whole arc, but 
merely the portion near t, as well as can be judged by eye. We thus avoid much use 
of the India-rubber, and dulling of the pencil-point. For larger radii we may dis¬ 
pense with the dividers, and use a straight strip of paper with the length of the rad 
marked on one edge; and by laying it from c toward s, and at the same time placing 
another strip (with one edge divided to a radius-scale) from n toward t, we can 
by trial find their exact point of intersection at the required point t. In such mat¬ 
ters, practice and some ingenuity are very essential to satisfactory results. We can¬ 
not devote more space to the subject. 

CHORDS TO A RADIUS 1. 



M. 

0° 

1° 

oo 

| 3° 

4° 

5° 

6° 

7° 

8° 

9° 

1 10° 

| M. 

0' 

.0000 

.0175 

.0349 

.0524 

.0898 

.0872 

.1047 

.1221 

.1395 

.1569 

.1743 

O' 

2 

.0006 

.0180 

j .0355 

.0529 

.0704 

.0878 

. 1053 

.1227 

.1401 

.1575 

.1749 

2 

4 

.0012 

.0186 

.0361 

.0535 

.0710 

.0884 

.1058 

.1233 

.1407 

.1581 

.1755 

4 

6 

.0017 

.0192 

.0366 

.0541 

.0715 

.0890 

.1064 

.1238 

.1413 

.1587 

.1761 

6 

8 

.0)23 

.0198 

.0372 

.0547 

.0721 

.0896 

.1070 

.1214 

.1418 

.1592 

.1766 

8 

10 

.0020 

.0204 

.0378 

.0553 

.0727 

.0901 

.1076 

. 1250 

.1424 

.1598 

.1772 

10 

1‘2 

.0035 

.0209 

.0381 

.0558 

.0733 

.0907 

.1082 

.1256 

.1430 

.1604 

.1778 

12 

14 

.0041 

.0215 

.0390 

.0564 

.0739 

.0.413 

.1087 

.1262 

.1436 

.1610 

.1784 

14 

16 

.0047 

.0221 

.0396 

.0570 

.0745 

.0919 

.1093 

.1267 

.1442 

.1616 

.1789 

16 

18 

.0052 

.0227 

.0101 

.0576 

.0750 

.0925 

.1099 

.1273 

.1447 

.1621 

.1795 

18 

20 

.0058 

.0233 

.0407 

.0582 

.0756 

.0931 

.1105 

.1279 

.1453 

.1627 

.1801 

20 

22 

.0064 

.0239 

.0413 

.0588 

.0762 

.0936 

.1111 

.1285 

.1459 

.1633 

.1807 

22 

2t 

.0070 

.0244 

.0419 

.0593 

.0768 

.0942 

.1116 

.1291 

.1465 

.1639 

.1813 

24 

26 

.0076 

.0250 

.0425 

.0599 

.0774 

.0948 

.1122 

.1296 

.1471 

.1645 

.1818 

26 

28 

.0081 

.0256 

.0430 

.0605 

.0779 

.0951 

.1128 

.1302 

.1476 

.1650 

.1824 

28 

30 

.0087 

.0262 

.0436 

.0611 

.0785 

.0960 

.1134 

.1308 

.1482 

.1656 

.ia3o 

30 

32 

.00.93 

.0268 

.0442 

.0617 

.0791 

.0965 

.1140 

.1314 

.1488 

.1662 

.1836 

32 

31 

.0099 

.0273 

.0418 

.0622 

.0797 

.0971 

.1145 

.1320 

.1494 

.1668 

.1842 

34 

36 

.0105 

.0279 

.0451 

.0628 

.0803 

.0977 

.1151 

.1.325 

.1500 

.1674 

.1847 

36 

38 

.0111 

.0285 

.0460 

.0634 

.0808 

.0983 

.1157 

.1331 

.1505 

.1679 

.1853 

38 

40 

.0116 

.0291 

.0165 

.0640 

.0814 

.0989 

.1163 

.1337 

.1511 

.1685 

.1859 

40 

42 

.0122 

.0297 

.0171 

.0646 

.0820 

.0994 

.1169 

.1343 

.1517 

.1691 

.1865 

42 

44 

.0128 

.0303 

.0477 

.0651 

.0826 

.1000 

.1175 

.1349 

.1523 

.1697 

.1871 

44 

46 

.0134 

.0308 

.0483 

.0657 

.0832 

.1006 

.1180 

.1.355 

.1529 

.1703 

.1876 

46 

4S 

.0140 

.0314 

.0489 

.0663 

.0838 

.1012 

.1186 

.1360 

.1534 

.1708 

.1882 

48 

50 

.0145 

.0.320 

.0494 

.0669 

.0843 

.1018 

.1192 

.1366 

.1540 

.1714 

.1888 

50 

52 

.0151 

.0326 

.0500 

.0675 

.0849 

.1023 

.1198 

.1.372 

.1546 

.1720 

.1894 

52 

54 

.0157 

.0332 

.0506 

.0681 

.0855 

.1029 

.1204 

.1378 

.1552 

.1726 

.1900 

54 

56 

.0163 

.0337 

.0512 

.0686 

.0861 

.1035 

.1209 

.1384 

.1558 

.1732 

.1905 

56 

58 

.0169 

.0343 

.0518 

.0692 

.0867 

.1041 

.1215 

.1389 

.15 <3 

.1737 

.1911 

58 

60 ; 

.0175 

.0349 

.0524 

.0698 

.0872 

.1047 

.1221 

.1395 

.1569 

.1743 

.1917 

60 
















































































106 


TABLE OF CHORDS 


Table of Chords, in parts of a rad 1; for protracting— Cont inued 


M . 

11 ° 

12 ° 

13 ° 

14 ° 

15 ° 

10 ° 

17 ° 

18 ° 

19 ° 

20 ° 

M. 

0 

.1017 

.2091 

.2264 

.2437 

.2611 

.2783 

.2956 

.3129 

.3301 

.3473 

O ' !] 

2 

.1923 

.2096 

.2270 

.2443 

.2616 

.2789 

.2962 

.3134 

.3307 

.3479 

2 k 

4 

.1928 

.2102 

.2276 

.2449 

.2622 

.2795 

.2968 

.3140 

.3312 

.3484 

4 

6 

.1934 

.2108 

.2281 

.2435 

.2628 

.2801 

.2973 

.3146 

.3318 

.3490 

6 

8 

.1940 

.2114 

.2287 

.2460 

.2634 

.2807 

.2979 

.3152 

.3324 

.3496 

8 

10 

.1946 

.2119 

.2293 

.2466 

.2639 

.2812 

.2985 

.3157 

.3330 

.3502 

10 

12 

.1952 

.2125 

.2299 

.2472 

.2645 

.2818 

.2991 

.3163 

.3335 

.3507 

12 

14 

14 

.1957 

.2131 

.2305 

.2478 

.2651 

.2824 

.2996 

.3169 

.3341 

.3513 

]fi 

.1963 

.2137 

.2310 

.2484 

.2657 

.2830 

.3002 

.3175 

.3347 

.3519 

16 

18 

.1969 

.2143 

.2316 

.2489 

.2662 

.2835 

.3008 

.3180 

.3353 

.3525 

18 

20 

.1975 

.2148 

.2322 

.2495 

.2668 

.2841 

.3014 

.3186 

.3358 

.3530 

20 

22 

.1981 

.2154 

.2328 

.2501 

.2074 

.2847 

.3019 

.3192 

.3364 

.3536 

22 

2>1 

.1986 

.2160 

.2333 

.2507 

.2680 

.2853 

.3025 

.3198 

.3370 

.3542 

24 

28 

.1992 

.2106 

.2339 

.2512 

.2685 

.2858 

.3031 

.3203 

.3376 

.3547 

26 

28 

.1998 

.2172 

.2345 

.2518 

.2691 

.2864 

.3037 

.3209 

.3381 

.3553 

28 l 

30 

.2004 

.2177 

.2351 

.2524 

.2697 

.2870 

.3042 

.3215 

.3387 

.3559 

30 1 

32 

.2010 

.2183 

.2357 

.2530 

.2703 

.2876 

.3048 

.3221 

.3393 

.3565 

32 

34 

.2015 

.2189 

.2362 

.2536 

.2709 

.2881 

.3054 

.3226 

.3398 

.3570 

34 

38 

.2021 

.2195 

.2368 

.2541 

.2714 

.2887 

.3060 

.3232 

.3404 

.3576 

36 

38 

.2027 

.2200 

.2374 

.2547 

.2720 

.2893 

.3065 

.3238 

.3410 

.3582 

38 / 

40 

.2033 

.2206 

.2380 

.2553 

.2726 

.2899 

.3071 

.3244 

.3416 

.3587 

40 ' 

42 

.2038 

.2212 

.2385 

.2559 

.2732 

.2904 

.3077 

.3249 

.3421 

.3593 

42 

44 

.2044 

.2218 

.2391 

.2564 

.2737 

.2910 

.3083 

.3255 

.3427 

.3599 

44 

48 

.2050 

.2224 

.2397 

.2570 

.2743 

.2916 

.3088 

.3261 

.3433 

.3605 

46 

48 

.2056 

.2229 

.2103 

.2576 

.2749 

.2922 

.3094 

.3267 

.3439 

.3610 

48 

50 

.2062 

.2235 

.2409 

.2582 

.2755 

.2927 

.3100 

.3272 

.3444 

.3616 

50 

52 

.2067 

.2241 

.2414 

.2587 

.2760 

.2933 

.3106 

.3278 

.3450 

.3622 

52 x 

54 

.2073 

.2247 

.2420 

.2593 

.2 706 

.2939 

.3111 

.3284 

.3456 

.3628 

54 

58 

.2079 

.2253 

.2426 

.2599 

.2772 

.2945 

.3117 

.3289 

.3462 

.8633 

56 

58 

.2085 

.2258 

.2432 

.2605 

.2778 

.2950 

.3123 

.3295 

.3467 

.3659 

58 1 

60 

.2091 

.2264 

.2437 

.2611 

.2783 

.2956 

.3129 

.3301 

.3473 

.8645 

60 


M . 

o|° 

>>oo 

fm* 

23° 

24° 

25° 

26° 

27° 

28° 

29° 

30° 

i 

M . ! 

0 ' 

.3645 

.3816 

.3987 

.4158 

.4329 

.4499 

.4669 

.4838 

. 501*8 

.5176 

0' 

2 

.3650 

.3822 

.3993 

.4164 

.4334 

.4505 

.4675 

.4844 

.5013 

.5182 

2 

4 

.3656 

.3828 

.3999 

.4170 

.4340 

.4510 

.4680 

.4850 

.5019 

.5188 

4 

6 

.3662 

.3833 

.4004 

.4175 

.4346 

.4516 

.4686 

.4855 

• 5 C 24 

.5193 

6 

8 

.3668 

.3839 

.4010 

.4181 

.4352 

.4522 

.4692 

.4861 

.5030 

.5199 

8 

10 

.3673 

.3845 

.4016 

.4187 

.4357 

.4527 

.4697 

.4867 

.5036 

.5204 

10 1 

12 

.3679 

.3850 

.4022 

.4192 

.4363 

.4533 

.4703 

.4872 

. 5(41 

.5210 

12 

14 

.3685 

.3856 

.4027 

.4198 

.4369 

.4539 

.4708 

.4878 

.5047 

.5216 

14 

16 

.3690 

.3862 

.4033 

.4204 

.4374 

.4544 

.4714 

.4884 

.5053 

.5221 

16 

18 

.3696 

.3868 

.4039 

.4209 

.4380 

.4550 

.4720 

.4889 

.5058 

.5227 

18 

20 

.3702 

.3873 

.4044 

.4215 

.4386 

.4556 

.4725 

.4895 

.5064 

.5233 

20 

22 

.3708 

.3879 

.4050 

.4221 

.4391 

.4561 

.4731 

.4901 

.5070 

.5238 

22 ! 

24 

.3713 

.3885 

.4056 

.4226 

.4397 

.4567 

.4737 

.4906 

.5075 

.5244 

24 

26 

3719 

.3890 

.4061 

.4232 

.4403 

.4573 

.4742 

.4912 

.5081 

.5249 

26 

28 

.3725 

.3896 

.4067 

.4238 

.4408 

.4578 

.4748 

.4917 

.5086 

.5265 

28 

30 

.3730 

.3902 

.4073 

.4244 

.4414 

.4584 

.4754 

.4923 

.5092 

.5261 

30 

32 

.3736 

.3908 

.4079 

.4249 

.4420 

.4590 

.4759 

.4929 

.5098 


32 

34 

.3742 

.3913 

.4084 

.4255 

.4425 

.4595 

.4765 

.4934 

.5103 

.5272 

34 

36 

.3748 

.3919 

.4090 

.4261 

.4431 

.4601 

.4771 

.4940 

.5109 

.5277 

36 

38 

.3753 

.3925 

.4096 

.4266 

.4437 

.4607 

.4776 

.4946 

.5115 

.5283 

38 

40 

.3759 

.3930 

.4101 

.4272 

.4442 

.4612 

.4782 

.4951 

.5120 

.5289 

40 

42 

.3765 

.3936 

.4107 

.4278 

.4448 

.4618 

.4788 

.4957 

.5126 

.5294 

42 

44 

.3770 

.3942 

.4113 

.4283 

.4454 

.4624 

.4793 

.4963 

.5131 

.5300 

44 

46 

.3776 

.3947 

.4118 

.4289 

.4459 

.4629 

.4799 

.4968 

.5137 

.5306 

46 1 

48 

.3782 

.3953 

.4124 

.4295 

.4465 

.4635 

.4805 

.4974 

.5143 

.5311 

48 ] 

50 

.3788 

.3959 

.4130 

.4300 

.4471 

.4641 

.4810 

.4979 

.5148 

.5317 

50 

52 

.3793 

.3965 

.4135 

.4306 

.4476 

.4646 

.4816 

.4985 

.5154 

.5322 

52 

54 

.3799 

.3970 

.4141 

.4312 

.4482 

.4652 

.4822 

.4991 

.5160 

.5328 

54 j 

56 

.3805 

.3976 

.4147 

.4317 

.4488 

.4658 

.4827 

.4996 

.5165 

.5334 

56 

58 

.3810 

.3982 

.4153 

.4323 

.4493 

.4663 

.4833 

.5002 

.5171 

.5339 

58 

60 

.3816 

.3987 

.4158 

.4329 

.4499 

.4669 

.4838 

.5008 

.5176 

.5345 

60 




































































































































TABLE OF CHORDS 


107 


Table of chords, in parts of a rad 1; for protracting: — Continued. 


M . 

31° 

32° 

33° 

34° 

35° 

36° 

37° 

3S° 

39° 

40° 

M . 

0 ' 

.5345 

.5513 

.5680 

.5847 

.6014 

.6180 

.6346 

.6511 

.6676 

.6840 

0 

2 

.5350 

.5518 

.5686 

.5853 

.6020 

.6186 

.6352 

.6517 

.6682 

.6846 

2 

4 

.5356 

.5524 

.5691 

.5859 

.6025 

.6191 

.6357 

.6522 

.6687 

.6851 

4 

6 

.5362 

.5530 

.5697 

.5864 

.6031 

.6197 

.6363 

.6528 

. 6691 V 

.6857 

6 

8 

.5367 

.5535 

.5703 

.5870 

.6036 

.6202 

.6368 

.6533 

. 6698 ’ 

.6862 

8 

10 

.5373 

.5541 

.5708 

.5875 

.6042 

.6208 

.6374 

.6539 

.6704 

.6868 

10 

12 

.5378 

.5546 

.5714 

.5881 

.6047 

.6214 

.6379 

.6544 

.6709 

.6873 

12 

14 

.5381 

.5552 

.5719 

.5886 

.6053 

.6219 

.6385 

.6550 

.6715 

.6879 

14 

16 

.5390 

.5557 

.5725 

.5892 

.6058 

.6225 

.6390 

.6555 

.6720 

.6884 

16 

18 

.5395 

.5563 

.5730 

.5897 

.6064 

.6230 

.6396 

.6561 

.6725 

.6890 

18 

20 

.5401 

.5569 

.5736 

.5903 

.6070 

.6236 

.6401 

.6566 

.6731 

.6895 

20 

22 

.5406 

.5574 

.5742 

.5909 

.6075 

.6241 

.6407 

.6572 

.6736 

.6901 

22 

24 

.5412 

.5580 

.5747 

.5914 

.6081 

.6247 

.6412 

.6577 

.6742 

.6906 

24 

26 

.5418 

.5585 

.5753 

.5920 

.6086 

.6252 

.6418 

.6583 

.6747 

.6911 

26 

28 

.5423 

.5591 

.5758 

.5925 

.6092 

.6258 

.6423 

.6588 

.6753 

.6917 

28 

30 

.5429 

.5597 

.5764 

.5931 

.6097 

.6263 

.6429 

.6594 

.6758 

.6922 

30 

32 

.5434 

.5602 

.5769 

.5936 

.6103 

.6269 

.643 4 

.6599 

.6764 

.6928 

32 

34 

.5440 

.5608 

.5775 

.5942 

.6108 

.6274 

.6440 

.6605 

.6769 

.6933 

34 

30 

.5446 

.5613 

.5781 

.59 47 

.6114 

.6280 

.6445 

6610 

.6775 

.6939 

36 

38 

.5451 

.5619 

.5786 

.5953 

.6119 

.6285 

.6451 

.6616 

.6780 

.6944 

38 

40 

.5457 

.5625 

.5792 

.5959 

.6125 

.6291 

.6456 

.6621 

.6786 

.6950 

40 

42 

.5462 

.5630 

.5797 

.5964 

.6130 

.6296 

.6462 

.6627 

.6791 

.6955 

42 

41 

.5408 

.5030 

.5803 

.5970 

.6136 

.6302 

.6467 

.6632 

.6797 

.6961 

44 

46 

.5474 

.5641 

.5808 

.5975 

.6142 

.6307 

.6473 

.6638 

.6802 

.6966 

46 

48 

.5479 

.5647 

.5814 

.5981 

.6147 

.6313 

.6478 

.6643 

.6808 

.6971 

48 

50 

.5485 

.5652 

.5820 

.5986 

.6153 

.6318 

.6484 

.6649 

.6813 

.6977 

50 

52 

.5490 

.5658 

.5825 

.5992 

.6158 

.6324 

.6489 

.6654 

.6819 

.6982 

52 

54 

.5496 

.5664 

.5831 

.5997 

.6164 

.6330 

.6495 

.6660 

.6824 

.6988 

54 

56 

.5502 

.5009 

.5836 

.6003 

.6169 

.6335 

.6500 

•6665 

.6829 

.6993 

56 

58 

.5507 

.5675 

.5842 

.6009 

.6175 

.6341 

.6506 

.6671 

.6835 

. 699.3 

58 

60 

.5513 

.5680 

.5847 

.6014 

.6180 

.6346 

.6511 

.6676 

.6840 

.7004 

60 


M . 

41° 

42° 

42° 

44° 

45° 

46° 

47° 

48° 

49° 

50° 

M . 

O ' 

.7004 

.7167 

.7330 

.7492 

.7654 

.7815 

.7975 

.8135 

.8294 

.8452 

0 ’ 

2 

.7010 

.7173 

.7335 

.7498 

.7659 

.7820 

.7980 

.8140 

.8299 

.8458 

2 

4 

.7015 

.7178 

.7341 

.7503 

.7664 

.7825 

.7986 

.8145 

.8304 

.8463 

4 

6 

.7020 

.7184 

.7346 

.7508 

.7670 

.7831 

.7991 

.8151 

.8310 

.8468 

6 

8 

.7026 

.7189 

.7352 

.7514 

.7675 

.7836 

.7996 

.8156 

.8315 

.8473 

8 

10 

.7031 

.7195 

.7357 

.7519 

.7681 

.7841 

.8002 

.8161 

.8320 

.8479 

10 

12 

.7037 

.7200 

.7362 

.7524 

.7686 

.7847 

.8007 

.8167 

.8326 

.8484 

12 

14 

.7042 

.7205 

.7368 

.7530 

.7691 

.7852 

.8012 

.8172 

.8331 

.8489 

14 

16 

.7048 

.7211 

.7373 

.7535 

.7697 

.7857 

.8018 

.8177 

.8336 

.8495 

16 

18 

.7053 

.7216 

.7379 

.7541 

.7702 

.7863 

.8023 

.8183 

.8341 

.8500 

18 

20 

.7059 

.7222 

.7384 

.7546 

.7707 

.7868 

.8028 

.8188 

.8347 

.8505 

20 

22 

.7064 

.7227 

.7390 

.7551 

.7713 

.7873 

.8034 

.8193 

.8352 

.8510 

22 

24 

.7069 

.7232 

.7395 

.7557 

.7718 

.7879 

.8039 

.8198 

.8357 

.8516 

24 

26 

.7075 

.7238 

.7400 

.7562 

.7723 

.7884 

.8044 

.8204 

.8363 

.8521 

26 

28 

.7080 

.7243 

.7406 

.7568 

.7729 

.7890 

.8050 

.8209 

.8368 

.8526 

28 

30 

.7086 

.7249 

.7411 

.7573 

.7734 

.7895 

.8055 

.8214 

.8373 

.8531 

30 

32 

.7091 

.7254 

.7417 

.7578 

.7740 

.7900 

.8060 

.8220 

.8378 

.8537 

32 

34 

.7097 

.7260 

.7422 

.7584 

.7745 

.7906 

.8066 

.8225 

.8384 

.8542 

34 

36 

.7102 

.7265 

.7427 

.7589 

.7750 

.7911 

.8071 

.8230 

.8389 

.8547 

36 

38 

.7108 

.7270 

.7433 

.7595 

.7756 

.7916 

.8076 

.8236 

.8394 

.8552 

38 

40 

.7113 

.7276 

.7438 

.7600 

.7761 

.7922 

.8082 

.8241 

.8400 

.8558 

40 

42 

.7118 

.7281 

.7443 

.7605 

.7766 

.7927 

.8087 

.8246 

.8405 

.8563 

42 

44 

.7124 

.7287 

.7449 

.7611 

.7772 

.7932 

.8092 

.8251 

.8410 

.8568 

44 

46 

.7129 

.7292 

.7454 

.7616 

.7777 

.7938 

.8098 

.8257 

.8415 

.8573 

46 

48 

.7135 

.7298 

.7460 

.7621 

.7782 

.7943 

.8103 

.8262 

.8421 

.8579 

48 

50 

.7140 

.7303 

.7465 

.7627 

.7788 

.7948 

.8108 

.8267 

.8426 

.8584 

50 

52 

.7146 

.7308 

.7471 

.7632 

.7793 

.7954 

.8113 

.8273 

.8431 

.8589 

52 

54 

.7151 

.7314 

.7476 

.7638 

.7799 

.7959 

.8119 

.8278 

.8437 

.8594 

54 

56 

.7156 

.7319 

.7481 

.7643 

.7804 

.7964 

.8124 

.8283 

.8442 

.8600 

56 

58 

.7162 

.7325 

.7487 

.7648 

.7809 

.7970 

.8129 

.8289 

.8447 

.8605 

58 

60 

.7167 

.7330 

.7492 

.7654 

.7815 

.7975 

.8135 

.8294 

.8452 

.8610 

60 
















































































































1 


108 TABLE OF CHORDS. 


Table of chords, in parts of a rad 1; for protracting— Continued 


M . 

51° 

52° 

53° 

54° 

55° 

56° 

57° 

58° 

159° 

60° 

M . 

O ' 

.8610 

.8767 

.8924 

.9080 

.9235 

.9389 

.9543 

.9696 

.9848 

1.0000 


2 

.8615 

.8773 

.8929 

.9085 

.9240 

.9395 

.9548 

.9701 

.9854 

1.0005 

2 

4 

.8621 

.8778 

.8934 

.9090 

.9245 

.9400 

.9553 

.9706 

.9859 

1 . 04)10 

4 

6 

.8626 

%.8783 

.8940 

.9095 

.9250 

.9405 

.9559 

.9711 

.9864 

1.0015 

6 

8 

.8631 

.8788 

.8945 

.9101 

.9256 

.9410 

.9564 

.9717 

.9869 

1.0020 

8 

10 

.8636 

.8794 

.8950 

.9106 

.9261 

.9415 

.9569 

.9722 

.9874 

1.0025 

10 

12 

.8642 

8799 

.8955 

.9111 

.9266 

.9420 

.9574 

.9727 

.9879 

1.0030 

12 

14 

.8647 

.8804 

.8960 

.9116 

.9271 

.9425 

.9579 

.9732 

.9884 

1.0035 

14 

16 

. 8 ( i 5‘2 

.8809 

.8966 

.9121 

.9276 

.9430 

.9584 

.9737 

.9889 

1.0040 

16 

18 

.8657 

.8814 

.8971 

.9126 

.9281 

.9436 

.9589 

.9742 

.9894 

1.0045 

18 

20 

.8663 

.8820 

.8976 

.9132 

.9287 

.9441 

.9594 

.9747 

.9899 

1.0050 

20 

22 

.8668 

.8825 

.8981 

.9137 

.9292 

.9446 

.9599 

.9752 

.9904 

1.0055 

22 

24 

.8673 

.8830 

.8986 

.9142 

.9297 

.9451 

.9604 

.9757 

.9909 

1.0060 

24 

26 

.8678 

.8835 

'.8992 

.9147 

.9302 

.9456 

.9610 

.9762 

.9914 

1.0065 

26 

28 

.8684 

.8841 

• 8 S 97 

.9152 

.9307 

.9461 

.9615 

.9767 

.9919 

1.0070 

28 

30 

.8689 

.8846 

.9002 

.9157 

.9312 

.9466 

.9620 

.9772 

.9924 

1.0075 

30 

32 

.8694 

.8851 

.9007 

.9163 

.9317 

.9472 

.9625 

.9778 

.8929 

1.0080 

32 

34 

.8699 

.8856 

.9012 

.9168 

.9323 

.9477 

.9630 

.9783 

.9934 

1.0086 

34 

36 

.8705 

.8861 

.9018 

.9173 

.9328 

.9482 

.9635 

.9788 

.9939 

1.0091 

36 

38 

.8710 

.8867 

.9023 

.9178 

.9333 

.9487 

.9640 

.9793 

.9945 

1.0096 

38 

40 

.8715 

.8872 

.9028 

.9183 

.9338 

.9492 

.9645 

.9798 

.9950 

1.0101 

40 

42 

.8720 

.8877 

.9033 

.9188 

.9343 

.9497 

.9650 

.9803 

.9955 

1.0106 

42 

44 

.8726 

.8882 

.9038 

.9194 

.9348 

.9502 

.9655 

.9808 

.9960 

1.0111 

44 

46 

.8731 

.8887 

.9044 

.9199 

.9353 

.9507 

.9661 

.9813 

.9965 

1.0116 

46 

48 

.8736 

.8893 

.9049 

.9204 

.9359 

.9512 

.9666 

.9818 

.9970 

1.0121 

48 

50 

.8741 

.8898 

.9054 

.9209 

.9364 

.9518 

.9671 

.9823 

.9975 

1.0126 

50 

52 

.8747 

.8903 

.9059 

.9214 

.9369 

.9523 

.9676 

.9828 

.9980 

1.0131 

52 

54 

.8752 

.8908 

.9064 

.9219 

.9374 

.9528 

.9681 

.9833 

.9985 

1.0136 

54 

56 

.8757 

.8914 

.9069 

.9225 

.9379 

.9533 

.9686 

.9838 

.9990 

1.0141 

56 

58 

.8762 

.8919 

.9075 

.9230 

.9384 

.9538 

.9691 

.9843 

.9995 

1.0146 

58 

60 

.8767 

.8924 

.9080 

.9235 

.9389 

.9543 

.9696 

.9848 

1.0000 

1.0151 

60 


M . 

01° 

62° 

63° 

64° 

65° 

66° 

67° 

68° 

69° 

70° 

M . 

0 

1.0151 

1.0301 

1.0450 

1.0598 

1.0746 

1.0893 

1.1039 

1.1184 

1.1328 

1.1472 

o - 

2 

1.0156 

1.0306 

1.0455 

1.0603 

1.0751 

1.0898 

1.1044 

1.1189 

1.1333 

1.1476 

2 

4 

1.0161 

1.0311 

1.0460 

1.0608 

1.0756 

1.0903 

1.1048 

1.1194 

1.1338 

1.1481 

4 

6 

1.0166 

1.0316 

1.0465 

1.0613 

1.0761 

1.0907 

1.1053 

1.1198 

1.1342 

1.1486 

6 

8 

1.0171 

1.0321 

1.0470 

1.0618 

1 0766 

1.0912 

1.1058 

1.1203 

1.1347 

1.1491 

8 

10 

1.0176 

1.0326 

1.0475 

1.0623 

1.0771 

1.0917 

1.1063 

1.1208 

1.1352 

1.1495 

10 

12 

1.0181 

1.0331 

1.0480 

1.0628 

1.0775 

1.0922 

1.1068 

1.1213 

1.1357 

1.1500 

12 

14 

1.0186 

1.0336 

1.0485 

1.0633 

1.0780 

1.0927 

1.1073 

1.1218 

1.1362 

1.1505 

14 

16 

1.0191 

1.0341 

1.0490 

1.0638 

1.0785 

1.0932 

1.1078 

1.1222 

1.1366 

1.1510 

16 

18 

1.0196 

1.0346 

1.0495 

1.0643 

1.0790 

1.0937 

1.1082 

1.1227 

1.1371 

1.1514 

18 

20 

1 0201 

l .0351 

1.0500 

1 0648 

1.0795 

1.0942 

1.1087 

1.1232 

1.1376 

1.1519 

20 

22 

1.0206 

1.0356 

1 . 05 i 4 

1.0653 

1.0800 

1.0946 

1.1092 

1.1237 

1.1381 

1.1524 

22 

24 

1.0211 

1.0361 

1.0509 

1.0658 

1.0805 

1.0951 

1.1097 

1.1242 

1.1386 

1.1529 

24 

26 

1.0216 

1.0366 

1.0514 

1.0662 

1.0810 

l.i 956 

1.1102 

1.1246 

1.1390 

1.1533 

26 

28 

1.0221 

1.0370 

1.0519 

1.0667 

1.0815 

1.0961 

1.1107 

1.1251 

1.1395 

1.1538 

28 

30 

1.0226 

1.0375 

1.0524 

1.0672 

1.0820 

1.0966 

1.1111 

1.1256 

1.1400 

1.1543 

30 

32 

1.0231 

1 0380 

1.0529 

1.0677 

1.0824 

1.0971 

1.1116 

1.1261 

1.1405 

1.1548 

32 

34 

1.0236 

1.0385 

1.0534 

1.0682 

1.0829 

1.0976 

1.1121 

1.1266 

1.1409 

1.1552 

34 

36 

1.0241 

1.0390 

1.0539 

1.0687 

1.0834 

1.0980 

1.1126 

1.1271 

1.1414 

1.1557 

36 

38 

1.0246 

1.0395 

1.0544 

1.0692 

1.0839 

1.0985 

1.1131 

1.1275 

1.1419 

1.1562 

38 

40 

1.0251 

1.0400 

1.0549 

1.0697 

1.0844 

1.0990 

1.1136 

1.1280 

1.1424 

1.1567 

40 

42 

1.0256 

1.0405 

1.0554 

1.0702 

1.0849 

1.0995 

1.1140 

1.1285 

1.1429 

1.1571 

42 

44 

1.0261 

1.0410 

1.0559 

1.0707 

1.0854 

1.1000 

1.1145 

1.1290 

1.1433 

1.1576 

44 

46 

1.0266 

1.0415 

1 0564 

1.0712 

1.0859 

1.1005 

1.1150 

1.1295 

1.1438 

1.1581 

46 

48 

1.0271 

1.0420 

1.0569 

1.0717 

1.0863 

1.1010 

1.1155 

1.1299 

1.1443 

1.1586 

48 

50 

1.0276 

1.0425 

1.0574 

1.0721 

1 0868 

1.1014 

1.1160 

1.1304 

1.1448 

1.1590 

50 

52 

1.0281 

1.0430 

1.0579 

1.0726 

1.0873 

1.1019 

1.1165 

1.1309 

1.1452 

1.1595 

52 

54 

1.0286 

1.0435 

1.0584 

1.0731 

1.0878 

1.1024 

1.1169 

1.1314 

1.1457 

1.1600 

54 

56 

1.0291 

1.0440 

1.0589 

1.0736 

1.0883 

1.1029 

1.1174 

1.1319 

1.1462 

1.1605 

56 

58 

1.0296 

1.0445 

1.0593 

1.0741 

1.0888 

1.1034 

1.1179 

t 1.1323 

1.1467 

1.1609 

58 

60 

1.0301 

1.0450 

1.0598 

1.0746 

1.0893 

1.1039 

1.1184 

1.1328 

1.1472 

1.1614 

60 































































































































TABLE OF CHORDS. 


109 


Table of Chords, in parts of a rad 1; for protracting 1 — Continued. 


M . 

71° 

72° 

7ii° 

74° 

75° 

76° 

77° 

78° 

79° 

0 

O 

M . 

0 

1.1614 

1.1756 

1.1896 

1.2036 

1.2175 

1.2313 

1.2450 

1.2586 

1.2722 

1.2856 

0 ' 

2 

1 UiiJ 

1.1760 

1.1901 

1.2041 

1.2180 

1.2318 

1.2455 

1.2591 

1.2726 

1.2860 

2 

4 

1.1524 

1.1765 

1.1906 

1.2046 

1.2184 

1.2322 

1.2459 

1.2595 

1.2731 

1.2865 

4 

6 

1.1628 

1.1770 

1.1910 

1.2050 

1.2189 

1.2327 

1.2464 

1.2600 


1.2869 

6 

8 

1 15.53 

1.1775 

1.1915 

1.2055 

1.2194 

1.2332 

1.2468 

1.2604 

1.2740 

1.2874 

8 

10 

1.1638 

1.1779 

1.1920 

1.2060 

1.2198 

1.2336 

1.2473 

1.2609 

1.2744 

1 2878 

10 

12 

1.1642 

1.1784 

1.1924 

1.2064 

1.2203 

1.2341 

1.2478 

1.2614 

1.2748 

1.2882 

12 

14 

1.1647 

1.1789 

1.1929 

1.2069 

1.2208 

1.2345 

1.2482 

1.2618 

1.2753 

1.2887 

14 

10 

1.1652 

1.1793 

1.1931 

1.2073 

1.2212 

1.2350 

1.2487 

1.2623 

1.2757 

1.2891 

16 

is 

1.1657 

1.1798 

1.1938 

1.2078 

1.2217 

1.2354 

1.2191 

1.2627 

1.2762 

1.2896 

18 

20 

1.1661 

1.1803 

1.1943 

1.2083 

1.2221 

1.2359 

1.2496 

1.2632 

1.2766 

1.2900 

20 

22 

1.1665 

1.1807 

1.1948 

1.2087 

1.2226 

1.2364 

1.2500 

1.2636 

1.2771 

1.2905 

22 

21 

1.1671 

1.1812 

1.1952 

1.2092 

1.2231 

1 2368 

1.2505 

1.2641 

1.2775 

1.2909 

24 

23 

1.1876 

1.1817 

1.1957 

1.2097 

1.2235 

1.2373 

1.2509 

1.2645 

1 2780 

1.2914 

26 

28 

1.1680 

) -1821 

1 1962 

1.2101 

1.2240 

1.2377 

1.2514 

1.2650 

1.2784 

1.2918 

28 

30 

1.1685 

1.1826 

1.1966 

1.2106 

1.2244 

1.2382 

1.2518 

1.2654 

1.2789 

1.2922 

30 

32 

1.1890 

1.1831 

1.1971 

1.2111 

1.2249 

1.2386 

1.2523 

1.2659 

1.2793 

1.2927 

32 

34 

1.1694 

1.1836 

1.1976 

1.2115 

1.2254 

1.2391 

1.2528 

1.2663 

1.2798 

1.2931 

34 

38 

1.1699 

1.1840 

1 1980 

1.2120 

1.2258 

1.2396 

1.2532 

1.2668 

1.2802 

1.2936 

36 

38 

1.1704 

1.1845 

1.1985 

1.2124 

1.2263 

1.2400 

1.2537 

1.2672 

1.2807 

1.2910 

38 

40 

1.1709 

1.1850 

1.1990 

1.2129 

1.2267 

1.2405 

1.2541 

1.2677 

1.2811 

1.2945 

40 

42 

1.1713 

1.1854 

1.1994 

1.2134 

1.2272 

1.2409 

1.2546 

1.2681 

1.2816 

1.2949 

42 

41 

1.1718 

1.1859 

1.1999 

1.2138 

1.2277 

1.2414 

1.2550 

1.2686 

1.2820 

1.2954 

41 

46 

1.1723 

1.1864 

1.2004 

1.2143 

1.2281 

1.2418 

1.2555 

1.2690 

1.2825 

1.2958 

46 

48 

1.1727 

1.1868 

1.2008 

1.2148 

1.2286 

1.2423 

1.2559 

1.2695 

1.2829 

1.2962 

48 

50 

1.1732 

1.1 <73 

1.2013 

1.2152 

1.2290 

1.2428 

1.2564 

1.2699 

1.2833 

1.2967 

50 

52 

1.1737 

1.1878 

1.2018 

1.2157 

1.2295 

1.2432 

1.2568 

1.2704 

1.2838 

1.2971 

52 

04 

l 1742 

1.1882 

1.2022 

1.2161 

1.2299 

1.2437 

1.2573 

1.2708 

1.2842 

1.2976 

54 

56 

1 1746 

1.1887 

1.2027 

1.2166 

1.2304 

1.2441 

1.2577 

1.2713 

1.2847 

1 2980 

56 

68 

1.1751 

1.1892 

1.2032 

1.2171 

1.2309 

1.2446 

1.2582 

1.2717 

1.2851 

1.2985 

58 

60 

1.1753 

1.1896 

1.2036 

1.2175 

1.2313 

1.2450 

1.2586 

1.2722 

1.2856 

1.2989 

60 


M . 

81° 

82° 

82° 

84° 

85° 

86° 

87° 

88° 

89° 

M . 

0 ' 

1 . 2)89 

1.3121 

1.3252 

1.3383 

1.3512 

1.3640 

1.3767 

1.3893 

1.4018 

O ' 

2 

1.2993 

1.3126 

1.3257 

1.3387 

1.3516 

1.3644 

1.3771 

1.3897 

1.4022 

2 

4 

1.2998 

1.3130 

1.3261 

1.3391 

1.3520 

1.3648 

1.3776 

1.3902 

1.4026 

4 

6 

1.3002 

1.3134 

1.3265 

1 -3396 

1.3525 

1 3653 

1.3780 

1.3906 

1.4031 

6 

8 

1.3007 

1.3139 

1.3270 

1.3400 

1.3529 

1.3657 

1.3784 

1.3910 

1 4035 

8 

10 

1.3011 

1.3143 

1.3274 

1.3404 

1.3533 

1.3661 

1.3788 

1.3914 

1.4039 

10 

12 

1.3015 

1.3147 

1.3279 

1.3409 

1.3538 

1.3665 

1.3792 

1.3918 

1.4043 

12 

14 

1.3020 

1.3152 

1.3283 

1.3413 

1.3542 

1.3670 

1.3797 

1.3922 

1.4017 

14 

16 

1.3024 

1 3156 

1.3287 

1.3417 

1.3546 

1.3674 

1.3501 

1.3927 

1.4051 

16 

18 

1.3029 

1.3161 

1.3292 

1.3421 

1.3550 

1.3678 

1.3805 

1 3931 

1.4055 

18 

20 

1.3033 

1.3165 

1.3296 

1.3426 

1.3555 

1.3682 

1.3809 

1.3935 

1.4060 

20 

22 

1.3038 

1.3169 

1.3300 

1.3430 

1.3559 

1.3687 

1.3813 

1.3939 

1.4064 

22 

21 

1.3012 

1.3174 

1.3305 

1.3434 

1.3563 

1.3691 

1.3818 

1.3943 

1.4068 

24 

26 

L .3016 

1.3178 

1.3309 

1.3439 

1.3567 

1.3695 

1.3822 

1.3947 

1.4072 

26 

28 

1.3051 

1.3183 

1.3313 

1.3443 

1.3572 

1.3699 

1.3826 

1.3952 

1.4076 

28 

30 

1.3055 

1.3187 

1.3318 

1.3447 

1.3576 

1.3704 

1.3830 

1.3956 

1.4080 

30 

32 

1.3060 

1.3191 

1.3322 

1.3452 

1.3580 

(.3708 

1.3834 

1.3960 

1.4084 

32 

31 

1 3061 

1.3196 

1.3326 

1.3456 

1.3585 

1.3712 

1.3839 

1 . 396 * 

1.4089 

34 

36 

1.3068 

1.3200 

1.3331 

1.3460 

1.3589 

1.3716 

1.3843 

1.3968 

1.4093 

36 

38 

1.3073 

1.3201 

1.3335 

1.3465 

1.3593 

1.3721 

1.3847 

1.3972 

1.4097 

38 

40 

1.3077 

1.3209 

1.3339 

1.3169 

1.3597 

1.3725 

1.3851 

1.3977 

1.4101 

40 

42 

1.3082 

1.3213 

1.3344 

1.3473 

1.3602 

1.3729 

1.3855 

1.3981 

1.4105 

42 

41 

1.3086 

1.3218 

1.3348 

1.3477 

1.3606 

1.3733 

1.3860 

1.3985 

1.4109 

44 

46 

1.3090 

1.3222 

1.3352 

1.3482 

1.3610 

1.3738 

1.3864 

1.3989 

1.4113 

46 

48 

1.3095 

1.3226 

1.3357 

1.3486 

1.3614 

1.3742 

1.3868 

1.3993 

1.4117 

48 

60 

1.3099 

1.3231 

1.3361 

1.3490 

1.3619 

1.3746 

1.3872 

1.3997 

1.4122 

50 

52 

1.3101 

1.3235 

1.3365 

1.3495 

1 3623 

1.3750 

1.3876 

1.4002 

1.4126 

52 

54 

1.3108 

1.3239 

1.3370 

1.3499 

1.3627 

1.3754 

1.3881 

1.4006 

1.4130 

54 

56 

1.3112 

1.3244 

1.3371 

1.3503 

1.3631 

1.3759 

1.3885 

1.4010 

1.4134 

66 

58 

1.3117 

1 . 32*8 

1.3378 

1.3508 

1.3636 

1.3763 

1.3889 

1.4014 

1.4138 

58 

60 

1.3121 

1.3252 

1.3383 

1.3512 

1.3640 

1.3767 

1.3893 

1.4018 

1.4142 

60 


































































































































110 


POLYGONS. 


POLYGONS. 


,:T| 


a 


a 0 b 




Heptagon. 

7 6LUZ6, 


Octagon. 

B sides. 



Any straight-sided fig is called a polygon. If all the sides and angles are equal, it is a regular 
polygon ; if not, it is Irregular. Of course the number of polygons is infinite. 


Table of Regular Polygons. 


Number 

of 

Sides. 

Name 

of 

Polygon. 

A ron ~~ 

(square of one 
side) mult by 

Radius of cir¬ 
cumscribing 
circle = side 
mult by 

Interior angle 

a It c contained 
between two 
sides. 

Angle at cen, 

subtended 
by a side. 

3 { 

Equilateral 

triangle. 

| .433013 

.577350 

60° 

120° 

4 

Square. 

1.000000 

.707107 

90° 

90° 

5 

Pentagon. 

1.720477 

.850651 

108° 

72° 

6 

Hexagon. 

2.598076 

1.000000 

120° 

60° 

7 

Heptagon. 

3.633912 

1.152382 

128° 34 2857' 

51° 25.7143' 

8 

Octagon. 

4.828427 

1.306563 

135° 

45° 

9 

Nonagon. 

6.181«24 

1.461902 

140° 

40° 

10 

Decagon. 

7.694209 

1.618034 

144° 

36° 

11 

Undecagon. 

9 365640 

1.774733 

147° 16.3636' 

32° 43.6364' 

12 

Dodecagon 

11.196152 

1.931854 

150° 

30° 


Area of any regular polygon = length of one side, a 5 X perp p drawn from cen of fig to 

cen of side X half the number of sides. , 

Sum of Interior angles, a b c, etc, of any polygon, regular or Irregular = 180° X 

(nuiniier of sides — 2). 



We speak here of plane triangles only ; or those having straight sides. 

A triangle Is equilateral when a'll its sides are equal, as a ; isosceles when only two sides 
arc equal, as B; scalene when all the sides are unequal,'as C. I), and K: acute-angled " hen 
all its angles are aouie. or each less than 90°, as A, 15, and C : right-angled "lieu it contains a 
right angle, as D : obtuse-angled when it contains au obtuse angle, or one greater thau 90°. as E. 

All the three ungles of any triangle are equal to two right 
angles, or ISO-’; therefore, if we know two oi them, we cau find the third by 
subtracting their sum from 180°. All triangles which have equal bases, 
and equal perp heights, have also equal areas; thus the areas olawc.itie d. and 
a w e. are equal to each other. The area of any triangle is equal to half 
that of any parallelogram which has an equal base,and unequal perp height. The 
areas of triangles which have equal bases, butdilf perp heights, are to 
eacu other as, or in proportion to. their perp heights; thus the triangle awn, 
with a perp height s n. equal to but one-half that (s e) of the three other trian¬ 
gles, but with the same base a w, has also but half the area of either of those ® W S 

others. 

Area of any triangle, Figs A, B, C, D, E, = half the base, S, X the height, or perp dist p to 
the opposite angle. Any side may be taken as the base of a triangle ; but the perp height must always 
be measured from the side so assumed; to do which, the side must sometimes be prolonged, as in 
Fig E : but the prolongation is not to be considered as a part of the base. 

Area of any equilateral triangle = .433013 X square of one side. 



To find area, bavins' the three Hides. 

Add them together; div the sum by 2; from the half sum. subtract each side separately; mult the 
half sum and the three remainders continuously together; take the sq rt of the prod. 

Ex.—The three sides — 20, 30. 40 ft. Here 20 -(- 30 -f- 40 =r 90 ; and — 45. And 45 — 20 — 25 . 

45 — 30 = 15 ; aud 45 — 40- 5. And 45 X 25 X 15 X 5 ~ 84375 ; and the sq rt of 84375 is 290.47 >q ft, 
area reqd. 













































TRIANGLES 


111 


To find area, having one side and file 2 angles at its ends. 

Add the 2 angles together: take the sum from 180° ; the rem will be the angle opp the given side. 
Kind the nat sine of this angle ; also find the nat sines of the other angles, and mult them together. 
Theu as the nat sine of the single angle, is to the prod of the nat sines of the other 2 angles, so is the 
square of the given side to double the reqd area. 

To find area, having two sides, and the included angle. 

Mult together the two sides, and the nat sine of the included angle ; div by 2. 

Ex—Sides 650 ft and 980 ft; included angle 69° 20'. By the table we find the nat sine .9356; 


therefore, 


650 X 980 X .9356 

Y 


— 297988.6 square ft area. 



To find area, having the three angles and the 
perp height, a b. 

Find the nat sines of the three angles ; mult together the sines of the angles 
d and o; div the sine of the angle b by the prod ; mult the quot by the square 
of the perp height a b ; div by 2. 

To find any side, as d o, having the three 
angles, d, b and o, and the area. 

(Sine of d X sine of o) • sine of b : t twice the area : square of d o. See Rem 2, p 119. 

The perp height of an equilateral triangle is equal to one side X .866025. Hence one of 
its sides is equal to the perp height div by .866025 or to perp height X 1.1547. Or, to find a side, 
mult the sq rt of its area by 1.51967. The side of an equilateral triangle, mult by .658037 = side of a 
square of the same area ; or mult by .742517 it gives the diam of a circle of the same area. 

The following apply to any plane triangle, whether oblique or right-angled : 

1 . The three angles amount to 180 3 , or two right angles. 

3. Any exterior angle, as A C n, is equal to the two interior and opposite 
ones, A and B. 

8. The greater side is opposite the greater angle. 

4. The sides are as the sines of the,opposite angles. Thus, the side a is to 
the side 6 as the sine of A is to the sine of B. 

£>. If any angle as s be bisected by a line s o, the two parts mo, on of 
the opposite side m n will be to each other as the other two sides am, sn ; 
or, m o: o n ::s m:a n. 

6. If lines be drawn from each angle r s t to the 

^ center of the opposite side, they will cross each 

other at one point, a, and the short part of each 
of the lines will be the third part of the whole line. 

Also, a is the cen of grav of the triangle. 

7, If lines be drawn bisecting the three angles, they will meet at a point 
perpendicularly equidistant from each side, and consequently the center 

, of the greatest circle that can be drawn in the triangle. 4 

t 8. If a line s n be drawn parallel to any side c a, 
the two triangles r a n, r c a, will be similar. 

9. To divide any triangle a c r into two equal parts by a line s n parallel to 
any one of its sides c a. On either one of the other sides, as a r, as a diam, 
describe a semicircle a o r; and find its middle o. From r (opposite c a), with 

radius r o, describe the arc o n. From n draw n s, par¬ 
allel to c a. 

10 . To find the greatest parallelogram that can be 
drawn in any given triangle o nb. Bisect the three sides at ace, and join 
a c, a e, e c. Then either a e b c, a e c o, or a c e n, each equal to half the 
triangle, will be the reqd parallelogram. Any of these parallelograms can 
plainly be converted into a rectangle of equal area, and the greatest that can be 
drawn in the triangle. 

1014. If a line a c bisects any two sides o b, o n, of a triangle, it will be par¬ 
allel to the third side n b. and half as long as it. 

11 . To find the greatest square that can be drawn in any triangle a xr. 
an angle as a draw a perp a n to the opposite side xr, and find its length 

r X fl tv 

on, or a side v t of the square will = 







From 

Theu 


x r an 

Rem.—If the triangle is such that two or three such perps can be drawn, then 
two or three equal squares may be found. 






















112 


PLANE TRIGONOMETRY 


♦> i 


Ki^tif-an^Iod Triangles. 

All the foregoing apply also to right-angled triangles; but what follow apply to them only. 

Call the right angle A, and the others B and C; and call the sides respectively 
opposite to them a, b, and c. Then is 



a ~ --j—- — c X Sec B = =—:-- — b X Sec C + c2 - 

Sine C Cosine C v • 


b = a X Sine B — a X Cos C — c X Cot C — c X Tang B. 
c — a X Sine C — a X Cos B — 6 X Tang C. 


CO c 

Also Sine of C = — ; Cos C ; Tang C = ^. 


b c b 

And Sine of B = - ; Cos B = - ; Tang B = —. 


And Sine of A or 90° =r 1. Cos A — 0. Tang A =: infinity. Sec A = infinity. 

If from the right angle o a line o to be drawn perp to the hypothenuse or long side h g , then the 
two small triangles o tv h, o to g, and the large one o h g, will oe similar. 
Or g to : tv o : : w o : to h; and g tv X tv A =r to o2. 

2. A line drawn from the right angle to the center of the long side will 
be half as long as sa ; d side. 

3 . If on the three sides oh, o g, g h we draw three squares t, u, v, or 
three circles, or triangles, or any other three (igs that are similar, then the 
area of the largest one is equal to the sum of the ureas of the two others. 

4. In a triangle whose sides are as 3, 4, and 5 (as are those of the tri¬ 
angle A B C), the angles are very approximately 90°; 53° 7'48.38": and 
36° 52' 11.62". Their Sines, 1.; .8; and .6. Their Tangs, infinity ; 1.3333; 
and .75. . 

5. One whose sides are as ", 7, and 9 9, has very approx one angle of 90° 
and two of 45° each, near enough for all practical purposes. 



I 




1 


PLANE TRIGONOMETRY. 


Plans trigonometry teaches how to find certain unknown parts of plane, or straight - sided tri¬ 
angles, by meaws of other parts which are known; aud thus enables us to measure inaccessible dis¬ 
tances, &c. A triangle consists of six parts, namely, three sides, and three angles; and if we know 
any three of these, (except the three angles, and in the ambiguous case under “ Case 2,’’) we can find 
the other three. The following four cases include the whole subject; the 6tudent should commit them 
to memory. 


Case 1. Having' any two angles, anil one side, 
to find the other sides and angle. 

Add the two angles together-; and subtract their sum from 180°; the rem 
will be the third angle. And for the sides, as 

Sine of the angle . Sine of the angle . . glven side : reqd side, 

opp the given side • opp the reqd side • • 6 H 

Use the side thus found, as the given one; and in the same manner find 
the third side. 



Case 2. Having two sides, b a , a e, Fig X. and the angle a b c, 
opposite to one of them, to find the other side and augles. 



Side a c opp 
the given an¬ 
gle a b c 


The other 
given side 
b a 


Sine of the 
given angle 
a b c 


Sine of angle b d a or 
b c a opposite the other 
given side b a. 


Having found the sine, take out the corresponding angle from the table of 
nat sines, but, in doing so, if the side ac opp the given angle is 

Bhorter than the other given side b a , bear in mind that an angle aDd its sup¬ 
plement have the same sine. Thus, in Fig X, the sine, as found above, is 

opp the angle 6 c a in (he table. But a c, if shorter than b a, can evidently be 
laid off in the opp direction, a rf, in which case b d a is the supplement of b c a. 
If a c is as long as, or longer than, b a , there can be no doubt; for in that case 
it cannot be drawn townrd b, but only toward n and the augle 6 c a will be 
found at once in the table, opp the sine as found above. 


Fig. X. 













PLANE TRIGONOMETRY 


113 


When the two angles, a b c, 6 e a, have been found, find the remaining side by Case 1. 
tor the remaining angle, b a c, add together the angle a b c first given, and the one, b c a, found 
as above. Deduct their sum from 180°. 

Case 3. Having- two sides, and the angle included 

between them. 


Take the angle from 180° ; the rem will be the sum of the two unknown angles. Div this sum by 
2; and find the nat tang of the quot. Then as 

The sum of the . Theirdiff . . Tang of half the sum of . Tang of half 

two given sides • • * the two unknown angles • their dill'. 

Take from the table of nat tang, the angle opposite this last tang. Add this augle to the half sum 
of the two unknown angles, and it will give the angle opp the longest given side: and subtract it 
from the same half sum, for the angle opp the shortest given side. Having thus found the angles, 
find the third side by Case 1. 

As a practical example of the use of Case 3, we can ascertain the dist n m across a deep pond, by 
measuring two lines n o and m o ; and the angle n o m. From these data we may calculate n m ; or 
by drawing the two sides, and the angle on paper, by a scale, we can afterward measure n m on 
the drawing. 



C 



Case 4. Having- Hie three sides. 


To find the three angles: upon one side a 5 as a base, draw (or suppose to be drawn) a perp c g from 
the opposite angle c. Find the diff between the other two sides, ac and c6 ; also their sum. Then,as 

The base • ® ura of the • • of other . Diff of the two 
1 e D • other two sides • • two sides • parts ag and b g, of the base. 

Add half this diff of the parts, to half the base a b; the sum will be the longest part op; which 
taken from the whole base, gives the shortest party b. By this means we get in each of the small tri¬ 
angles a eg and egb, two sides, (namely, a c and a g; and c b aud g b;) and an augle (namely, the 
right angle eg a, or egb) opposite to one of the given sides. Therefore, use Case 2 for finding the 
angles a and b. When that is done, take their sum from 180°, for the augle a c b. 



Or, 58d mode : call half the sum of the three sides, g; and call the 
two sides which form either angle, m and n. Then the nat sine of 


half that angle will be equal to 


V (s — m) X (s 
m X n 


—w) 


Ex. 1. To find the dist from n to an inac¬ 
cessible object c. 

Measure a line ah ; and from its ends measure the angles cab and 
c b a. Thus having found one side and two angles of the triangle a b c, 
calculate a c by means of Case 1. Or if extreme accuracy is uot reqd, 
draw the line a b on paper to auy convenient scale; then by means of a 
protractor lay off the angles c a b, c b a ; and draw a c and cb ; then 
measure a c by the same scale. 


n 



Ex. 2. To find the height of a vertical 
object, n a. 

Place the instrument for measuring angles, at any conve- 
nient spot o ; also meas the dist o a : or if on cannot be actually 
measd in consequence of some obstacle, calculate it by the 
same process as a c in Fig 1. Then, first directing the instru 
ment horizontally,# as o s. measure the angle of depression. 
s o a, say 12°; also the angle son, say 30°. These two angles 
added together, give the angle a o n, 42°. Now, in the small 
triangle o s a we have the angle os a equal to 90°, because a « 
is vert, aud os hor; and since the three angles of any triangle 
are equal to 180°, if we subtract the angles os a (90°). and so a 
(12°) from 180°, the rem (78°) will be the angle o a s or o a n. 
Therefore, in the triangle o n a, we have one side o a\ and two 
angles a o n, and o a n, to calculate the side a a by Case 1. 


Angles and distson sloping- ground must be measured hor¬ 
izontally. The graduated hor 



circle of the instrument evidently meas¬ 
ures the angle between two objects hori¬ 
zontally, no matter how much higher one 
of them may be than the pther; one per¬ 
haps requiring the telescope of the instru¬ 
ment to be directed upward toward it; 
and the other downward. If, therefore, 
the sides of triangles lying upon stoning 
ground, are not also measd hnr. there can 
be no accordance between thp two. Thus 


s 


















































114 


PLANE TRIGONOMETRY 


Rkm. If, as in Fig 3, it should be necessary to ascertain the vert height an from a point o, entirely 
above it, then both the angles measd at o, namely, s o n, and s o a. will be angles of depression, or 
below the hor line o s assumed to measure them from. In this case we have the side o a as before; 

the angle no a = soa — son; and the angle o a n = 180° — s a (90°), and $ o a;) to calculate 
an by Case 1. 




Or If, as In Fig4, the observations are to be taken from a point o, entirely below the object 
both the augies s o a, s o n. will be angles of elevation, or above the assumed hor line os. 
have in the triangle o n a, the given side o a as 
before; the angle a o n — s o n — soo; and the 

angle o n a — 180°— ^os » (90°), and no s, ^ to 
calculate an by Case 1. 

If the object on, as in Fig 5, instead of being 
vert, is inclined; and instead of its vert height, 
we wish to tiud its length a n, we must first as¬ 
certain its angle y t i of inclination to the hori¬ 
zon ; to which angle each of the angles o*n will 
be equal. To find this angle yti, suspend a plumb- 
line i y, of any convenient kuown length, from the 
object a n : and measure also y t horizontally. 

Then say as 

y t : i y : : 1 : nat tang of angle yti. 

From the table of nat tangs take out the angle 
yti found opposite this nat tang; aud use it for 
the augles o s n or os a; instead of the 90° of Figs 
3 and 4. Also when the object inclines, the side 
a o of the triangle must be measd in line, or in 
rangewith the inclination. If the object, as the 
rock a n, Fig 6, is curved or irregular, a pole a s 
may be planted sloping in the direction a n ; aud 


a n, then 
Here we 



A'WVO 


Fig. 5 . 


in the triangle ah c. upon sloping ground, the instrument at o. measures the hor angle ion; and not 
the angle b a c. Therefore, the side which corresponds with this hor augle t o n. is the hor dist t n ; 
and not the sloping dist b c. In other words, when sides and angles are on sloping ground, we do 
not seek their actual measures; but their hor ones. This remark applies to all surveying for farms, 
railroads, triangnlations of countries. Ac, Ac; and the want of a strict attention to it, is one cause 
of the small errors, almost unavoidable, (and fortunately, of but trilling consequence in practice), 
which occur in all ordinary field operations. See p 176. 

WheiR a sextant is used, angles between objects at diff altitudes, as p and 

q, may be measd hor, by first planting two vert rods 
o and s. in range with the objects; aud then taking 
the hor angle o n s, subtended by the rods. 

Angles may be measil without 
any inst, thus: Measure 100 ft toward 

each object, and drive stakes; measure tlie dist across 
from one stake to the other. Half this dist will be 
the sine of half the angle to a rad of 100: and if we move 
the decimal point * wo places to the left, we get the nat 
fine of this one half of the angle to a rad of 1, as in the 
tables. Thus, suppose the dist to be 80.64 feet; then 
40.32 is the sine of half the angle; and .4032 will be 
the nat sine, opposite to which in the table of nat 
sines we find the angle 23° 47’ ; which mult by 2 gives 
47° 34’, the reqd angle. If obstacles prevent measuring toward the objects, we may measure directly 
from them ; because, when two lines intersect, the opposite angles are equal. A rough measurement 
may be made by sticking three pins vert, and a few ins apart, into a small piece of board, nailed hor 
to the top of a post. The pins would occupy the positions nos, of the last figure. Pencil-lines may 
then be drawn, connecting the pin-holes ; and the angle be measd with a protractor. By nailing a 
piece of board vert to a tree, and then drawing upon it a short hor line, by means of a pocket car¬ 
penters’ spirit-level, vert angles of elevation and depression mav he taken roughly in the same way. 
In this way the writer has at times availed himself of the outer door of a house, by opening it until it 
pointed toward some mountain-peak, the dist of which he knew approximately ; but of the height of 



<z 






























PLANE TRIGONOMETRY, 


115 






its angle yti of inclination with the horizon found as before; 
in which case the dist a n is calculated. Or if the vert height c n 
is sought, the poiut c uiay first be found by sighting upward 
along a plumb-line held above the head. 


Ex. 3. To find the approximate height, 
s x, of a mountain. 


Of which, perhaps, only the very summit, x, is visible above 
interposing forests, or other obstacles; but the dist, mi, of which 
is known. In this case, first direct the instrument hor, as m h; 

and then measure the angle i m x. 
% Then in the triangle i m x we have 

one side mi; the tneasd augle imx, 
and the angle mix (90°), to fiud i x 
by Case 1. But to this i x we must 
add i o, equal to the height ym of the 
instrument above the ground; and 
also o s. Now, o s is apparently due 
entirely to the curvature of the earth, 
which is equal to very nearly 8 ins, or 
.867 ft in one mile; and increases as 
the squares of the dists; being 4 
times 8 ins in 2 miles; 9 times 8 ins 
in 3 miles, &c. But this is somewhat diminished by the refraction of the atmosphere ; which varies 
with temperature, moisture, &c; but always tends to make the object x appear higher than it 

actually is. At an average, this deceptive elevation amounts to about—th part of the curvature of 

< 1 

the earth; and like the latter.it varies with the squares of the dists. Consequently if we subtract — 


part from 8 ins, or .667 ft, we have at once the combined effect of curvature and refraction for one 
mile, equal to 6.857 ins, or .5714 ft; and for other dists, as shown in the following table, by the use 
of which we avoid the necessity of making separate allowances for curvature and refraction. 

Table of allowances to be added for curvature of the earth; 
and for refraction; combined. 


Dist. 
in yards. 

Allow. 

feet. 

Dist. 
in miles. 

Allow. 

feet. 

Dist. 
in miles. 

Allow. 

feet. 

Dist. 
in miles. 

Allow. 

feet. 

100 

.002 


.036 

6 

20.6 

20 

229 

150 

.004 

X 

.143 

7 

28.0 

22 

277 

200 

.007 

% 

.321 

8 

36.6 

25 

357 

300 

.017 

l 

.572 

9 

46.3 

30 

514 

400 

.030 

i M 

.893 

10 

57.2 

35 

700 

500 

.046 

VA 

1.29 

11 

69.2 

40 

915 

600 

.066 

m 

1.75 

12 

82.3 

45 

1158 

700 

.090 

2 

2.29 

13 

96.6 

50 

1429 

800 

.118 

2 A 

3.57 

14 

112 

55 

1729 

900 

.149 

3 

5.14 

15 

129 

60 

2058 

1000 

.185 

3'A 

7.00 

16 

146 

70 

2801 

1200 

.266 

4 

9.15 

17 

165 

80 

3659 

1500 

.415 

4 ^ 

11.6 

18 

185 

90 

4631 

2000 

.738 

5 

14.3 

19 

206 

100 

5717 


lienee, if a person whose eye is 5.14 ft, or 112 ft above the sea, sees an object just at the sea’s 
horizon, that object will be about 3 miles, or 14 miles distant from him. 

A horizontal line is not a level one, for a straight line cannot be a 

level one. The curve of the earth, as exemplified in an expanse of quiet water, is level. In Fig 7, 
if we suppose the curved line t y s g to represent the surface of the sea, then the points t y s and g are 
on a level with eacii other. They need not be equidistant from the center of the earth, for the sea at 
the poles is about 13 miles nearer it than at the equator; yet its surface is everywhere on a level. 

Up, and down, refer to sea level. Eevel means parallel to the curvature 
of the sea; and horizontal means tangential to a level. 


Ex. 4. If the inaccessible vert height c d, Fig 8, 

Is so situated that we cannot reach it at all, then place the instrument for measuring angles, at any 
convenient spot n ; and in range between n and d, plant two staffs, whose tops o and i shall range 
precisely, with n, though they need not be on the same level or hor line with it. Measure n o: also 
from n measure the augles o n d and o n c. Then move the instrument to the precise spot previously 


which he had no idea. For allowance for curvature and refraction see above Table. 

A triangle whose sides are as 3, 4, and 5, is right angled; and one 

whose sides are as 7; 7; and 9. 9; contains 1 right angle; and 2 angles of 45° each. As it is fre¬ 
quently necessary to lay down angles of 45° and 90° on the ground, these proportions may be used for 
the purpose, by shaping a portion of a tape-line or chain into such a triangle, and driving a stake at 
each augle. See p. 58. 








































PLANE TRIGONOMETRY, 


116 


w>_. 




Fig. 8. 


Fig:. 9. 



occupied by the top o of the staff; aud from o measure the angles tod and doc. This being done, sub 

tract the angle toe from 
180°; the rent will be the 
angle c o n. Consequent¬ 
ly in the triaugle noc, we 
have one side « o, and two 
angles, c n o and c o n, to 
find by Case 1 the side o c. 

Again, take the angle iod 
from 180°; the remainder 
will be the angle nod, so 
that in the triangle duo 
we have one side n o, and 
the two angles duo and 
nod, to find by Case 1 
the side od. Finally, in 
the triangle cod, we have 
two sides c o and o d, and 
their included angle cod, 
to find c d, the reqd vert 
height. 

Rkm. If cd were in a valley, or on a hill, and the observations reqd to be made from either higher 
or lower grouud, the operation would be precisely the same. 

Ex. 5. See Ex 10. 

To lint! tlie dist ao, Fig 1 9, between two entirely inaccessible 

objects. 

Measure a side n m; at n measure the angles anm and onm; also at m measure the angles o m n, and 
a m n. This being done, we have in the triaugle anm, one side n m, Fig f>, and the angles a n m, and 
nm a; hence, by Case 1, we can calculate the side an. 

Again, in the triaugle o m n we have one side n m, and 
the two angles o m n, and mno; hence, by Case 1, we can 
calculate the side n o. This being done, we have in the 
triangle ano, two sides an, aud no; and their included 
angle ano; hence, bv Case 3, we can calculate the side 
ao, which is the reqd dist. It is plain that in this manner 
we may obtain also the position or direction of the inacces¬ 
sible line a o; for we can calculate the angle nao ; and can 
therefrom deduce that of ao; aud thus be enabled to run 
a line parallel to it, if required. By drawing n m on pa¬ 
per by a scale, nnd laying down the four nieasd angles, 
the dist a o may be measd upon the drawing by the same scale. 

If the position of the inaccessible dist c n. Fig 10. be such that 
we can place a stake p in line with it,we may proceed thus : Place 
the instrument at any suitable point s, and take the angles p s c 
atidcsn. Also find the angle cps, and measure the distps. Then 
in the triangle p s c find s c by Case 1; again, the exterior angle 
n c s, being equal to the two interior and opposite angles cps, 
and p s c, we have in the triangle c s n, one side and two angles 
to find c n by Case 1. 

Ex. (i. To find a dint ah, Fig: 11. of which 
the ends only are accessible. 

From a and h, measure any two lines a c. b c, meeting at c; also 
measure the angle a c b. Then in the triangle a b c we have two 
sides, and the included angle, to find the third side a 6 by Case 3. 

Ex. 7. To find the vert height o in, of a 
hill, above a given point i. 

Place the instrument at i; measure a m. Directing 
the instrument hor, as an, take the angle nam. Then, 
since a n m is 90° Fig 12, we have one side a m, and 
two angles, nam and an m, to find n m by Case 1. 

Add no, equal to at, the height of the instrument. 

Also, if the hill is a long one, add for curvature of the 
earth, and for refraction, as explained in Example 3, 

Fig 7. Or the instrument may be placed at the top of 
the hill; and an angle of depression measured ; instead 
of the angle of elevation n a m. 

Rem. 1. It is plain, that if the height o m be previously 
known, and we wish to ascertain the dist from its sum¬ 
mit m to any point i, the same measurement as before, 
of the angle w a in, will enable us to calculate a. m by 
Case 1. So in F.x. 2, if the height na be known, the’ angles measd in that example, will enable us 
to compute the dist a o; so also in Figs 3, 4, 5, and 7 ; in all of which the process is so plain as to 
require no further explanation. 

Rkm. 2. The height of a vert object by means of its shadow. Plant one end of 
a straight stick vert in the ground ; and measure its shadow : also measure the leugth of the shadow 
#f the object. Then, as the length of the shadow of the stick is to the leugth of the stick above 


Fig - . 10. 



m 


















PLANE TRIGONOMETRY. 


ground, so is the length of the shadow of the object, to its height. If the object is inclined, the stick 
must be equally inclined. 


Fi ^. 12 1 4 


Rem. 3. Or the height of a vert object mn, 

. Fig 12*4, whose distance r m is known, may be found by 
its reflection in a vessel of water, or in a piece of 

' looking glass placed perfectly horizontal at r; for as r ais to the height 
a i of the eye above the reflector r, so is r m to — 
the height m n of the object above r. ft' 

Item. 4. Or let o c, Fig 12)4, be 
a planted pole, or a rod held vert by an assistant. Then 

stand at a proper dist back from it, and keeping the eyes steady, let marks be 
made at o and c. where the lines of sight i n and t m strike the rod. Then as 
t c is to c o, so is i m to m n. 





d 

Fig - . 13. 


The following examples may be regarded as substitutes for strict trigonome¬ 
try : and will at times be useful, in case a table of sines, &c, is not at hand for 
making trigonometrical calculations. 


Ex. 8. To find the dist ab, of which one end only 

is accessible. 

Drive a stake at any convenient point a; from a lay off any angle h a c. In 
the line a c, at any convenient point c, drive a stake; and from c lav off an angle 
acd, equal to the angle 6 a c. In the line c d, at any convenient point, as d, 
drive a stake. Then, standing at d, and looking at b, place a stake o in range 
with d h ; and at the same time in the line o c. Measure ao, o c, and c a!; then, 
from the principle of similar triangles, as 

o c : c d : : a o : a b. 



Fig. 14. 


Or thus: 

Fig 14, n h being the dist, place a stake at n ; and lay off the angle h n m 90°. 
At any convenient dist n m, place a stake m. Make the angle limy — 90°; and 
place a stake at y, in range with h n. Measure n y and n m ; then, from the 
principle of similar triaugles, as 

» y : n m : : n rn : » h. 

Or thus. Fig 14. Lay off the angle hnm = 90°, placing a stake 

m, at any convenient dist n m. Measure n m. Also measure the augle n m h. 
Find nat tang of n m A by Table Mult this nat tang by n m. The prod 

will be n h. 

Or thus. Lay off angle h n m = 90°. From m measure the 

angle n m h, and lay off angle n m y equal to it, placing a stake at y in range 
with h n. Then is »i y= n It. 



Or thus, without measuring; 

any angle; 

t u being the dist. Make u v of any convenient 
length, in range with t u. Measure any v o ; and 
o x equal to It, in range. Measure u o ; and o y 
equal to it in range. Place a stake z in range with 
both x y, and t o. Then will y z be both equal to 
t u, and parallel to it. 



Or thns, without measuring any angle. 

Drive two stakes t and u, in range with the object s. From Mav off any 
onvenient dist f x, in any direction. From u lay off u w parallel to It, 
.lacing w in range with z s. Make u v equal to t x. Measure w v, v x, and 
; f. Then, as 


tv v : v x : s xtits. 

Or thus. At a lav off angle nac = 5° 43'. Lay 
off oc at right angles to ao. Measure oc. Then 
no = 10 oc, too long only 1 part, in 935.6, or 5.643 feet 
in a mile, or .1069 foot (full H inches) in 100 feet. 






































118 


PLANE TRIGONOMETRY 



Fig. 17. 


Ex. 9. To find the dist a b, of wliieli the 
cuds only are accessible. 

From o lay off the angle ft a c; and from 6, the angle ah d. each 
90°. Make a c and h d equal to each other. Then. cd~a h. Or 
a h may be considered as the dist across the river in Figs 15, 13. or 
14: and be ascertained in the same way. Or measure anv dist. Fig 
17, no; and make on in line and equal to it. Also measure ho; 
and make om in line and equal to it. Then will ran be both paral¬ 
lel to a 6, and equal to it. 


Ex. lO. See Ex. 4. To find the entirely 
inaccessible dist y z, and also 
its direction. 

At any two convenient points a and ft, from each of which 
y and z can be seen, drive stakes. Theu we have the four 
corners of a four-sided figure, in which are given the directions 
of three of its sides, and of its two dings. These data enable us 
to lay out on the ground, the small four-sided fig acoi. exactly 
similar to the large one. Thus, in the line a ft place a stake 
c: and make co parallel to bz; o being at the same time in 
range of the diag a z. Also, from c make c i parallel to 6 y; 
i being at the same time in rauge of a y. Then tvill i o be in 
the same direction as y z, or parallel to it. Measure a c, aft, 
and to; then evidently, from the principle of similar figures, as 

a c : a b : i i o : y z. 

If y z were a visible line such as a fence or road, we could 
from a divide it into any required portions. Thus, if we wish 
to place a stake halfway between y and z, first place one half¬ 
way between i and o; then standing at a, by means of signals, 
place a person in range on y z. Or, to find along a 6, a point t 
perp to y z at y, first make ois- 90°; and measure a s. Then, 
as , . 

o i t a s : : y z : a t. 



Ex. 11. To find tiie position of a point, n, Fig 19, 

By means of two angles a n b and b n c. taken from it to the three objects a b c, whose positions 

and dists apart are known. 

The use of this problem is more frequent in marine than in land surveying. It is chiefly employed 
for determining the position n of a 
boat from which soundings are being 
taken along a coast. As the boat 
moves from point to point to take 
fresh soundings, it becomes necessary 
to make a fresh observation at each 
point, in order to define its position 
on the chart. An observation consists 
in the measurement by a sextant of 
the two angles an 6, 6 n c, to the sig¬ 
nals a ft c. previously arranged on the 
shore. When practicable, this method 
should be rejected; and the observa¬ 
tions taken to the boat at the same 
instant, by two observers on shore, at 
two of the stations. The boat to show 
a signal at the proper moment. The 
most expeditious mode of fixing the 
point n upon the map, is to draw three 
lines, forming the two angles, and ex¬ 
tended indefinitely, on a piece of trans¬ 
parent paper. Place the paper upon the map, and move it about until the three lines passthrough 
the three stations ; then prick through the point n wherever it happens to come. 

Instead of the transparent paper, an instrument called a station pointer may be used when there 
are many points to be fixed. 

But the position of the point n cnn be found more correctly by describing two circles, as in Fig 19, 
each of which shall pass through n and two of the station points. The questiou is to find the centers 
o and x of two such circles. This is very simple. We know that the angle a o ft at the center of a circle is 
twice as great as any angle a n ft at the circumf of tbe same circle, when both are subtended by the 
same chord a ft. Consequently, if the angle a n ft, observed from the boat, is say, 50°, the angle aob 
must be 100°. And, since the three angles of every plane triangle are equal to 180°, the two angles 
o a ft and o 6 a are together equal to 180° — 100° — 80°. And, since the two sides a o and 6 o are 
equal (being radii of the same circle), therefore, the angles o a 6 and o ft a are equal; and each equal to 
80° 

— - = 40°. Consequently, on the map we have only to lay down at a and ft, two angles of 40°; the 

point o of intersection will be the center of the circle ah n. Proceed in the same way with the angle 
ft n c, to find the center s. Then the intersection of the two eircles at n will be the point sought. 

















PARALLELOGRAMS 


119 


PARALLELOGRAMS. 


Square. 


Rectangle. 



Rhombus. Rhomboid. 



A parallelogram is any figure of four straight sides, the opposite ones of which 
are parallel. There are but four, as in the above figs. The rhombus, like the rhom- 
bohedron, Fig 3, p 155, is sometimes called “ rhomb.” In the square and rhombus 
all the four sides are equal; in the rectangle and rhomboid only the opposite ones 
are equal. In any parallelogram the four angles amount to four right angles, or 
360°; and any two diagonally opposite angles are equal to each other; hence, having 
one angle given, the other three can readily be found. In a square, or a rhombus, a 
diag divides each of two angles into two equal parts ; but in the two other parallel¬ 
ograms it does not. 


To find tlie area of any parallelogram. 

Multiply any side, as S, by the perp height, or dist p to the opposite side. Or, multiply together 
two sides and nat sine.of their included angle. 

The diag a b of any square is equal to one side mult by 1.41421; and a side is equal to 
diagonal 

1 4L42i ~ ’ or * t0 mult ^ .707107. 

The side of a square equal In area to a given circle, is equal to diam X .886227. 

The side of the greatest square, ihat can be inscribed, in 
a given circle, is equal to diam X .707107. 

The side of a square mult by 1.-51967 gives the side of an equi¬ 
lateral triangle of the same area. Ail parallelograms as A 
and C, which have equal bases, a c, aud equal perp heights n 
c, have also equal areas ; aud the area of each is twice that of a tri¬ 
angle having the same base, and perp height. The area of a 
square inscribed in a circle is equal to twice the squaie of the 
rad. 

In every parallelogram, the 4 squares drawn on its sides have a united area equal to that of 
the two squares drawn on its 2 diags. If a larger square he drawn on the diag a b of a smaller 
square, its area will be twice that of said smaller square. Either diag of any parallelogram 
divides it into two equal triangles, and the 2 diags div it into 4 triangles of equal areas. The two 
diags of any parallelogram divide each other into two equal parts. Any line drawn through 
the center of a diag divides the parallelogram into two equal parts. 

Remark 1. —The area of any fig whatever as B that is enclosed by four straight 
lines, may be found thus : Mult together the two diags a m, n b; aud the nat sine of the least angle 
a o b ; or n o m, formed by their intersection. Div the product by 2. This is useful in land surveying, 
when obstacles, as is often the case, make it difficult to measure the sides of the fig or field; while it 
may be easy to measure the diags; and after finding their point of intersection o, to measure the re¬ 
quired angle. But if the fig Is to be drawn, the parts o a, oh, on, o m of the diags must also 
be measd. 

Rem. 3.— The sides of a parallelogram, triangle, and many other figs may be 
found, when only the area and angles are given, thus ; Assume some particular one of its 
sides to be of the length 1 ; and calculate what its area would be if that were the case. Then as the 
sq rt of the area thus found is to this side 1, so is the sq rt of the actual given area, to the corre¬ 
sponding actual side of the fig. 




On a given line was,to draw a square, 

tv as ti tn,. 

From w and x, with rad w x, describe the arcs xry and lore. 
From their intersection r, and with rad equal to y 2 of wx, describe 
8 $ s. From w and x draw w n and xm tangential to s s «, and 
endiug at the other arcs; join n m. 
















120 


TRAPEZOIDS AND TRAPEZIUMS 


TRAPEZOIDS. 

m t n 

\U. 

as c a 

A trapezoid a c n to, is any figure with four straight sides, only two of which, as ae and n to, arc 
parallel. 

To fiml the area of any trapezoid. 

Add together the two parallel sides, ac and wi n; mult the sum by the perp dist * / between 
them ; div the prod by 2. See the following rules for trapeziums, which are all equally applicable 
to trapezoids; also see Kemarks after Parallelograms. 

TRAPEZIUMS. 



A trapezium a b c o, is any fig with four straight sides, of which no two are parallel. 

To find the area of any trapezium, having- given the diag 
bo, or a c, between either pair of opposite angles; and also 
the two perps, n, n, from the other two angles. 

Add together these two perps ; mult the sum by the diag; div the prod by 2. 

Having the four sides; and either pair of opposite angles, 

as a be, a o c; or b a o, and b e o. 

Consider the trapezium as divided into two triangles, in each of which are given two sides and the 
Included angle. Find the area of each of these triangles as directed under the preceding head “ Tri¬ 
angles," and add them together. 

Having the four angles, and either pair of opposite sides. 

Begin with one of the sides, and the two angles at its ends. If the sum of these two angles exceeds 
180°, subtract each of them from 180°. and make use of the reins instead of the angles themselves. 
Then consider this side and its two adjacent angles (or the two rems, as the case may be) as those 
of a triangle; and find its area as directed for that case under the preceding head “ Triangle.” Do 
the same with the other given side, and its two adjacent angles, (or their rems, as the case may be.) 
Subtract the least of the areas thus found, from the greatest; the rem will be the reqd area. 

Having throe sides ; and the two included angles. 

Mult together the middle side, and oue of the adjacent sides; mult the prod by the nat siue of their 
included angle; call the result u. Do the same with the middle side and its other adjacent side, 
and the nat siue of the other included angle; call the result b. Add the two angles together; find 
the dill between their sum and 180°, whether greater or less; find the nat sine of this diff; mult 
together the two given sides which are opposite oue another; mult the prod by the nat sine just foand; 
call the result c. Add together the results o and b ; then, if the sum of the two given angles is less 
than 180°, subtract c from the sum of a and 6; Aa//the rem will be the area of the trapezium. But 
if the sum of the two given angles be greater than 180°, add together the three results a, b, and c: 
half their sum will be the area. 

Having the two diagonals, and citlicr angle formed by their 

intersection. 

See Remarks after Parallelograms, p 119. 

In railroad measurements 

Of excavation and embaukment, the trapezium 
l m n o frequently occurs; as well as the two 5-sided 
figures l in n o t aud Imno t\ in all of which to n 
represents the roadway ; rs, r c, and r t the center- 
depths or heights; l u aud o v the side-depths or 
heights, as given by the level; I to and n o the side- 
slopes. 

The same general rule for area applies to all three 
of these figs; namely, mult the extreme hor width 
u v by half the center depth rs, rc.or r t, as the 
case may be. Also mult one fourth of the width of 
roadway to n, by the sum of the two side-depths l u 
and o v. Add the two prods together; the sum is the 
reqd area. This rule applies whether the two side- 
slopes to l and n o have the same angle of inclination or not. In railroad work, etc., the mid¬ 
way hor width, center depth, and side depths (see page 161) of a prismoid are respectively ~ The 
half sums of the corresponding end ones, and thus can be fouud without actual measurement. 





















POLYGONS. 


121 


To draw a hexagon, each side of whieh shall 
be equal to a given line, a b. 

J _ Prom a and b, with rad a b, describe the two arcs; from their intersection, 
| *> with the same rad, describe a circle; around the circumf of which, step off 
I the same rad. 

Side of a hexagon = «nX .57735. 



To draw an octagon, with each side 
equal to a given line, c e. 

From c and e draw two perps, cp, ep. Also prolong c e toward 
! / and g ; and from c and e, with rad equal c e, draw the two 
quadrants ; and find their centers h h ; join c h, and e h ; draw 
h s and h t parallel to cp; and make each of them equal to c e; 
make c o, and e o, each equal to h h ; join o o, o s, aud o t. 

Side of an octagon = n « X .41421354. 



To draw an octagon in a given square. 

From each corner of the square, and with a rad equal to half its diag, 
[ describe the four arcs; and join the points at which they cut the sides of the 
square. 

To draw any regular polygon, with each side 

equal to m n. 

Div 3G0 degrees by the number of sides; take the quot from 180°; div the 
rem by 2. This will give the angle cm n, ore nm. At m and n lay down these 
angles by a protractor: the sides of these angles will meet at a point, c, from 
which describe the circle mny, and around its circumf step off dists equal to 
m 71. 

In any circle, m n y, to draw any regular 

polygon. 

Div 360° by the number of sides ; the quot will be the angle men, at the center. 
Lay off this angle by a protractor; and its chord m n will be one side; which 
step off around the circumf. 




To reduce any polygon, as a b c d e f a, to a triangle of the 

same area. 




If we produce the side fa toward to; and draw b g parallel to a c, and join g c, we get equal tri¬ 
angles a c ft, aud a c g, both on the same base a c; and both of the same perp height, inasmuch as 
they are between the two parallels a c and g b. But the part act forms a portion of both these tri¬ 
angles, or in other words, is common to both. Therefore, if it be taken away from both triangles, 
the remaining parts, i c b of one of them, aud i g a of the other, are also equal. Therefore, if the 
part i c b be left off from the polygon, and the part i g a be taken into it, the polygon g fed c i g will 
have the same area as a/e d c b a; but it will have but five sides, while the other has six. Again, 
if e s be drawn parallel to d /, and d s joined, we have upon the same base e s, aud betweeu the same 
parallels e s and df, the two equal triangles e s d, and e sf. with the part e o s common to both ; and 
consequently the remaining part e o d of oue, and o sf of the other, are equal. Therefore, if os f be 
left off from the polygon, aud eod be taken into it, the new polygon g a d eg, Fig 2, will have the same 
area as g / e d c g; but it has but four sides, while the other has five. Finally, if g $, Fig 2, be 
extended toward n; and d n drawn parallel toes; aud c n joined, we have on the same base c s, and 
betweeu the same parallels c s aud d n, the two equal triangles c s », aud c s d, with the part c s t 
common to both. Therefore, if we leave out cd t, and take in s t n, we have the triangle g n c equal 
to the polygon g s d c g, Fig 2; or to a/e d c b a. Fig 1. 

This simple method is applicable to polygons of any number of sides. 
























122 


POLYGONS. 


fo reduce a large fig, a b c d e fg, to a smaller 
similar one. 

From any interior point o, which had better be near the center, draw lines 
to all the angles a, b, c, &c. Join these lines by others parallel to the sides 
of the fig. If it should be reqd to enlarge a small fig, draw, from any poiut 
o within it, lines extending beyond its angles ; and join these lines by others 
parallel to the sides of the small fig. 



To reduce a map to one on a smaller scale. 

The best method is by dividing the large map into squares by faint lines, with a very 
pencil, and then drawing the reduced map upon a sheet of 
smaller squares. A pair of proportional dividers will assist 
much in fixing points intermediate of the sides of the squares. 

If the large map would be injured by drawing, and rubbing 
out the squares, threads may be stretched across it to form the 
squares. 

Maps, plans, and drawings of all kinds, are now copied, 
reduced, enlarged, and multiplied, cheaply and expeditiously, by 
photography. Most of the newer illustrations in this work are 
from electrotypes made by the “wax process” of Struthers, 

Servoss & Co., 34 New Chambers St, New York, and American 
Bank Note Co, 78 Trinity Place, New York. 

In a rectangular fig, g h s d, 

Representing an open panel, to find the points o o o o in its 
sides; and at equal dists from the angles g. and s; for inserting 
a diag piece o o o o, of a given width l l, measured at right 
angles to its length. From g and s as centers, describe several 
concentric arcs, as in the Fig. Draw upon transparent paper, 
two parallel lines a a, c c, at a distance apart equal to 11: and 
placing these lines on top of the panel, move them about until it 
is shown bv the arcs that the four dists g o. go, s o, s o. are 
equal. Instead of the transparent paper, a strip of common 
paper, of the width l1 may be used. 

Rem. Many problems which would otherwise be very difficult, 
may be thus solved with an accuracy sufficient for practical 

purposes, by means of transparent paper. 


soft; 



c 



To find the area of any irregular poly* 

goti, a n b c in. 

Div it into triangles, as a n b, a m c, and a b c; in each ol 
which find the perp dist o, betweeu its base a b, a c, or 6 c; ant 
the opposite angle n, m, or a; mult each base by its perp dist 
add all the prods together; div by 2. 

To fiml approx the area of a long ir* 
regular fig, as abed. Between its ends ab, c d 



space off equal dists, (the shorter they are the more accurate will be the result,) through whic 
draw the intermediate parallel lines 1, 2, 3, &c, across the breadth of the fig. Measure the length 
of these intermediate lines ; add them together: to the sum add half the sum of the two end breadth 
a b and c d. Mult the entire sum by one of the equal spaces between the parallel lines. The pro 
will be the area. This rule answers as well if either one or both the ends terminate in points, as at» 
and n. In the last of these cases, both a b and c d will be included in the intermediate lines; an 
half the two end breadths will be 0, or nothing. 

To find file area of a fig whose outline is extremely 

irregular. 

Draw lines around it which shall enclose withi 
them (as nearly as can be judged by eye) as muc 
space not belonging to the fig, as they exclude spac 
belonging to it. The area of the simplified fig thu 
formed,beiugin this manner rendered equal to tha 
of the complicated one,may be calculated by dividin 
it into triangles, &c. By using a piece of fine thread 
the proper position for the newboundarv lines mai 
be found, before drawing them in. Small irregula 
areas may be found from a drawing, by laying upon 
it a piece of transparent paper carefully ruled iut 
small squares, each of a given area, say 10. 20. o 
100 sq ft each; and by first counting the whol 
squares, and then addiug the fractions of squares 



























CIRCLES. 


123 


C I RCL ES. 

A circle Is the area Included within a carved line of Much a character that every point in it is 
erjuitlljr distant from a certain point within it, called its center. The curved line itself is called the 
circumference, or periphery of the circle; or very commonly it is called the circle. 

To find the circumference. 

Mult dlam by 3.1416, which gives too much by only .148 of an inch in a mile. Or, as 113 is to 355 
to Is diarn to circumf; too great 1 inch in 186 miles. Or, mult diam by 3y; too great by about 1 
part in '4485. Or, rault area by 12.566, and take sq root of prod. Or, use tables pp. 125 tic. The 
Greek letter n, also p, is used by writers to denote this 3.1416; and p 2 = 9.86960. 

To find the diam. 

Div the oircumf by 3.1416 ; or, as 355 is to 113, so is circumf to diam ; or, mult the circumf. by 7: 
sud div the prod by 22, which gives the diam too small hy only about one part in 2485; or, mult the 
srea by 1.2T32; and take the sq rt of the prod ; or use tables of circles, pp 125, he. 

The diam is to the circumf more exactly as 1 to 3.14159265. 

To find the urea of a circle. 

Square the diam; mult this square by .7854; or more accurately by .78539816; or square the cir¬ 
cumf; mult this square by .07958: or more accurately by .07957747 : or mult half the diam by half the 
'circumf; or refer to the following table of areas of circles. Also area — sq of rad X 3.1416. 

The area of a circle is to the area of any circumscribed straight-sided fig, as the circumf of the 
circle is to the circumf or periphery of the fig. The area of a square inscribed in a circle, is equal to 
Uice the square of the rad. Of a circle in a square, — square X .7854. 

It is convenient to remember, in rounding off a square corner a b c, by a quarter of 
a circle, that the shaded area a b c is equal to about 1 part (correctly .2116j of the 
whole square abed. 5 



For tables of circumferences and areas of cir¬ 
cles, see pages 125 to 140. 


To find the diarn of a circle equal in area to a g-iven square. 

Mult one side of the square by 1.12838. 

To find the rad of a circle to circumscribe a given square. 

Mult one side hy .7071 ; or take the diag. 

To find the side of a square equal In area to a gi%’en circle. 

Mult the diam by .88623. 

To find the side of the greatest square in a given circle. 

Mult diam by .7071. The area of the greatest square that can be inscribed in a circle is equal t* 
twice the -.quareof the rad. The diam X by 1.3468 gives the side of an equilateral triangle of equal area. 




* 



t 

To find the center r, of a given circle. 

Draw any chord a b ; and from the middle of it o, draw at right angles to 
it, a diam d g ; find the center c of this diam. 


r 



To describe a circle through any three 
points, ahc, not in a straight line. 

Join the points by the lines ah, be; from the centers of these lines draw 
the dotted perps meeting, as at o, which will be the center of the circle. 
Or from b, with any convenient rad. draw the arc m nj and from a and c, 
with the same ra/1," draw arcs y and z\ then two lines drawn through the 
intersections of these arcs, will meet at the center o. 

To describe a circle to touch the three 
angles of a triangle is plainly the same as this. 

To inscribe a ei rele i n a triangle draw two linos 

bisecting any two of the augles. Where these lines meet is the center of 
the circle. 









124 


CIRCLES. 





any 


To draw a tangent, i e i, to a cirele, from 
given point, e, in its cirenmf. 

Through the center n, anii the given point e, draw n o; make e o equal to 
en; from n and o, with any rad greater than half of o n, describe the two 
pairs of arc i i; join their intersections t i. 

Here, and in the following three figs, the tangents are ordinary or geo¬ 
metrical ones; and may end where we please. But the trigonometrical 
tangent of a given angle, must end in a secant, as in the top fig of p 59. 

Or from e lay off two equal distances e c, el; and draw i i 

parallel to c t. 


r*( 


r 

Hit 


To draw a tang, a s b, to a circle, from a point, 
a, which is outside of the circle. 

Draw a c, and on it describe a semicircle; through the intersection, s, draw 


ash. Here c is the center of the circle. 


04/ " 


Ax 


To draw a tang, g h, from a circular are, g a c, 

Of which n a is the rise. With rad g a, describe an arc, s a o. Make t a 
equal to s a. Through t draw g h. 


1 = 


n s 


To draw a tang to two circles. 


First draw the line m n, just touching the two 
circles; this gives the direction of the tang. Then 
from the centers of the circles draw the radii, o o, perp 
to m n. The points t t are the tang points. If the 
tang is in the position of the dotted line, s y, the ope¬ 
ration is the same. 

Rem. This empirical method is at 

least as accurate as the scientific ones, especially if 
a correct triangular ruler is used for the radii. 



V 





If any two chords, asab.o c, cross each other. 

then as 0 n : n b :: a n : n c. Hence, nb X an = onXnc. That 
is, the product of the two parts of one of the lines, is = the pro¬ 
duct of the two parts of the other line. 


• ! 












CIRCLES 


125 


TABLE 1 OF CIRCLES. 

Diameters in units and eighths, Ac. 

Circumferences or areas intermediate of those in this table, may be found by sim¬ 
ple arithmetical proportion. No errors. 


Diam 

Circumf. 

Area. 

Diam. 

Circumf. 

Area. 

Diam 

Circumf. 

Area. 

Diam. 

Circumf. 

Area. 

1-64 

.0191)87 

.00019 

3. A 

10.9956 

9.6211 

10X 

31.8086 

80.516 

1914 

60.4757 

291.04 

1-32 

.038175 

.00077 

9-16 

11.1919 

9.9678 

A 

32.2013 

82.516 

% 

60.8684 

294.83 

3-64 

.147262 

.00173 

% 

11.3883 

10.321 

% 

32.5940 

84.541 

bs 

61.2611 

298.65 

1-16 

.196350 

.00307 

11-16 

11.5846 

10.680 


32.9867 

86.590 

% 

61.65^8 

302.49 

3-32 

.291524 

.00690 

A 

11.7810 

11.045 

A 

33.3794 

88.664 

A 

62.0435 

306.35 

A 

.392699 

.01227 

13-16 

11.9773 

11.416 

A 

33.7721 

90.763 

% 

62.4392 

310.24 

5-32 

.490874 

.01917 

% 

12.1737 

11.793 

y» 

34,1648 

92.886 

20. 

62.8319 

314.16 

3-16 

589049 

.02761 

15-16 

12.3700 

12.177 

ii. 

34.5575 

95.033 

A 

63.2246 

318.10 

7-32 

.687223 

.03758 

4. 

12.5664 

12.566 

A 

34.9502 

97.205 

A 

63.6173 

322.06 

A 

.785398 

.04909 

1-16 

12.7627 

12.962 

A 

35.3429 

89.402 

% 

64.0100 

326.05 

9-32 

.883573 

.06213 

A 

12 9591 

13.364 

% 

35.7356 

101.62 

A 

64.4026 

330.06 

5 16 

.981748 

.07670 

3-16 

13.1554 

13.772 

A 

36.1283 

103.87 

A 

64.7953 

334.10 

11-32 

1.07992 

.09281 

l A 

13.3518 

14.186 

% 

36.5210 

106.14 

A 

65.1880 

338.16 

% 

1.17810 

.11045 

5-16 

13.5481 

14.607 

A 

36.9137 

108.43 

A 

65.5807 

342.25 

13-32 

1.27627 

.12962 

% 

13.7445 

15.033 

y» 

37.3064 

110.75 

21. 

65.9734 

346.36 

7-16 

1.37445 

.15033 

7-16 

13 9408 

15.466 

12. 

37.6991 

113.10 

A 

66.3661 

350.50 

15-32 

1.47262 

.17257 

A 

14.1372 

15.904 

A 

38.0918 

115.47 

A 

66.7588 

354.66 

A 

1.57080 

.19635 

9-16 

14.3335 

16.349 

A 

38.4845 

117.86 

3 A 

67.1515 

358.84 

17-32 

1.66897 

.22166 

% 

14.5299 

16.800 

% 

38.8772 

120.28 

Vi 

67.5442 

363.05 

9 16 

1.76715 

.24850 

11-16 

14.7262 

17.257 

A 

39.2699 

122.72 

% 

67.9369 

367.28 

19-32 

1.86532 

.27688 

A 

14.9226 

17.721 

% 

39.6626 

125.19 

A 

68.3290 

371.54 

% 

1.96350 

.30680 

13-16 

15.1189 

18.190 

A 

40.0553 

127.68 

% 

68.7223 

375.83 

21-32 

2.06167 

.33824 

y» 

15.3153 

18.665 

A 

40.4480 

130.19 

22. 

69.1150 

380.13 

11-16 

2.15984 

.37122 

15-16 

15.5116 

19.147 

13. 

40.8107 

132.73 

A 

69.5077 

384.46 

23-32 

2.25802 

.40574 

5. 

15.7080 

19.635 

A 

41.2334 

135.30 

A 

69.9004 

388.82 

A 

2.35619 

.44179 

1-16 

15.9043 

20.129 

A 

41.6261 

137.89 

% 

70.2931 

393.20 

25-32 

2.45437 

.47937 

A 

16.1007 

20.629 

% 

42.0188 

140.50 

A 

70.6858 

397.61 

13-16 

2.55254 

.51819 

3-16 

16.2970 

21.135 

A 

42.4115 

143.14 

A 

71.0785 

402.04 

27-32 

2.65072 

.55914 

A 

16.4934 

21.648 

% 

42.8042 

145.80 

A 

71.4712 

406.49 

V» 

2.74889 

.60132 

5-16 

16.6897 

22.166 

A 

43.1969 

148.49 

y» 

71.8639 

410.97 

29-32 

2.84707 

.61504 

% 

16.8861 

22.691 

% 

43.5896 

151.20 

23. 

72.2566 

415.48 

15-16 

2.94524 

.69029 

7-16 

17.0824 

23.221 

14. 

43.9823 

153.94 


72.6493 

420.00 

31-32 

3.04342 

.73708 

A 

17.2788 

23.758 

A 

44.3750 

156.70 

A 

73.0420 

424.56 

I. 

3.14159 

.78510 

9-16 

17.4751 

24.301 

A 

44.7677 

159.48 

H 

73.4347 

429.13 

1-16 

3.33791 

.88661 

H 

17.6715 

24.850 

% 

45.1604 

162.30 

A 

73.8274 

433.74 

A 

3.53429 

.99102 

11-16 

17.8678 

25.406 

A 

45.5531 

165.13 

A 

74.2201 

438.36 

3-16 

3.73064 

1.1075 

3 4 

18.0642 

25.967 

% 

45.9458 

167.99 

A 

74.6128 

443.01 

A 

3.92699 

1.2272 

13-16 

18.2605 

26.535 

3 4 

46.3385 

170.87 

A 

75.0055 

447.69 

5-16 

4.12334 

1.3530 

% 

18.4569 

27.109 

% 

46.7312 

173.78 

24. 

75.3982 

452 39 

% 

4 31969 

1.4849 

15-16 

18.6532 

27.688 

15. 

47.1239 

176.71 

A 

75.7909 

457.11 

7-16 

4.51604 

1.6230 

6. 

18.8496 

28.274 

A 

47.5166 

179.67 

A 

76.1836 

461.86 

*4 

4.71239 

1.7671 

A 

19.2423 

29.465 

A 

47.9093 

182.65 

3 A 

76 5763 

466.64 

9-16 

4.90874 

1.9175 

% 

19.6350 

30.680 

3 A 

48.3020 

185.66 

A 

76.9690 

471.44 

% 

5.10509 

2.0739 

% 

20.0277 

31.919 

A 

48.6917 

188.69 

A 

77.3617 

476.26 

11-16 

5.30114 

2.2365 

A 

20.4204 

33.183 

A 

49.0874 

191.75 

A 

77.7544 

481.11 

H 

5.49779 

•2.4053 

% 

20.8131 

34.472 

A 

49.4801 

194.83 

% 

78 1471 

485.98 

13-16 

5 69414 

2.5802 

A 

21.2058 

35.785 

A 

49.8728 

197.93 

25. 

78.5398 

490.87 

% 

5.89049 

2.7612 

% 

21.5984 

37.122 

16. 

50.2655 

201.06 

A 

78.9325 

495.79 

15-16 

6.08684 

2.9483 

7. 

21.9911 

38.485 

A 

50.6582 

204.22 

A 

79.3252 

500.74 

2. 

6.28319 

3.1416 

A 

22.3838 

39.871 

H 

51.0509 

207.39 

A 

79.7179 

505.71 

"1-16 

6.47953 

3.3110 

% 

22.7765 

41.282 

% 

51.4436 

210.60 

A 

80.1106 

510.71 

A 

6.67588 

3.5166 

% 

23.1692 

42.718 

A 

51.8363 

213.82 

A 

80.5033 

515.72 

3-16 

6 87223 

3.7583 

A 

23.5619 

44.179 

% 

52.2290 

217.08 

A 

80.8960 

520.77 

y. 

7.06858 

3.9761 

% 

23.9546 

45 664 

3 4 

52.6217 

220.35 

A 

81.2887 

525.84 

5 16 

7.26493 

4.2000 

A 

24.3473 

47.173 

% 

53.0144 

223.65 

26. 

81.6814 

530.93 

y» 

7.46128 

4.4301 

y» 

24.7400 

48.707 

17. 

53.4071 

226.98 

A 

82.0741 

536.05 

7-16 

7.65763 


8. 

25.1327 

50.265 

A 

53.7998 

230.33 

A 

82.4668 

541.19 

A 

7.85398 

4 9087 

A 

25.5254 

51.849 

A 

54.1925 

233.71 

A 

82.8595 

546.35 

9-16 

8.05033 

5.1572 

A 

25 9181 

53.450 

3 4 

54.5852 

237.10 

A 

83.2522 

551.55 

% 

8.24668 

5.4119 

% 

26.3108 

55.088 

A 

54.9779 

240.53 

A 

83.6449 

556.76 

11-16 

8.44303 

5.6727 

A 

26.7035 

56.745 

% 

55*3706 

243.98 

A 

84.0376 

562.00 

3 4 

8.63938 

5.9396 

% 

27.0962 

58.426 

A 

55.7633 

247.45 

A 

84.4303 

567.27 

13-16 

8.83573 

6.2126 

*4 

27.4889 

60.132 

A 

56.1560 

250.95 

27. 

84.8230 

572.56 

y» 

9.03208 

6.4918 

A 

27.8816 

61.862 

18. 

56.5487 

254.47 

A 

85.2157 

577.87 

15-16 

9.22843 

6.7771 

9. 

28.2743 

63.617 

A 

56.9414 

258.02 

A 

85.6084 

583.21 

■{ 

9.42478 

7.0686 

A 

28.6670 

65.397 

A 

57.3341 

261.59 

% 

86.0011 

588.57 

1-16 

9.62113 

7.3662 

V* 

29.0597 

67.201 

% 

57.7268 

265.18 

K 

86-3938 

593.96 

H • 

9.81748 

7.6699 

a 

29.4524 

69.029 

A 

58.1195 

268.80 


86.7865 

599.37 

3-16 

10.0138 

7.9798 

V, 

29.8451 

70.882 

A 

58.5122 

272.45 

A 

87.1792 

604.81 


10.2102 

8.2958 

% 

30.2378 

72.760 

A 

58.9049 

276.12 

y» 

87.5719 

610.27 

5-16 

10.4065 

8.6179 

a 

30.6305 

74 662 

% 

59.2976 

279.81 

28. 

87.9646 

615.75 

3/ n 

10 6029 

8 9462 

% 

31.0232 

76.589 

19. 

59.6903 

283.53 

A 

88.3573 

621.26 

7-itt, 10.7992 

9.2806 

10. 

31.4159 

78.540 

A 

60.0830 

287.27 

A 

88.7500 

626.80 

































126 


CIRCLES, 


jL 


TABLE 1 OF CIRCLES— (Continued). 
Diameters in units and eighths, A'C. 


Diam. 

Circumf. 

Area. 

[Mam- 

Circumf. 

Area. 

Diam. 

Circumf. 

Area. 

Diam- 

Circumf. 

Area. 

28% 

89.1427 

632.36 

38. 

119 381 

1134.1 

47% 

149.618 

1781.4 

57 A 

179.856 

2574.2 

A 

89.5354 

637.94 

A 

A 

119.773 

1141.6 

A 

150.011 

1790.8 

A 

180.249 

2585-4 

% 

89.9281 

643 55 

120.166 

1149.1 

A 

150.404 

1800.1 

A 

180.642 

2596.7 

h 

90.3208 

649.18 

% 

120.559 

1156.6 

48. 

150.796 

1809.6 

% 

181.034 

2608.0 

% 

90.7135 

654.84 

A 

120.951 

1164.2 

A 

151.189 

1819.0 

A 

181.427 

2619.4 

29. 

91.1062 

660.52 

% 

121.344 

1171.7 

A 

151.582 

1828.5 

A 

181.820 

2630.7 1 

% 

91.4989 

666.23 

A 

121.737 

1179.3 

% 

151.975 

1837.9 

58. 

182.212 

2642.1 j 

V* 

91.8916 

671.96 

A 

122.129 

1186.9 

A 

152.367 

1847.5 

% 

182.605 

2653.5 

% 

92.2843 

677.71 

39. 

122.522 

1194.6 

% 

152.760 

1857.0 

A 

182.998 

2664.9 

A 

92.6770 

683.49 

A 

122.915 

1202.3 

A 

153.153 

1866.5 

A 

183.390 

2676.4 

% 

93.0697 

689.30 

A 

123.308 

1210.0 

A 

153.545 

1876.1 

A 

183.783 

2687.8 

A 

93.4624 

695.13 

% 

123.700 

1217.7 

49. 

153.938 

1885.7 

% 

184.176 

2699.3 

a 

93.8551 

700.98 

A 

124.093 

1225.4 

A 

154.331 

1895.4 

A 

184.569 

2710.9 

30. 

94.2478 

706.86 

% 

124.486 

1233.2 

A 

154.723 

1905.0 

A 

184.961 

2722.4 

A 

94.6405 

712.76 

A 

124.878 

1241.0 

% 

155.116 

1914.7 

59. 

185.354 

2734.0 

H 

95.0332 

718.69 

A 

125.271 

1248.8 

A 

155.509 

1924.4 

A 

185.747 

2745.6 

H 

95.4259 

724.64 

40. 

123.664 

1256.6 

A 

155.902 

1934.2 

h 

186.139 

2757.2 

A 

95.8186 

730.62 

A 

126.056 

1264.5 

A 

156.294 

1943.9 

A 

186.532 

2768.8 

% 

96.2113 

736.62 

A 

126.449 

1272.4 

A 

156.687 

1953-7 

A 

186.925 

2780.5 

A 

96.6040 

742.64 

As 

126.842 

1280.3 

50. 

157.080 

1963.5 

A 

187.317 

2792.2 

a 

96.9967 

748.69 

A 

127.235 

1288.2 

A 

157.472 

1973.3 

A 

187.710 

2803.9 

31. 

97.3894 

754.77 

% 

127.627 

1296.2 

A 

157.865 

1983.2 

A 

188.103 

2815.7 

% 

97.7821 

760.87 

A 

128.020 

1304.2 

A 

158.258 

1993.1 

60. 

188.496 

2827.4 

hi 

98.1748 

766.99 

A 

128.413 

1312.2 

A 

158.650 

2003-0 

% 

188.888 

2839.2 

A 

98.5675 

773.14 

41. 

128.805 

1320.3 

% 

159.043 

2012.9 

A 

189.281 

2851.0 | 

A 

98.9602 

779.31 

A 

129 198 

1328.3 

A 

159.436 

2022.8 

A 

189.674 

2862 9 

% 

99.3529 

785.51 

A 

129.591 

1336.4 

A 

159.829 

2032.8 

A 

190.066 

2874.8 

A 

99.7456 

791.73 

A 

129.983 

1344.5 

51. 

160.221 

2042.8 

% 

190.459 

2886.6 

a 

100.138 

797.98 

A 

130.376 

1352.7 

A 

160 614 

2052.8 

A 

190.852 

2898.6 

32. 

100.531 

804.25 

A 

130.769 

1360.8 

A 

161.007 

2062 9 

A 

191.244 

2910.5 

A 

100.924 

810.54 

A 

131.161 

1369.0 

% 

161.399 

2073.0 

61. 

191.637 

2922.5 

A 

101.316 

816.86 

A 

131.554 

1377.2 

A 

161.792 

2083.1 

A 

192.030 

2934.5 

% 

101.709 

823.21 

42. 

131.947 

1385.4 

% 

162.185 

2093.2 

A 

192.423 

2946.5 

A 

102.102 

829.58 

A 

132.310 

1393.7 

A 

162.577 

2103.3 

A 

192.815 

2958.5 

% 

102.494 

835.97 

A 

132.732 

1402.0 

A 

162.970 

2113.5 

A 

193.208 

2970.6 4 

A 

102.887 

842.39 

% 

133.125 

1410.3 

52. 

163.363 

2123.7 

A 

193.601 

2982.7 

a 

103.280 

848.83 

A 

133.518 

1418.6 

% 

163.756 

2133.9 

A 

193.993 

2994.8 

33. 

103.673 

855.30 

% 

133.910 

1427.0 

A 

164.148 

2144.2 

A 

194.386 

3006.9 

A 

101.065 

861.79 

A 

134.303 

1435.4 

h 

164.541 

2154.5 

62. 

194.779 

3019.1 

A 

104.458 

868.31 

A 

134.696 

1443.8 

A 

164.934 

2164.8 

A 

195.171 

3031.3 

% 

101.851 

874.85 

43. 

135.088 

1452.2 

A 

165.326 

2175.1 

A 

195.564 

3043.5 

A 

105.213 

881.41 

A 

135.481 

1460.7 

A 

165.719 

2185.4 

A 

195.957 

3055.7 

% 

105.636 

888.00 

A 

135.874 

1469.1 

A 

166.112 

2195.8 

A 

196.350 

3068.0 

A 

106.029 

894.62 

% 

136.267 

1477.6 

53. 

166.504 

2206.2 

% 

196.742 

3080.3 

A 

106.421 

901.26 

A 

136.659 

1486.2 

A 

166.897 

2216.6 

A 

197.135 

3092.6 

34. 

106.814 

907.92 

A 

137.052 

1494.7 

A 

167.290 

2227.0 

A 

197.528 

3104.9 

A 

107.207 

914.61 

A 

137.445 

1503.3 

A 

167.683 

2237.5 

63. 

197.920 

3117.2 

A 

107.600 

921.32 

A 

137.837 

1511.9 

A 

168.075 

2248.0 

A 

198.313 

3129.6 

% 

107.992 

928.06 

44. 

138.230 

1520.5 

% 

168.468 

2258.5 

A 

198.706 

3142.0 

A 

108.385 

934.82 

A 

138.623 

1529.2 

A 

168.861 

2269.1 

A 

199.098 

3154.5 

% 

108.778 

941.61 

A 

139.015 

1537.9 

A 

169.253 

2279.6 

A 

199.491 

3166.9 

% 

109.170 

948.42 

A 

139.408 

1546.6 

54. 

169.646 

2290.2 

% 

199.884 

3179.4 

% 

109.563 

955.25 

A 

139.801 

1555.3 

A 

170.039 

2300.8 

A 

200.277 

3191.9 

15. 

109.956 

962.11 

% 

140.194 

1564.0 

A 

170.431 

2311.5 

A 

200.669 

3204.4 

A 

110.348 

969.00 

A 

140 586 

1572.8 

A 

170.824 

2322.1 

64. 

201.062 

3217.0 

A 

110.741 

975.91 

A 

140.979 

1581.6 

A 

171.217 

2332.8 

A 

201.455 

3229.6 

% 

111.134 

982-84 

45. 

141.372 

1590.4 

% 

171.609 

2343.5 

A 

201.847 

3242.2 

A 

111.527 

989.80 

A 

141.764 

1599.3 

A 

172.002 

2354.3 

% 

202.240 

3254.3 

% 

111.919 

996.78 

A 

142.157 

1608.2 

A 

172.395 

2365.0 

A 

202.633 

3267.5 

A 

112.312 

1003.8 

% 

142.550 

1617.0 

55. 

172.788 

2375.8 

A 

203.025 

3280.1 

A 

112.705 

1010.8 

A 

142.942 

1626.0 

A 

173.180 

2386.6 

A 

203.418 

3292.8 

36. 

113.097 

1017.9 

A 

143.335 

1634.9 

A 

173.573 

2397.5 

A 

203.811 

3305.6 

A 

113.490 

1025-0 

A 

143.728 

1643.9 

A 

173.966 

2408.3 

65. 

204.204 

3318.3 

A 

113.883 

1032 1 

A 

144.121 

1652.9 

A 

174.358 

2419.2 

A 

204.596 

3331.1 

A 

114.275 

1039.2 

46. 

144.513 

1661.9 

% 

174.751 

2430.1 

A 

204.989 

3343.9 

A 

114.668 

1046.3 

A 

144.906 

1670.9 

A 

175.144 

2441.1 

A 

205.382 

3356.7 

% 

115.061 

1053.5 

A 

145.299 

1680.0 

A 

175.536 

2452.0 

A 

205.774 

3369.6 

A 

115.454 

1060.7 

A 

145.691 

1689.1 

56. 

175.929 

2463.0 

% 

206.167 

3382.4 

A 

115.846 

1068 0 

A 

146.084 

1698.2 

A 

176.322 

2474.0 

A 

206,560 

3395.3 

37. 

116.239 

1075.2 

A 

146.477 

1707.4 

A, 

176.715 

2485.0 

A 

206.952 

3408.2 

A 

116.632 

1082.5 

A 

146.869 

1716.5 

A 

177.107 

2496.1 

66. 

207.345 

3421.2 

A 

117.024 

10K9.8 

A 

147.262 

1725.7 

A 

177.500 

2507.2 

A 

207.738 

3434.2 

A 

117.417 

1097.1 

47. 

147.655 

1734.9 

% 

177.893 

2518.3 

A 

208.131 

3447.2 

A 

117.810 

1104.5 

A 

148.048 

1744.2 

A 

178.285 

2529.4 

A 

208.523 

3460.2 

% 

118.202 

1111.8 

A 

148.440 

1753.5 

A 

178.678 

2540.6 

A 

208.916 

3473.2 

% 

118.596 

1119.2 

% 

148.833 

1762.7 

57. 

179.071 

2551.8 

% 

209.309 

3486.3 

A 

118.988 

1126.7 

A 

149.226 

1772.1 

A 

179.463 

2563.0 

A 

209.701 

3499.4 












































CIRCLES 


127 


TABLE 1 OF CIRCLES— (Continued). 
Diameters in units and eighths, «&c. 


Diam. 

Circumf. 

Area. 

Diam. 

Circumf. 

Area. 

Diam. 

Circumf. 

Area. 

Diam. 

Circumf. 

Area. 

% 

210.094 

3512.5 

75M 

236.405 

4447.4 

83% 

262.716 

5492.4 

92. 

289.027 

6647.6 

67. 

210.487 

3525.7 

% 

236.798 

4462.2 

X 

263.108 

5508.8 

X 

289.419 

6665.7 

x 

210.879 

3538.8 


237.190 

4477.0 

X 

263.501 

5525.3 

x 

289.812 

6683.8 

x 

211.272 

3552.0 

% 

237.583 

4491.8 

84. 

263.894 

5541.8 

X 

2S0.205 

6701.9 

% 

‘211. Hfi5 

3565.2 

X 

237.976 

4506.7 

X 

264.286 

5558.3 

X 

290.597 

6720.1 

% 

212.058 

3578.5 

X 

238.368 

4521.5 

X 

264.679 

5574.8 

% 

290.990 

6738.2 

% 

212.450 

3591.7 

76. 

238.761 

4536.5 

% 

265.072 

5591.4 

X 

291,383 

6756.4 

X 

212.843 

3605.0 

X 

239.154 

4551.4 

X 

265.465 

5607.9 

X 

291.775 

6774.7 

X 

213.236 

3618.3 

X 

239.546 

4566.4 

% 

265.857 

5624.5 

93. 

292.168 

6792.9 

68. 

213.628 

3631.7 

% 

239.939 

4581.3 

X 

266.250 

5641.2 

X 

292.561 

6811.2 

X 

214.021 

3645.0 

X 

240.332 

4596.3 

X 

266.643 

5657.8 

X 

292.954 

6829.5 

X 

214.414 

3658.4 

X 

240.725 

4611.4 

85. 

267.035 

5674.5 

% 

293.346 

6847.8 

X 

214.806 

3671.8 

X 

241.117 

4626.4 

X 

267.428 

5691.2 

X 

293.739 

6866.1 

X 

215.199 

3685. '6 

X 

241.510 

4641.5 

X 

267.821 

5707.9 

% 

294.132 

6884.5 

X 

215.592 

3698.7 

77. 

241.903 

4656.6 

Vs 

268.213 

5724.7 

X 

294.524 

6902.9 

X 

215.984 

3712.2 

X 

242.295 

4671.8 

X 

268.606 

5741.5 

X 

294.917 

6921.3 

X 

216.377 

3725.7 

X 

242.688 

4686.9 

% 

268.999 

5758.3 

94. 

295.310 

6939.8 

69. 

216.770 

3739.3 

% 

243.081 

4702.1 

X 

269.392 

5775.1 

X 

295.702 

6958.2 

X 

217.163 

3752.8 

X 

243.473 

4717.3 

X 

269.784 

5791.9 

X 

296.095 

6976.7 

X 

217.555 

3766.4 

% 

243.866 

4732.5 

86. 

270.177 

5808 8 

X 

296.488 

6995.3 

% 

217.948 

3780.0 

X 

244.259 

4747.8 

X 

270.570 

5825.7 

X 

296.881 

7013.8 

X 

218.341 

3793.7 

X 

244.652 

4763.1 

X 

270.962 

5842.6 

% 

297.273 

7032.4 

% 

218.733 

3807.3 

78. 

245.044 

4778.4 

X 

271.355 

5859.6 

X 

297.666 

7051.0 

X 

219.126 

3821.0 

X 

245.437 

4793.7 

X 

271.748 

5876.5 

X 

298.059 

7069.6 

% 

219.519 

3834.7 

X 

245.830 

4809.0 

% 

272.140 

5893.5 

95. 

298.451 

7088.2 

70. 

219.911 

3848.5 

% 

246.222 

4824.4 

X 

272.533 

5910.6 

X 

298.844 

7106.9 

X 

220.304 

3862.2 

X 

246.615 

4839.8 

X 

272.926 

5927.6 

X 

299.237 

7125.6 

X 

220.697 

3876.0 

X 

247.008 

4855.2 

87. 

273.319 

5944.7 

% 

299.629 

7144.3 

% 

221.090 

3889.8 

X 

247.400 

4870.7 

X 

273.711 

5961.8 

X 

300.022 

7163.0 

X 

221.482 

3903.6 

X 

247.793 

4886.2 

X 

274.104 

5978.9 

% 

300.415 

7181.8 

% 

221.875 

3917.5 

79. 

248.186 

4901.7 

X 

274.497 

5996.0 

X 

300.807 

7200.6 

X 

222.268 

3931.4 

X 

248.579 

4917.2 

X 

274.889 

6013.2 

X 

301.200 

7219.4 

X 

222.660 

3945.3 

X 

248.971 

4932.7 

% 

275.282 

6030.4 

96. 

301.593 

7238.2 

71. 

223.053 

3959.2 

x 

249.364 

4948.3 

X 

275-675 

6047.6 

X 

301.986 

7257.1 

X 

223.446 

3973.1 

X 

249.757 

4963.9 

X 

276.067 

6064.9 

X 

302.378 

7276 0 

X 

223.838 

3987.1 

% 

250.149 

4979.5 

88. 

276.460 

6082.1 

% 

302.771 

7294.9 

X 

224.231 

4001.1 

X 

250.542 

4995.2 

X 

276.853 

6099.4 

X 

303.164 

7313.8 

X 

224.624 

4015.2 

X 

250.935 

5010.9 

X 

277.246 

6116.7 

% 

303.556 

7332.8 

% 

*225.017 

4029.2 

80. 

251.327 

5026.5 

X 

277.638 

6134.1 

X 

303.949 

7351.8 

X 

225.109 

4043.3 

X 

251.720 

5042.3 

X 

278.031 

6151.4 

X 

304.342 

7370.8 

x 

225.802 

4057.4 

X 

252.113 

5058.0 

% 

278.424 

6168.8 

97. 

304.734 

7389.8 

72. 

226.195 

4071.5 

% 

252.506 

5073.8 

X 

278.816 

6186.2 

X 

305.127 

7408.9 

X 

226.587 

4085.7 

X 

252.898 

5089.6 

X 

279.209 

6203.7 

X 

305 520 

7428.0 

X 

226.980 

4099.8 

% 

253.291 

5105.4 

89. 

279.602 

6221.1 

• X 

305.913 

7447.1 

% 

227.373 

4114.0 

X 

253.684 

5121.2 

X 

279.994 

6238.6 

X 

306.305 

7466.2 

X 

227.765 

4128.2 

X 

254.076 

5137.1 

X 

280.387 

6256.1 

% 

306.698 

7485.3 

% 

228.158 

4142.5 

81. 

254.469 

5153.0 

% 

280.780 

6273.7 

X 

307.091 

7504.5 

X 

228.551 

4156.8 

X 

254.862 

5168.9 

X 

281.173 

6291.2 

X 

307.483 

7523.7 

% 

228 914 

4171.1 

X 

255.254 

5184.9 

% 

281.565 

6308.8 

98. 

307.876 

7543.0 

73. 

229.336 

4185.4 

% 

255.647 

5200.8 

X 

281.958 

6326.4 

X 

308.269 

7562.2 

X 

229.729 

4199.7 

X 

256.040 

5216.8 

X 

282.351 

6344.1 

X 

308.661 

7581.5 

X 

230.122 

4214 1 

% 

256.433 

5232.8 

90. 

282.743 

6361.7 

X 

309.054 

7600.8 

y» 

230.514 

4228.5 

X 

256.825 

5248.9 

X 

283.136 

6379.4 

X 

309.447 

7620.1 

X 

230.907 

4242.9 

X 

257.218 

5264.9 

x 

283.529 

6397.1 

% 

309.840 

7639.5 

% 

231.300 

4257.4 

82. 

257.611 

5281.0 

% 

283.921 

6414.9 

X 

310.232 

7658.9 

X 

231.692 

4271.8 

X 

258.003 

5297.1 

34 

284.314 

6432.6 

X 

310.625 

7678.3 

y» 

232.085 

4286.3 

X 

258.396 

5313.3 

% 

284.707 

6450.4 

99. 

311.018 

7697.7 

74. 

232.478 

4300.8 

% 

258.789 

5329.4 

X 

285.100 

6468.2 

X 

311.410 

7717.1 

x 

232.871 

4315.4 

X 

259.181 

5345.6 

X 

285.492 

6486.0 

X 

311.803 

7736.6 

X 

233 263 

4329.9 

% 

259.574 

5361.8 

91. 

285.885 

6503.9 

% 

312.196 

7756.1 

% 

233.656 

4344.5 

X 

259.967 

5378.1 

X 

286.278 

6521.8 

X 

312.588 

7775.6 

X 

234.019 

4359.2 

X 

260.359 

5394.3 

X 

286.670 

6539.7 

% 

312.981 

7795.2 

% 

231.441 

4373.8 

83. 

260.752 

5410.6 

x 

287.063 

6557.6 

X 

313.374 

7814.8 

X 

234.834 

4388.5 

X 

261.145 

5426.9 

X 

287.456 

6575.5 

X 

313.767 

7834.4 

% 

235.227 

4403.1 

X 

261.538 

5443.3 

% 

287.848 

6593.5 

100. 

314.159 

7854.8 

75. 

235.619 

4417.9 

X 

261930 

5459 6 

X 

288.241 

6611.5 




X 

236.012 

4432.6 

X 

262.323 

5476.0 

X 

288.634 

6629.6 







































128 CIRCLES. 


TABLE 2 OF CIRCLES. 
Diameters in units and tenths. 


Din. 

Circumf. 

Area. 

Ilia. 

Circumf. | 

Area. 

Dia. 

Circumf. 

Area. 

0 1 

.314159 

.007854 

6.3 

19.79203 

31.17245 

12.5 

39.26991 

122.7185 1 

2 

’.s 

4 

.628319 

.942478 

1 256637 

.031416 

.070686 

.125664 

.4 

.5 

.6 

20.10619 

20.42035 

20.73451 

32.16991 

33.18307 

34.21194 

.6 

.7 

.8 

39.58407 

39.89823 

40.21239 

124.6898 
126.6769 
128.6796 | 

.5 

1 570796 

.196350 

.7 

21.04867 

35.25652 

.9 

40.52655 

130.6981 


1 884956 

.282743 

.8 

21.36283 

36.31681 

13.0 

40.31070 

132.7323 

134.7822 

.7 

2.199115 

.381845 

.9 

21.67699 

37.39281 

.1 

41.15486 

.8 

2.513274 

.502655 

7.0 

21.99115 

38.48451 

.2 

41.46902 

13b.cS4 / 3 

9 

2.827433 

.636173 

.1 

22.30531 

39.59192 

.3 

41.78318 

138.9291 

1 0 

3.141593 

.785398 

.2 

22.61947 

40.71504 

.4 

42.09734 

141.0261 

.1 

3.455752 

.950332 

.3 

22.93363 

41.85387 

.5 

42.41150 

143.1338 

.2 

3.769911 

1.13097 

.4 

23.24779 

43.00840 

.6 

42.72566 

145.2672 

.3 

4.084070 

1.32732 

.5 

23.56194 

44.17865 

.7 

43.03982 

147.4114 } 

.4 

4.398230 

1.53938 

.6 

23.87610 

45.36460 

.8 

43.35398 

149.5712 

.5 

4.712389 

1.76715 

.7 

24.19026 

46.56626 

.9 

43.66814 

lol.7468 

.6 

5.026548 

2.01062 

.8 

24.50442 

47.78362 

14.0 

43.98230 

153.9380 

.7 

5.340708 

2.26980 

.9 

24.81858 

49.01670 

.1 

44.29646 

156.1450 *| 

.8 

5.654867 

2.54469 

8.0 

25.13274 

50.26548 

.2 

44.61062 

158.3677 

.9 

5.96)9026 

2.83529 

.1 

25.44690 

51.52997 

.3 

44.92477 

160.6061 

2.0 

6.283185 

3.14159 

.2 

25.76106 

52.81017 

.4 

45.23893 

162.8602 

.1 

6.597345 

3.46361 

.3 

26.07522 

54.10608 

.5 

45 55309 

165.1300 

.2 

6.911504 

3.80133 

.4 

26.38938 

55.41769 

.6 

45.86725 

167.4155 

.3 

7.225663 

4.15476 

.5 

26.70354 

56.74502 

.7 

46.18141 

169.7167 

.4 

7.539822 

4.52389 

.6 

27.01770 

58.08805 

.8 

46.49557 

172.0336 

.5 

7.853982 

4.90874 

.7 

27.33186 

59.44679 

.9 

46.80973 

174.3662 

.6 

8.168141 

5.30929 

.8 

27.64602 

60.82123 

15.0 

47.12389 

176.7146 

.7 

8.482300 

5.72555 

.9 

27.96017 

62.21139 

.1 

47.43805 

179.0786 

.8 

8.796159 

6.15752 

9.0 

28.27433 

63.61725 

.2 

47.75221 

181.4584 

.9 

9.110619 

6.60520 

.1 

28.58849 

65.03882 

.3 

48.06637 

183.8539 

3.0 

9.424778 

7.06858 

.2 

28.90265 

66.47610 

.4 

48.38053 

186.2650 

.1 

9.738937 

7.54768 

.3 

29.21681 

67.92909 

.5 

48.69469 

18^.6919 

.2 

10.05310 

8.04248 

.4 

29.53097 

69.39778 

.6 

49.00885 

191.1345 

.3 

10.36726 

8.55299 

.5 

29.84513 

70.88218 

.7 

49.32300 

193.5928 

.4 

10.68142 

9.07920 

.6 

30.15929 

72.38229 

.8 

49.63716 

196.0668 

.5 

10.99557 

9.62113 

.7 

30.47345 

73.89811 

.9 

49.95132 

198.5565 

.6 

11.30973 

10.17876 

.8 

30.78761 

75.42964 

16.0 

50.26548 

201.0619 

.7 

11.62389 

10.75210 

.9 

31.10177 

76.97687 

.1 

50.57964 

203.5831 

.8 

11.93805 

11.34115 

10.0 

31.41593 

78.53982 

.2 

50.89380 

206.1199 

.9 

12.25221 

11.94591 

.1 

31.73009 

80.11847 

.3 

51.20796 

208.6724 ; 

4.0 

12.56637 

12.56637 

.2 

32.04425 

81.71282 

.4 

51.52212 

211.2407 

.1 

12.88053 

13.20254 

.3 

32.35840 

83.32289 

.5 

51.83628 

213.8246 

.2 

13.19469 

13.85442 

.4 

32.67256 

84.94867 

.6 

52.15044 

216.4243 

.3 

13.50885 

14.52201 

.5 

32.98672 

86.59015 

.7 

52.46460 

219.0397 

.4 

13.82301 

15.20531 

.6 

33.30088 

88.24734 

.8 

52.77876 

221.6708 

.5 

14.13717 

15.90431 

.7 

33.61504 

89.92024 

.9 

53.09292 

224.3176 

.6 

14.45133 

16.61903 

.8 

33.92920 

91.60S84 

17.0 

53.40708 

226.9801 

.7 

14.76549 

17.34945 

.9 

34.24336 

93.31316 

.1 

53.72123 

229.6583 

.8 

15.07964 

18.09557 

11.0 

34.55752 

95.03318 

.2 

54.03539 

232.3522 

.9 

15.39380 

18.85741 

.1 

34.87168 

96.76891 

.3 

54.34955 

235.0618 

5.0 

15.70796 

19.63495 

.2 

35.18584 

98.52035 

.4 

54.66371 

237.7871 

.1 

16.02212 

20.42821 

.3 

35.50000 

100.2875 

.5 

54.97787 

240.5282 

.2 

16.33628 

21.23717 

.4 

35.81416 

102.0703 

.6 

55.29203 

243.2849 

.3 

16.65044 

22.06183 

.5 

36.12832 

103.8689 

.7 

55.60619 

246.0574 

.4 

16.96460 

22.90221 

.6 

36.44247 

105.6832 

.8 

55.92035 

248.8456 

.5 

17.27876 

23.75829 

.7 

36.75663 

107.5132 

.9 

56.23451 

251.6494 

.6 

17.59292 

24.63009 

.8 

37.07079 

109.3.588 

18.0 

56.54867 

254.4690 

.7 

17.90708 

25.51759 

.9 

37.38495 

111.2202 

.1 

56.86283 

257.3043 

.8 

18.22124 

26.42079 

12.0 

37.69911 

113.0973 

.2 

57.17699 

260.1553 

.9 

18.53540 

27.33971 

.1 

38.01327 

114.9901 

.3 

57.49115 

263.0220 

6.0 

18.84956 

28.27433 

.2 

38.32743 

116.8987 

A 

57.80530 

265.9044 

.1 

19.16372 

29.22467 

.3 

38.64159 

118.8229 

.5 

58.11946 

268.8025 

.2 

19.47787 

30.19071 

.4 

38.95575 

120.7628 

.6 

58.43362 

271.7163 















































CIRCLES, 


129 


TABLE 2 OF CIRCLES—(Continued). 
Diameters in units and tenths. 


Dia. 

Circumf. 

Area. 

Dia. 

Circumf. 

Area. 

Dia. 

Circumf. 

Area. 

18.7 

58.74778 

274.6459 

24.9 

78.22566 

486.9547 

31.1 

97.70353 

759.6450 

.8 

59.00194 

277.5911 

25.0 

78.53982 

490.8739 

.2 

98.01769 

764.5380 

.9 

59.37610 

280.5521 

.1 

78.85398 

494.8087 

.3 

98.33185 

769.4467 

19.0 

59.69026 

283.5287 

.2 

79.16813 

498.7592 

.4 

98.64601 

774.3712 

.1 

60.00442 

286.5211 

.3 

79.48229 

502.7255 

.5 

98.96017 

779.3113 

2 

60.31858 

289.5292 

.4 

79.79645 

506.7075 

.6 

99.27433 

784.2672 

.3 

60.63274 

292.5530 

.5 

80.11061 

510.7052 

.7 

99.58849 

789.2388 

.4 

60.94690 

295.5925 

.6 

80.42477 

514.7185 

.8 

99.90265 

794.2260 

.5 

61.26106 

298.6477 

.7 

80.73893 

518.7476 

.9 

100.2168 

799.2290 

.6 

61.57522 

301.7186 

.8 

81.05309 

522.7924 

32.0 

100.5310 

804.2477 

.7 

61.88938 

304.8052 

.9 

81.36725 

526.8529 

.1 

100.8451 

809.2821 

.8 

62.20353 

307.9075 

26.0 

81.68141 

530.9292 

.2 

101.1593 

814.3322 

.9 

62.51769 

311.0255 

.1 

81.99557 

535.0211 

.3 

101.4734 

819.3980 

20.0 

62.83185 

314.1593 

.2 

82.30973 

539.1287 

.4 

101.7876 

824.4796 

.1 

63.14601 

317.3087 

.3 

82.62389 

543.2521 

.5 

102.1018 

829.5768 

.2 

63.46017 

320.4739 

.4 

82.93805 

547.3911 

.6 

102.4159 

834.6898 

.3 

63.77433 

323.6547 

.5 

83.25221 

551.5459 

.7 

102.7301 

839.8184 

.4 

64.08849 

326.8513 

.6 

83.56636 

555.7163 

.8 

103.0442 

844.9628 

.5 

64.40265 

330.0636 

.7 

83.88052 

559.9025 

.9 

103.3584 

850.1228 

.6 

64.71681 

333.2916 

.8 

84.19468 

564.1044 

33.0 

103.6726 

855.2986 

.7 

65.03097 

336.5353 

.9 

84.50884 

568.3220 

.1 

103.9867 

860.4901 

.8 

65.34513 

339.7947 

27.0 

84.82300 

572.5553 

.2 

104.3009 

865.6973 

.9 

65.65929 

343.0698 

.1 

85.13716 

576.8043 

.3 

104.6150 

870.9202 

21.0 

65.97345 

346.3606 

.2 

85.45132 

581.0690 

.4 

104.9292 

876.1588 

.1 

66.28760 

349.6671 

.3 

85.76548 

585.3494 

.5 

105.2434 

881.4131 

.2 

66.60176 

352.9894 

.4 

86.07964 

589.6455 

.6 

105.5575 

886.6831 

.3 

66.91592 

356.3273 

.5 

86.39380 

593.9574 

.7 

105.8717 

891.9688 

.4 

67.23008 

359.6809 

.6 

86.70796 

598.2849 

.8 

106.1858 

897.2703 

.5 

67.54424 

363.0503 

.7 

87.02212 

602.6282 

.9 

106.5000 

902.5874 

.6 

67.85840 

366.4354 

.8 

87.33628 

606.9871 

34.0 

106.8142 

907.9203 

.7 

68.17256 

369.8361 

.9 

87.65044 

611.3618 

.1 

107.1283 

913.2688 

.8 

68.48672 

373.2526 

28.0 

87.96459 

615.7522 

.2 

107.4425 

918.6331 

.9 

68.80088 

376.6848 

.1 

88.27875 

620.1582 

.3 

107.7566 

924.0131 

22.0 

69.11504 

380.1327 

.2 

88.59291 

624.5800 

.4 

108.0708 

929.4088 

.1 

69.42920 

383.5963 

.3 

88.90707 

629.0175 

.5 

108.3849 

934.8202 

.2 

69.74336 

387.0756 

.4 

89.22123 

633.4707 

.6 

108.6991 

940.2473 

.3 

70.05752 

390.5707 

.5 

89.53539 

637.9397 

.7 

109.0133 

945.6901 

.4 

70.37168 

394.0814 

.6 

89.84955 

642.4243 

.8 

109.3274 

951.1486 

.5 

70.68583 

397.6078 

.7 

90.16371 

646.9246 

.9 

109.6416 

956.6228 

.6 

70.99999 

401.1500 

.8 

90.47787 

651.4407 

35.0 

109.9557 

962.1128 

.7 

71.31415 

404.7078 

.9 

90.79203 

655.9724 

.1 

110.2699 

967.6184 

.8 

71.62831 

408.2814 

29.0 

91.10619 

660.5199 

.2 

110.5841 

973.1397 

.9 

71.94247 

411.8707 

.1 

91.42035 

665.0830 

.3 

110.8982 

978.6768 

23.0 

72.25663 

415.4756 

.2 

91.73451 

669.6619 

.4 

111.2124 

984.2296 

.1 

72.57079 

419.0963 

.3 

92.04866 

674.2565 

.5 

111.5265 

989.7980 

.2 

72.88495 

422.7327 

.4 

92.36282 

678.8668 

.6 

111.8407 

995.3822 

.3 

73.19911 

426.3848 

.5 

92.67698 

683.4928 

.7 

112.1549 

1000.9821 

.4 

73.51327 

430.0526 

.6 

92.99114 

688.1345 

.8 

112.4690 

1006.5977 

.5 

73.82743 

433.7361 

.7 

93.30530 

692.7919 

.9 

112.7832 

1012.2290 

.6 

74.14159 

437.43.54 

.8 

93.61946 

697.4650 

36.0 

113.0973 

1017.8760 

.7 

74.45575 

441.1503 

.9 

93.93362 

702.1538 

.1 

113.4115 

1023.5387 

.8 

74.76991 

444.8809 

30.0 

94.24778 

706.8583 

.2 

113.7257 

1029.2172 

9 

75.08406 

448.6273 

.1 

94.56194 

711.5786 

.3 

114.0398 

1034.9113 

24 0 

75.39822 

452.3893 

.2 

94.87610 

716.3145 

.4 

114.3540 

1040.6212 

1 

75.71238 

456.1671 

.3 

95.19026 

721.0662 

.5 

114.6681 

1046.3467 

9 

76.02654 

459.9606 

.4 

95.50442 

725.8336 

.6 

114.9823 

1052.0880 

3 

76.34070 

463.7698 

.5 

95.81858 

730.6166 

.7 

115.2965 

1057.8449 

4 

76.65486 

467.5947 

.6 

96.13274 

735.4154 

.8 

115.6106 

1063.6176 

5 

76.96902 

471.4352 

.7 

96.44689 

740.2299 

.9 

115.9248 

1069.4060 

6 

77.28318 

475.2916 

.8 

96.76105 

745.0601 

37.0 

116.2389 

1075.2101 


77.59734 

479.1636 

.9 

97.07521 

749.9060 

.1 

116.5531 

1081.0299 

.8 

77.91150 

483.0513 

31.0 

97.38937 

754.7676 

.2 

116.8672 

1086.8654 


9 
























130 


CIRCLES 


TABLE 2 OF CIRCLES—(Continued). 
Diameters in units ami tenths. 


Dia. 

Circunif. 

Area. 

Dia. 

Circunif. 

Area. 

Dia. 

Circunif. 

Area. 

37.3 

117.1814 

1092.7166 

43.5 

136.6593 

1486.1697 

49.7 

156.1372 

1940.0041 

.4 

117.4956 

1098.5835 

.6 

136.9734 

1493.0105 

.8 

156.4513 

1947.8189 

.5 

117.8097 

1104.4662 

.7 

137.2876 

1499.8670 

.9 

156.7655 

1955.6493 

.6 

118.1239 

1110.3645 

.8 

137.6018 

1506.7393 

50.0 

157.0796 

1963.4954 

.7 

118.4380 

1116.2786 

.9 

137.9159 

1513.6272 

.1 

157.3938 

1971.3572 

.8 

118.7522 

1122 2083 

44.0 

138.2301 

1520.5308 

.2 

157.7080 

1979.2318 

.9 

119.0664 

1128.1538 

.1 

138.5442 

1527.4502 

.3 

158.0221 

1987.1280 

38.0 

119.3805 

1134.1149 

.2 

138.8584 

1534.3853 

.4 

158.3363 

1995.0370 

.1 

119.6947 

1140.0918 

.3 

139.1726 

1541.3360 

.5 

158.6504 

2002.9617 

.2 

120.0088 

1146.0844 

.4 

139.4867 

1548.3025 

.6 

158.9646 

2010.9029 

.3 

120.3230 

1152.0927 

.5 

139.8009 

1555.2847 

.7 

159.2787 

2018.8581 

.4 

120.6372 

1158.1167 

.6 

140.1150 

1562.2826 

.8 

159.5929 

2026.8299 

.5 

120.9513 

1164.1564 

.7 

140.4292 

1569.2962 

.9 

159.9071 

2034 8174 

.6 

121.2655 

1170.2118 

.8 

140.7434 

1576.3255 

51.0 

160.2212 

2042.8206 

.7 

121.5796 

1176.2830 

.9 

141.0575 

1583.3706 

.1 

160.5354 

2050.8395 

.8 

121.8938 

1182.3698 

45.0 

141.3717 

1590.4313 

.2 

160.8495 

2058.8742 

9 

122.2080 

1188.4724 

.1 

141.6858 

1597.5077 

.3 

161.1637 

2066.9245 

89.0 

122.5221 

1194.5906 

.2 

142.0000 

1604.5999 

.4 

161.4779 

2074 9905 

.1 

122.8363 

1200.7246 

.3 

142.3141 

1611.7077 

.5 

161.7920 

2083.0723 

.2 

123.1504 

1206.8742 

.4 

142.6283 

1618.8313 

.6 

162.1062 

2091.1697 

.3 

123.4646 

1213.0396 

.5 

142.9425 

1625.9705 

.7 

162.4203 

2099.2829 

.4 

123.7788 

1219.2207 

.6 

143.2566 

1633.1255 

.8 

162.7345 

2107.4118 

.5 

124.0929 

1225.4175 

.7 

143.5708 

1640.2962 

.9 

163.0487 

2115.5563 

.6 

124.4071 

1231.6300 

.8 

143.8849 

1647.4826 

52.0 

163.3628 

2123.7166 

.7 

124.7212 

1237.8582 

.9 

144.1991 

1654.6847 

.1 

163.6770 

2131.8926 

.8 

125.0354 

1244.1021 

46.0 

144.5133 

1661.9025 

2 

163.9911 

2140.0843 

.9 

125.3495 

1250.3617 

.1 

144.8274 

1669.1360 

.3 

164.3053 

2148.2917 

40.0 

125.6637 

1256.6371 

.2 

145.1416 

1676.3853 

.4 

164.6195 

2156.5149 

.1 

125.9779 

1262.9281 

.3 

145.4557 

1683.6502 

.5 

164.9336 

2164 7537 

.2 

126.2920 

1269.2348 

.4 

145.7699 

1690.9308 

.6 

165.2478 

2173.0082 

.3 

126.6062 

1275.5573 

.5 

146.0841 

1698.2272 

.7 

165.5619 

2181.2785 

.4 

126.9203 

1281.8955 

.6 

146.3982 

1705.5392 

.8 

165.8761 

2189.5644 

.5 

127.2345 

1288.2493 

.7 

146.7124 

1712.8670 

.9 

166.1903 

2197.8661 

.6 

127.5487 

1294.6189 

.8 

147.0265 

1720.2105 

53.0 

166.5044 

2206.1834 

.7 

127.8628 

1301.0042 

.9 

147.3407 

1727.5697 

.1 

166.8186 

2214.5165 

.8 

128.1770 

1307.4052 

47.0 

147.6549 

1734.9445 

.2 

167.1327 

2222.8653 

.9 

128.4911 

1313.8219 

.1 

147.9690 

1742.3351 

.3 

167.4469 

2231.2298 

41.0 

128.8053 

1320.2543 

.2 

148.2832 

1749.7414 

.4 

167.7610 

2239.6100 

.1 

129.1195 

1326.7024 

.3 

148.5973 

1757.1635 

.5 

168.0752 

2248.0059 

.2 

129.4336 

1333.1663 

.4 

148.9115 

1764.6012 

.6 

168.3894 

2256 4175 

.3 

129.7478 

1339.6458 

.5 

149.2257 

1772.0546 

.7 

168.7035 

2264 8448 

.4 

130.0619 

1346.1410 

.6 

149.5398 

1779.5237 

.8 

169.0177 

2273 2879 

.5 

130.3761 

1352.6520 

.7 

149.8540 

1787.0086 

.9 

169.3318 

2281 7466 

.6 

130.6903 

1359.1786 

.8 

150.1681 

1794.5091 

54.0 

169.6460 

2290 2210 

.7 

131.0044 

1365.7210 

.9 

150.4823 

1802.0254 

.1 

169.9602 

2298 7112 

.8 

131.3186 

1372.2791 

48.0 

150.7964 

1809.5574 

.2 

170.2743 

2307 ‘>171 

.9 

131.6327 

1378.8529 

.1 

151.1106 

1817.1050 

.3 

170.5885 

2315 7386 

42.0 

131.9469 

1385:4424 

.2 

151.4248 

1824.6684 

.4 

170.9026 

2324 2759 

.1 

132.2611 

1392.0476 

.3 

151.7389 

1832.2475 

.5 

171.2168 

2332 8289 

.2 

132.5752 

1398.6685 

.4 

152.0531 

1839.8423 

.6 

171.5310 

2341 3976 

.3. 

132.8894 

1405.3051 

.5 

152.3672 

1847.4528 

.7 

171.8451 

2349 98' ; 0 

.4 

133.2035 

1411.9574 

.6 

152.6814 

1855.0790 

.8 

172.1593 

2358 5821 

•5 

135,5l / / 

1418.6254 

.7 

152.9956 

1862.7210 

.9 

172.4734 

2367 1979 

.6j 

133.8318 

1425.3092 

.8 

153.3097 

1870.3786 

55.0 

172.7876 

2375 8‘>94 

.7 

134.1460 

1432.0086 

.9 

153.6239 

1878.0519 

.1 

173.1018 

2384 4767 

.8 

134.4602 

1438.7238 

49.0 

153.9380 

1885.7410 

.2 

173.4159 

2393 1396 

.9 

134. / / 43 

1445.4546 

.1 

154.2522 

1893.4457 

.3 

173.7301 

2401 8183 

43.0 

135.0885 

1452.2012 

.2 

154 5664 

1901.1662 

.4 

174.0442 

2410 5126 

.1 

135.4026 

1458.9635 

.3 

154.8805 

1908.9024 

.5 

174 3584 

2419 2‘>27 

.2 

135.7168 

1465.7415 

.4 

155.1947 

1916.6543 

.6 


9JOT CUvT 

.3 

136.0310 

1472.5352 

.5 

155.5088 

1924.4218 

.7 


24H6 (;xqQ 

.4 

136.3451 

1479.3446 

.6 

155.8230 

1932.2051 

.8 

175.3009 

2445.4471 






























CIRCLES 


131 


TABLE 2 OF CIRCLES—(Continued). 


Diameters in units and tenths. 


Dia. 

Circumf. 

Area. 

Dia. 

Circumf. 

Area. 

Dia. 

Circumf. 

Area. 

55.9 

175.6150 

2454.2200 

62.1 

195.0929 

3028.8173 

68.3 

214.5708 

3663.7960 

50.0 

175.9292 

2463.0086 

.2 

195.4071 

3038.5798 

.4 

214.8849 

3674.5324 

.1 

176.2433 

2471.8130 

.3 

195.7212 

3048.3580 

.5 

215.1991 

3685.2845 

.2 

176.5575 

2480.6330 

.4 

196 0354 

3058.1520 

.6 

215.5133 

3696.0523 

.3 

176.8717 

2489.4687 

.5 

196.3495 

3067.9616 

.7 

215.8274 

3706.8359 

.4 

177.1858 

2498.3201 

.6 

196.6637 

3077.7869 

.8 

216.1416 

3717.6351 

.5 

177.5000 

2507.1873 

.7 

196.9779 

3087.6279 

.9 

216.4557 

3728.4500 

.6 

177.8141 

2516.0701 

.8 

197.2920 

3097.4847 

69.0 

216.7699 

3739.2807 

.7 

178.1283 

2524.9687 

.9 

197.6062 

3107.3571 

.1 

217.0841 

3750.1270 

.8 

178.4425 

2533.8830 

63.0 

197.9203 

3117.2453 

.2 

217.3982 

3760.9891 

.9 

178.7566 

2542.8129 

.1 

198.2345 

3127.1492 

.3 

217.7124 

3771.8668 

67.0 

179.0708 

2551.7586 

.2 

198.5487 

3137.0688 

.4 

218.0265 

3782.7603 

.1 

179.3849 

2560.7200 

.3 

198.8628 

3147.0040 

.5 

218.3407 

3793.6695 

.2 

179.6991 

2569.6971 

.4 

199.1770 

3156.9550 

.6 

218.6548 

3804.5944 

.3 

180.0133 

2578.6899 

.5 

199.4911 

3166.9217 

.7 

218.9690 

3815.5350 

.4 

180.3274 

2587.6985 

.6 

199.8053 

3176.9042 

.8 

219.2832 

3826.4913 

.5 

180 6416 

2596.7227 

.7 

200.1195 

3186.9023 

.9 

219.5973 

3837.4633 

.6 

180.9557 

2605.7626 

.8 

200.4336 

3196.9161 

70.0 

219.9115 

3848.4510 

.7 

181.2699 

2614.8183 

.9 

200.7478 

3206.9456 

.1 

220.2256 

3859.4544 

.8 

181.5841 

2623.8896 

64.0 

201.0619 

3216.9909 

.2 

220.5398 

3870.4736 

.9 

181.8982 

2632.9767 

.1 

201.3761 

3227.0518 

.3 

220.8540 

3881.5084 

6S.0 

182.2124 

2642.0794 

.2 

201.6902 

3237.1285 

.4 

221.1681 

3892.5590 

.1 

182.5265 

2651.1979 

.3 

202.0044 

3247.2209 

.5 

221.4823 

3903.6252 


182.8407 

2660.3321 

.4 

202.3186 

3257.3289 

.6 

221.7964 

3914.7072 

.3 

183 1549 

2669.4820 

.5 

202.6327 

3267.4527 

.7 

222.1106 

3925.8049 

.4 

183.4690 

2678.6476 

.6 

202.9469 

3277.5922 

.8 

222.4248 

3936.9182 

.6 

183.7832 

2687.8289 

.7 

203.2610 

3287.7474 

.9 

222.7389 

3948.0473 

.6 

184.0973 

2697.0259 

.8 

203.5752 

3297.9183 

71.0 

223.0531 

3959.1921 

.7 

184.4115 

2706.2386 

.9 

203.8894 

3308.1049 

.1 

223.3672 

3970.3526 

.8 

184.7256 

2715.4670 

65.0 

204.2035 

3318.3072 

.2 

223.6814 

3981.5289 

.9 

185.0398 

2724.7112 

.1 

204.5177 

3328.5253 

.3 

223.9956 

3992.7208 

69.0 

185.3540 

2733.9710 

.2 

204.8318 

3338.7590 

.4 

224.3097 

4003.9284 

.1 

185.6681 

2743.2466 

.3 

205.1460 

3349.0085 

.5 

224.6239 

4015.1518 

.2 

185.9823 

2752.5378 

.4 

205.4602 

3359.2736 

.6 

224.9380 

4026.3908 

.3 

186.2964 

2761.8448 

.5 

205.7743 

3369.5545 

.7 

225.2522 

4037.6456 

.4 

186.6106 

2771.1675 

.6 

206.0885 

3379.8510 

.8 

225.5664 

4048.9160 

.5 

186.9248 

2780.5058 

.7 

206.4026 

3390.1633 

.9 

225.8805 

4060.2022 

.6 

187.2389 

2789.8599 

.8 

206 7168 

3400.4913 

72.0 

226.1947 

4071.5041 

.7 

187 5531 

2799.2297 

.9 

207.0310 

3410.8350 

.1 

226.5088 

4082.8217 

.8 

187.8672 

2808.6152 

66.0 

207.3451 

3421.1944 

.2 

226.8230 

4094.1550 


188.1814 

2818.0165 

.1 

207.6593 

3431.5695 

.3 

227.1371 

4105.5040 

60.0 

188.4956 

2827.4334 

.2 

207.9734 

3441.9603 

.4 

227.4513 

4116.8687 

.1 

188.8097 

2836.8660 

.3 

208.2876 

3452.3669 

.5 

227 7655 

4128.2491 

o_\ 

189.1239 

2846.3144 

.4 

208.6018 

3462.7891 

.6 

228.0796 

4139.6452 

.3 ! 

189.4380 

2855.7784 

.5 

208.9159 

3473.2270 

.7 

228.3938 

4151.0571 

.4 

189.7522 

2865.2582 

.6 

209.2301 

3483.684)7 

.8 

228.7079 

4162.4846 

.5 

190.0664 

2874.7536 

.7 

209.5442 

3494.1500 

.9 

229.0221 

4173.9279 

.6 

190 3805 

2884.2648 

.8 

209.8584 

3504.6351 

73.0 

229.3363 

4185.3868 

./ | 

11)0.6947 

2893.7917 

.9 

210.1725 

3515.1359 

.1 

229.6504 

4196.8615 

.8 

191.0088 

2903.3343 

67.0 

210.4867 

3525.6524 

.2 

229.9646 

4208.3519 

.9 

191.3230 

2912.8926 

.1 

210.8009 

3536.1845 

.3 

230.2787 

4219.8579 

61.0 

191.6372 

2922.4666 

.2 

211.1150 

3546.7324 

.4 

230.5929 

4231.3797 

.1 

191.9513 

2932.0563 

.3 

211.4292 

3557.2960 

.5 

230.9071 

4242.9172 

.2 

192.2655 

2941.6617 

.4 

211.7433 

3567.8754 

.6 

231.2212 

4254.4704 

.3 

192.5796 

2951 2828 

.5 

212.0575 

3578.4704 

.7 

231.5354 

4266.0394 

.4 

192.8938 

2960.9197 

.6 

212.3717 

3589.0811 

.8 

231.8495 

4277.6240 

.5 

193.2079 

2970.5722 

.7 

212.6858 

3599.7075 

.9 

232.1637 

4289.2243 

.6 

193.5221 

2980.2405 

.8 

213.0000 

3610.3497 

74.0 

232.4779 

4300.8403 

.7 

193.8363 

2989.9244 

.9 

213.8141 

3621.0075 

.1 

232.7920 

4312.4721 

.8| 

194.1504 

2999.6241 

68.0 

213.6283 

3631.6811 

.2 

233.1062 

4324.1195 

.9' 

194.4646 

3009.3395 

.1 

213.9425 

3642.3704 

.3 

233.4203 

4335.7827 

6*2.0 

194.7787 

3019.0705 

.2 

214.2566 

3653.0754 

.4 

233.7345 

1 

4347.4616 









































132 CIRCLES. 


TABLE 2 OF CIRCLES—(Continued). 
Diameters in units and tenths 


Din. 

Circumf. 

Area. 

Dia. 

Circumf. 1 

Area. 

Dia. 

Circumf. 

A rea. 

1 

74.5 

234.0487 

4359.1562 

80.7 

253.5265 

5114.8977 

86.9 

273.0044 

5931.0206 

.6 

234.3628 

4370.8664 

.8 

253.8407 

5127.5819 

87.0 

273.3186 

5944.6787 

.7 

234.6770 

4382.5924 

.9 

254.1548 

5140.2818 

.1 

273.6327 

5958.3525 

.8 

234.9911 

4394.3341 

81.0 

254.4690 

5152.9974 

2 

273.9469 

5972.0420 

.9 

235.3053 

4406.0916 

.1 

254.7832 

5165.7287 

.3 

274.2610 

5985.7472 

75.0 

235.6194 

4417.8647 

.2 

255.0973 

5178.4757 

.4 

274.5752 

5999.4681 

.1 

235.9336 

4429.6535 

.3 

255.4115 

5191.2384 

.5 

274.8894 

6013.2047 

.2 

236.2478 

4441.4580 

.4 

255.7256 

5204.0168 

.6 

275.2035 

6026.9570 

.3 

236.5619 

4453.2783 

.5 

256.0398 

5216.8110 

.7 

275.5177 

6040 7250 f 

.4 

236.8761 

4465.1142 

.6 

256.3540 

5229.6208 

.8 

275.8318 

6054.5088 

.5 

237.1902 

4476.9659 

.7 

256.6681 

5242.4463 

.9 

276.1460 

6068.3082 

.6 

237.5044 

4488 8332 

.8 

256.9823 

5255.2876 

88.0 

276.4602 

6082.1234 

.7 

237.8186 

4500.7163 

.9 

257.2964 

5268.1446 

.1 

276.7743 

6095.9542 

.8 

238.1327 

4512.6151 

82.0 

257.6106 

5281.0173 

.2 

277.08,'-5 

6109 8008 

.9 

238.4469 

4524.5296 

.1 

257.9248 

5293.9056 

.3 

277.4026 

6123.6631 II 

76.0 

238.7610 

4536.4598 

.2 

258.2389 

5306.8097 

.4 

277.7168 

6137.5411 

.1 

239.0752 

4548.4057 

.3 

258.5531 

5319.7295 

.5 

278.0309 

6151.4348 

.2 

239.3894 

4560.3673 

.4 

258.8672 

5332.6650 

.6 

278.3451 

6165.3442 

.3 

239.7035 

4572.3446 

.5 

259.1814 

5345.6162 

.7 

278.6593 

6179.2693 

.4 

240.0177 

4584.3377 

.6 

259.4956 

5358.5832 

.8 

278.9734 

6193 2101 

.5 

240.3318 

4596.3464 

.7 

259.8097 

5371.5658 

.9 

279.2876 

6207.1666 

.6 

240 6460 

4608.3708 

.8 

260.1239 

5384.5641 

89.0 

279.6017 

6221.1389 

.7 

240.9602 

4620.4110 

.9 

260.4380 

5397.5782 

.1 

279.9159 

6235.1268 

.8 

241.2743 

4632.4669 

83.0 

260.7522 

5410.6079 

.2 

280.2301 

6249.1304 

.9 

241.5885 

4614.5384 

.1 

261.0663 

5423.6534 

.3 

280.5442 

6263.1498 

77.0 

241.9026 

4656.6257 

.2 

261.3805 

5436.7146 

.4 

280.8584 

6277.1849 

.1 

242.2168 

4668.7287 

.3 

261.6947 

5449.7915 

.5 

281.1725 

6291 2356 

.2 

242.5310 

4680.8474 

.4 

262.0088 

5462.8840 

.6 

281.4867 

6305.3021 

.3 

242.8451 

4692.9818 

.5 

262.3230 

5475.9923 

.7 

281.8009 

6319.3843 

.4 

243.1593 

4705.1319 

.6 

262.6371 

5489.1163 

.8 

282.1150 

6333.4822 

.5 

243.4734 

4717.2977 

.7 

262.9513 

5502.2561 

.9 

282.4292 

6347.5958 

.6 

243.7876 

4729.4792 

.8 

263.2655 

5515.4115 

90.0 

282.7433 

6361.7251 

.7 

244.1017 

4741.6765 

.9 

263.5796 

5528.5826 

.1 

283.0575 

6375.8701 

.8 

244.4159 

4753.8894 

84.0 

263.8938 

5541.7694 

.2 

283.3717 

6390.0309 

.9 

244.7301 

4766.1181 

.1 

264.2079 

5554.9720 

.3 

283.6858 

6404.2073 

78.0 

245.0442 

4778.3624 

.2 

264.5221 

5568.1902 

.4 

284.0000 

6418.3995 

.1 

245.3584 

4790.6225 

.3 

264.8363 

5581.4242 

.5 

284.3141 

6432.6073 

.2 

245.6725 

4802.8983 

.4 

265.1504 

5594.6739 

.6 

284.6283 

6446.8309 

.3 

245.9867 

4815.1897 

.5 

265.4646 

5607.9392 

.7 

284.9425 

6461.0701 

.4 

216.3009 

4827.4969 

.6 

265.7787 

5621.2203 

.8 

285.2566 

6475.3251 

.5 

246.6150 

4839.8198 

.7 

266.0929 

5634.5171 

.9 

285.5708 

6489.5958 

.6 

246.9292 

4852.1584 

.8 

266.4071 

5647.8296 

91.0 

285.8849 

6503.8822 

.7 

247.2433 

4864.5128 

.9 

266.7212 

5661.1578 

.1 

286.1991 

6518.1843 

.8 

247.5575 

4876.8828 

85.0 

267.03-54 

5674.5017 

.2 

286.5133 

6532.5021 

.9 

247.8717 

4889.2685 

.1 

267.3495 

5687.8614 

.3 

286.8274 

6546.8356 

79.0 

248.1858 

4901.6699 

.2 

267.6637 

5701.2367 

.4 

287.1416 

6561.1848 

.1 

248.5000 

4914.0871 

.3 

267.9779 

5714.6277 

.5 

287.4557 

6575.5498 

.2 

248.8141 

4926.5199 

.4 

268.2920 

5728.0345 

.6 

287.7699 

6589.9304 

.3 

249.1283 

4938.9685 

.5 

268.6062 

5741.4569 

.7 

288.0840 

6604.3268 

.4 

249.4425 

4951.4328 

.6 

268.9203 

5754.8951 

.8 

288.3982 

6618.7388 

.5 

249.7566 

4963.9127 

.7 

269.2345 

5768.3490 

.9 

288.7124 

6633.1666 

.6 

250.0708 

4976.4084 

.8 

269.5486 

5781.8185 

92.0 

289.0265 

6647.6101 

.7 

250.3849 

4988.9198 

.9 

269.8628 

5795.3038 

.1 

289.3407 

6662.0692 

.8 

250.6991 

5001.4469 

86.0 

270.1770 

5808.8048 

.2 

289.6548 

6676.5441 

.9 

251.0133 

5013.9897 

.1 

270.4911 

5822.3215 

.3 

289.9690 

6691.0347 

80.0 

251.3274 

5026.5482 

.2 

270.8053 

5835.8539 

.4 

290.2832 

6705.5410 

.1 

251.6416 

5039.1225 

.3 

271.1194 

5849.4020 

.5 

290.5973 

6720.0630 

.2 

251.9557 

5051.7124 

.4 

271.4336 

5862.9659 

.6 

290.9115 

6734.6008 

.3 

252.2699 

5064.3180 

.5 

271.7478 

5876.54,54 

.7 

291.2256 

6749.1542 

.4 

252 5840 

5076.9394 

.6 

272.0619 

5890.1407 

.8 

291.5398 

6763.7233 

.5 

252.8982 

5089.5764 

.7 

272.3761 

5903.7516 

.9 

291.8540 

6778.3082 

.6 

253.2124 

5102.2292 

.8 

272.6902 

5917.3783 

93.0 

292.1681 

6792.9087 














































CIRCLES, 


133 


TABLE 2 OF CIRCLES— (Continued). 


Diameters in units and tenths. 


I)ia. 

Circumf. 

Area. 

Dia. 

Circumf. 

Area. 

l>ia. 

Circumf. 

Area. 

93.1 

292.4823 

6807.5250 

95.5 

300.0221 

7163.0276 

97.8 

307.2478 

7512.2078 

.2 

292.7964 

6822.1569 

.6 

300.3363 

7178.0366 

.9 

307.5619 

7527.5780 

.3 

293.1106 

6836.8046 

.7 

300.6504 

7193.0612 

98.0 

307.8761 

7542.9640 

.4 

293.4248 

6851.4680 

.8 

300.9646 

7208.1016 

.1 

308.1902 

7558.3656 

.5 

293.7389 

6866.1471 

.9 

301.2787 

7223.1577 

.2 

308.5044 

7573.7830 

.6 

294.0531 

6880.8419 

90.0 

301.5929 

7238.2295 

.3 

308.8186 

7589.2161 

.7 

294.3672 

6895.5524 

.1 

301.9071 

7253.3170 

.4 

309.1327 

7604.6648 

.8 

294.6814 

6910.2786 

.2 

302.2212 

7268.4202 

.5 

309.4469 

7620.1293 

.9 

294.9956 

6925.0205 

.3 

302.5354 

7283.5391 

.6 

309.7610 

7635.6095 

94.0 

295.3097 

6939.7782 

.4 

302.8495 

7298.6737 

.7 

310.0752 

7651.1054 

.1 

295.6239 

6954.5515 

.5 

303.1637 

7313.8240 

.8 

310.3894 

7666.6170 

.2 

295.9380 

6969.3406 

.6 

303.4779 

7328.9901 

.9 

310.7035 

7682.1444 

.3 

296.2522 

6984.1453 

.7 

303.7920 

7344.1718 

99.0 

311.0177 

7697.6874 

.4 

296.5663 

6998.9658 

.8 

304.1062 

7359.3693 

.1 

311.3318 

7713.2461 

.5 

296.8805 

7013.8019 

.9 

304.4203 

7374.5824 

,2 

311.6460 

7728.8206 

.6 

297.1947 

7028.6538 

97.0 

304.7345 

7389.8113 

.3 

311.9602 

7744.4107 

.7 

297.5088 

7043.5214 

.1 

305.0486 

7405.0559 

.4 

312.2743 

7760.0166 

.8 

297.8230 

7058.4047 

.2 

305.3628 

7420.3162 

.5 

312.5885 

7775.6382 

.9 

298.1371 

7073.3037 

.3 

305.6770 

7435.5922 

.6 

312.9026 

7791.2754 

95.0 

298.4513 

7088.2184 

.4 

305.9911 

7450.8839 

.7 

313.2168 

7806.9284 

.1 

298.7655 

7103.1488 

.5 

306.3053 

7466.1913 

.8 

313.5309 

7822.5971 

.2 

299.0796 

7118.0950 

.6 

306.6194 

7481.5144 

.9 

313.8451 

7838.2815 

.3 

.4 

299.3938 

299.7079 

7133.0568 

7148.0343 

.7 

306.9336 

7496.8532 

100.0 

•* 

314.1593 

7853.9816 


Circumferences when the diameter has more than one 

place of decimals. 


Diam. 

Circ. 

Diam. 

Circ. 

Diam. 

Circ. 

Diam. 

Circ. 

1 

Diam. 

Circ. 

.1 

.314159 

.01 

.031416 

.001 

.003142 

.0001 

.000314 

.00001 

.000031 

.2 

.628319 

.02 

.062832 

.002 

.006283 

.0002 

.000628 

.00002 

.000063 

.3 

.942478 

.03 

.094248 

.003 

.009425 

.0003 

.000942 

.00003 

.000094 

.4 

1.256637 

.04 

.125664 

.004 

.012566 

.0004 

.001257 

.00004 

.000126 

.5 

1 570796 

.05 

.157080 

.005 

.015708 

.0005 

.001571 

.00005 

.000157 

.6 

1.884956 

.06 

.188496 

.006 

.018850 

.0006 

.001885 

.00006 

.000188 

.7 

2.199115 

1 .07 

.219911 

.007 

.021991 

.0007 

.002199 

.00007 

.000220 

.8 

2.513274 

1 .08 

251327 

.008 

.025133 

.0008 

.002513 

.00008 

.000251 

.9 

2.827433 

.09 

.282743 

.009 

.028274 

.0009 

.002827 

.00009 

.000283 


Examples. 


Diameter = 3.12699 
Circumference = 

Sum of 

Circumfee = 
Diameter = 

9.823729 

Sum of 

Circ for dia of 3.1 

= 9.738937 

Dia for circ of 

9.738937 = 

3.1 

“ .02 

“ .006 

= .062832 
= .018850 

U 

.084792 
.062832 = 

.02 

“ .0009 

“ .00009 

= .002827 
= .000283 

a 

.021960 
.018850 = 

.006 


9.823729 

«( 

.003110 
.002827 = 

.000283 
.000283 = 

.0009 

.00009 

3.12699 































































134 


CIRCLES 




TABLE 3 OF CIRCLES. 

Diains in units ami twelfths; as in feet and inches. 




I)i». 

Circumf. 

Area. 

Dia. 

Circumf. 

Area. 

. 

Dia. 

Circumf. 

| 

I Area. 

Ft.I n. 

Feet. 

Sq. ft. 

Ft. I n. 

Feet. 

Sq. ft. 

Ft.In. 

Feet. 

Sq. ft. 




5 

0 

15.70796 

19.63495 

10 0 

31.41593 

78.53982 

0 1 

.261799 

.005454 


1 

15.96976 

20.29491 

1 

31.67773 

79.85427 

2 

.523599 

.021817 


2 

16.23156 

20.96577 

2 

31.93953 

81.1796? 

3 

.785398 

.049087 


3 

16.49336 

21.64754 

3 

32.20132 

82.51589 

4 

1.047198 

.087266 


4 

16.75516 

22.34021 

4 

32.46312 

83.86307 

5 

1.308997 

.136354 


5 

17.01696 

23.04380 

5 

32.72492 

86.22115 

6 

1.570796 

.196350 


6 

17.27876 

23.75829 

6 

32.98672 

86.59015 

7 

1.832596 

.267254 


7 

17.54056 

24.48370 

7 

33.24852 

87.97005 1 

8 

2.094395 

.349066 


8 

17.80236 

25.22001 

8 

33.51032 

89.36086 

9 

2.356195 

•441786 


9 

18.06416 

25.96723 

9 

33.77212 

90.76258 

10 

2.617994 

.545415 


10 

18.32596 

26.72535 

10 

34.03392 

92.17520 

11 

2.879793 

.659953 


11 

18.58776 

27.49439 

11 

34.29572 

93.59874 

1 0 

3.14159 

.785398 

6 

0 

18.84956 

28.27433 

11 0 

34.55752 

95.03318 

1 

3.40339 

.921752 


1 

19.11136 

29.06519 

1 

34.81932 

96.47853 

2 

3.66519 

1.06901 


2 

19.37315 

29.86695 

2 

35.08112 

97.93479 

3 

3.92699 

1.22718 


3 

19.63495 

30.67962 

3 

35.34292 

99.40196 

4 

4.18879 

1.39626 


4 

19.89675 

31.50319 

4 

35.60472 

100.8800 

5 

4.45059 

1.57625 


5 

20.15855 

32.33768 

5 

35.86652 

102.3690 

6 

4.71239 

1.76715 


6 

20.42035 

33.18307 

6 

36.12832 

103.8689 

7 

4.97419 

1.96895 


7 

20.68215 

34.03937 

7 

36.39011 

105.3797 

8 

5.23599 

2.18166 


8 

20.94395 

34.90659 

8 

36.65191 

106.9014 

9 

5.49779 

2.40528 


9 

21.20575 

35.78470 

9 

36.91371 

108.4340 

10 

5.75959 

2.63981 


10 

21.46755 

36.67373 

10 

37.17551 

109.9776 

11 

6.02139 

2.88525 


11 

21.72935 

37.57367 

11 

37.43731 

111.5320 

2 0 

6.28319 

3.14159 

7 

0 

21.99115 

38.48451 

12 0 

37.69911 

113.0973 

1 

6.54498 

3.40885 


1 

22.25295 

39.40626 

1 

37.96091 

114.6736 

2 

6.80678 

3.68701 


2 

22.51475 

40.33892 

2 

38.22271 

116.2607 

3 

7.06858 

3.97608 


3 

22.77655 

41.28249 

3 

38.48451 

117.8588 

4 

7.33038 

4.27606 


4 

23.03835 

42.23697 

4 

38.74631 

119.4678 

5 

7.59218 

4.58694 


5 

23.30015 

43.20235 

5 

39.00811 

121.0877 

6 

7.85398 

4.90874 


6 

23.56194 

44.17865 

6 

39.26991 

122.7185 

7 

8.11578 

5.24144 


7 

23.82374 

45.16585 

7 

39.53171 

124.3602 

8 

8.37758 

5.58505 


8 

24.08554 

46.16396 

8 

39.79351 

126.0128 

9 

8.63938 

5.93957 


9 

24.34734 

47.17298 

9 

40.05531 

127.6763 

10 

8.90118 

6.30500 


10 

24.60914 

48.19290 

10 

40.31711 

129.3507 

11 

9.16298 

6.68134 


11 

24.87094 

49.22374 

11 

40.57891 

131.0360 

3 0 

9.42478 

7.06858 

8 

0 

25.13274 

50.26548 

13 0 

40.84070 

132.7323 

1 

9.68658 

7.46674 


1 

25.39454 

51.31813 

1 

41.10250 

134.1394 

2 

9.94838 

7.87580 


2 

25.65634 

52.38169 

2 

41.36430 

136.1575 

3 

10.21018 

8.29577 


3 

25.91814 

53.45616 

3 

41.62610 

137.8865 

4 

10.47198 

8.72665 


4 

26.17994 

54.54154 

4 

41.88790 

139.6263 

5 

10.73377 

9.16813 


5 

26.44174 

55.63782 

5 

42.14970 

141.3771 

6 

10.99557 

9.62113 


6 

26.70354 

56.74502 

6 

42.41150 

143.1388 

7 

11.25737 

10.08473 


7 

26.96534 

57.86312 

7 

42.67330 

144.9114 

8 

11.51917 

10.55924 


8 

27.22714 

58.99213 

8 

42.93510 

146.6949 

9 

11.78097 

11.04466 


9 

27.48894 

60.13205 

9 

43.19690 

148.4893 

10 

12.04277 

11.54099 


10 

27.75074 

61.28287 

10 

43.45870 

150 2947 

11 

12.30457 

12.04823 


11 

28.01253 

62.44461 

11 

43.72050 

152 1109 

4 0 

12.56637 

12.56637 

9 

0 

28.27433 

63.61725 

14 0 

43.98230 

153.9380 

1 

12.82817 

13.09542 


1 

28.53613 

64.80080 

1 

44.24410 

155.7761 

2 

13.08997 

13.63538 


2 

28.79793 

65.99526 

2 

44.50590 

157.6250 

3 

13.35177 

14.18625 


3 

29 05973 

67.20063 

3 

44.76770 

159 4849 

4 

13.61357 

14.74803 


4 

29.32153 

68.41691 

4 

45.02949 

161 3557 

5 

l3.8/f>37 

15.32072 


5 

29.58333 

69.64409 

5 

45.29129 

163.2374 

6 

14.13717 

15.90431 


6 

29.84513 

70.88218 

6 

45.55309 

165 1300 

7 

14.39897 

16.49882 


7 

30.10693 

72.13119 

7 

45.81489 

167 0335 

8 

14.66077 

17.10423 


8 

30.36873 

73.39110 

8 

46.07669 

168.9479 

9 

14.92257 

17.72055 


9 

30.63053 

74.66191 

9 

46.33849 

170 8732 

10 

1 ■>. 1X436 

18.34777 


10 

30.89233 

75.94364 

10 

46.60029 

172 8094 


15.44616 

18.98591 


11 

31.15413 

77.23627 

11 

46.86209 

174.7565 








































1* 

n. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

LI 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

L0 

LI 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 


CIRCLES 


135 


TABLE 3 OF CIRCLES— (Continued). 


units and twelfths; as in feet and incites. 


Area. 

Dia. 

Circmuf. 

Area. 

Dia. 

Cireumf. 

Area. 

Sq. ft. 

Ft.In. 

Feet. 

Sq. ft. 

Ft.Iu. 

Feet. 

Sq. ft. 

176.7146 

‘20 0 

62.83185 

314.1593 

25 0 

78.53982 

490.8739 

178.6835 

1 

63.09365 

316.7827 

1 

78.80162 

494.1518 

180.6634 

2 

63.35545 

319.4171 

2 

79.06342 

497.4407 

182.6542 

3 

63.61725 

322.0623 

3 

79.32521 

500.7404 

184.6558 

4 

63.87905 

324.7185 

4 

79.58701 

504.0511 

186.6684 

5 

64.14085 

327.3856 

5 

79.84881 

507.3727 

188.6919 

6 

64.40265 

330.0636 

6 

80.11061 

510.7052 

190.7263 

7 

64.66445 

332.7525 

7 

80.37241 

514.0486 

192.7716 

8 

64.92625 

335.4523 

8 

80.63421 

517.4029 

194.8278 

9 

65.18805 

338.1630 

9 

80.89601 

520.7681 

196.8950 

10 

65.44985 

340.8816 

10 

81.15781 

524.1442 

198.9730 

11 

65.71165 

343.6172 

11 

81.41961 

527.5312 

201.0619 

21 0 

65.97345 

346.3606 

26 0 

81.68141 

530.9292 

203.1618 

1 

66.23525 

349.1149 

1 

81.94321 

534.3380 

205.2725 

2 

66.49704 

351.8802 

2 

82.20501 

537.7578 

207.3942 

3 

66.75884 

354.6564 

3 

82.46681 

541.1884 

209.5268 

4 

67.02064 

357.4434 

4 

82.72861 

544.63(H) 

211.6703 

5 

67.28244 

360.2414 

5 

82.99041 

518.0825 

213.8246 

6 

67.54424 

363.0503 

6 

83.25221 

551.5459 

215.9899 

7 

67.80604 

365.8701 

7 

83.51400 

555.0202 

218.1662 

8 

68.06784 

368.7008 

8 

83.77580 

558.5054 

220.3533 

9 

68.32964 

371.5424 

9 

84.03760 

562.0015 

222.5513 

10 

68.59144 

374.3949 

10 

84.29940 

565.5085 

224.7602 

11 

68.85324 

377.2584 

11 

84.56120 

569.026)4 

226.9801 

22 0 

69.11504 

380.1327 

27 0 

84.82:100 

572.5553 

229.2108 

1 

69.37684 

383.0180 

1 

85.08480 

576.0950 

231.4525 

2 

69.63864 

385.9141 

2 

85.34660 

579.6457 

233.7050 

3 

69.90044 

388.8212 

3 

85.60840 

583.2072 

235.9685 

4 

70.16224 

391.7392 

4 

85.87020 

586.7797 

238.2429 

5 

70.42404 

394.6680 

5 

86.13200 

590.3631 

240.5282 

6 

70.68583 

397.6078 

6 

86.39380 

593.9574 

242.8244 

7 

70.94763 

400.5585 

7 

86.65560 

597.5626 

245.1315 

8 

71.20943 

403.5201 

8 

86.91740 

601.1787 

247.4495 

9 

71.47123 

406.4926 

9 

87.17920 

601.8057 

249.7784 

10 

71.73303 

409.4761 

10 

87.44100 

608.4436 

252.1183 

11 

71.99483 

412.4704 

11 

87.70279 

612.0924 

254.4690 

ti 

09 

© 

72.25663 

415.4756 

28 0 

87.96459 

615.7522 

256.8307 

1 

72.51843 

418.4918 

1 

88.22639 

619.4228 

259.2032 

2 

72.78023 

421.5188 

2 

88.48819 

623.1044 

261.5867 

3 

73.04203 

424.5568 

3 

88.74999 

626.7968 

263.9810 

4 

73.30383 

427.6057 

4 

89.01179 

630.5002 

266.3863 

5 

73.56563 

430.6654 

5 

89.27359 

634.2145 

268.8025 

6 

73.82743 

433.7361 

6 

89.53539 

637.9397 

271.2296 

7 

74.08923 

436.8177 

7 

89.79719 

641.6758 

273.6676 

8 

74.35103 

439.9102 

8 

90.05899 

645.4228 

276.1165 

9 

74.61283 

443.0137 

9 

90.32079 

649.1807 

278.5764 

10 

74.87462 

446.1280 

10 

90.58259 

652.9495 

281.0471 

11 

75.13642 

449.2532 

11 

90.84439 

656.7292 

283.5287 

24 0 

75.39822 

452.3893 

29 0 

91.10619 

660.5199 

286.0213 

1 

75.66002 

455.5364 

1 

91.36799 

664.3214 

288.5247 

2 

75.92182 

458.6943 

2 

91.62979 

668.1339 

291.0391 

3 

76.18362 

461.8632 

3 

91.89159 

671.9572 

293.5644 

4 

76.44542 

465.0430 

4 

92.15338 

675.7915 

296.1006 

5 

76.70722 

468.2337 

5 

92.41518 

679.6367 

298.6477 

6 

76.96902 

471.4352 

6 

92.67698 

683.4928 

301.2056 

7 

77.23082 

474.6477 

7 

92.93878 

687.3597 

303.7746 

8 

77.49262 

477.8711 

8 

93.20058 

691.2377 

306.3544 

9 

77.75442 

481.1055 

9 

93.46238 

695.1265 

308.9451 

10 

78.01622 

484.3507 

10 

93.72418 

699.0262 

311.5467 

11 

78.27802 

487.6068 

11 

93.98598 

702.9368 































136 


CIRCLES, 


TABLE 3 OF CIRCLES—(Continued). 


Diams in units and twelfths; as in feet and inches. 


Ilia. 

j Circumf, 

Area. 

Dia. 

Circumf. 

Area. 

Dia. 

Circumf. 

A rea. 

Ft. In 

30 0 
1 
2 

3 

4 

5 

6 

7 

8 
9 

10 

n 

31 0 
1 
2 

3 

4 

5 

6 

7 

8 
9 

10 

11 

32 0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

33 0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

34 0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

Feet. 

94.24778 
94.50958 
94.77138 
95.0:1318 
95.29498 
95.55678 
95.81858 
96.080:18 
96.34217 
96.60397 
96.86577 
97.12757 
97.38937 
97.65117 
97.91297 
98.17477 
98.43657 
98.69837 
98.96017 
99.22197 
99.48377 
99.74557 
100.0074 
100.2692 
100.5310 
100.7928 
101.0546 
101.3164 
101.5782 
101.8400 
102.1018 
102.3636 
102.6254 
102.8872 
103.1490 
103.4108 
103.6726 
103,9314 
104.1962 
104.4580 
101.7198 
104.9816 
105.24:14 
105.5052 
105.7670 
106.0288 
106.2906 
106 5524 
106.8142 
107.0759 
107.3377 
107.5995 
107.8613 
108.1231 
108.3849 
108.6467 
108.9085 
109.1703 
109.4321 
109.6939 

Sq. ft. 

706.8583 

710.7908 

714.7341 

718.6884 

722.6536 

726.6297 

730.6166 

731.6145 

738.6233 

742.6431 

746 6737 

750.7152 

754.7676 

758.8310 

762.9052 

766.9904 

771.0865 

775.1934 

779.3113 

783.4401 

787.5798 

791.7304 

795.8920 

800.0614 

804.2477 

808.4420 

812.6471 

816.8632 

821.0901 

825.3280 

829.5768 

833.8365 

838.1071 

812.3886 

846.6810 

850.9844 

855.2986 

859.6237 

863.9598 

868.3068 

872.6616 

877.0334 

881.4131 

885.8037 

890.2052 

894.6176 

899.0409 

903.4751 

907.9203 

912.3763 

916.8433 

921.3211 

925.8099 

930.3096 

934.8202 

939.3417 

943.8741 

948.4174 

952.9716 

957.5367 

Ft. In 

So 0 
1 
2 

3 

4 

5 

6 

7 

8 
9 

10 

11 

36 0 
1 
2 

3 

4 

5 

6 

7 

8 
9 

10 

11 

37 0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

38 0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

39 0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

F'eet. 

109.9557 

110.2175 

110.4793 

110.7411 

111.0029 

111.2647 

111.5265 

111.7883 

112.0501 

112.3119 

112.5737 

112.8355 

113.0973 

113.3591 

113.6209 

113.8827 

114.1445 

114.4063 

114.6681 

114.9299 

115.1917 

115.4535 

115.7153 

115.9771 

116.2389 

116.5007 

116.7625 

117.0243 

117.2861 

117.5479 

117.8097 

118.0715 

118.3333 

118.5951 

318.8569 

119.1187 

119.3805 

119.6423 

li 9.9041 

120.1659 

120.4277 

120.6895 

120.9513 

121.2131 

121.4749 

121.7367 

121.9985 

122.2603 

122.5221 

122.7839 

123.0457 

123.3075 

123.5693 

123.8311 

124.0929 

124.3517 

124.6165 

124.8783 

125.1401 

125.4019 

Sq. ft. 

962.1128 

966.6997 

971.2975 

975.9063 

980.5260 

985.1566 

989.7980 

994.4504 

999.1137 

1003.7879 

1008.4731 

1013.1691 

1017.8760 

1022.5939 

1027.3226 

1032.0623 

1036.8128 

1041.5743 

1046.3467 

1051.1300 

1055.9242 

1060.7293 

1065.5453 

1070.3723 

1075.2101 

1080.0588 

1084.9185 

1089.7890 

1091.6705 

1099.5629 

1104.4662 

1109.3804 

1114.3055 

1119.2415 

1124.1884 

1129.1462 

1134.1149 

1139.0946 

1144.0851 

1149.0866 

1154.0990 

1159.1222 

1164.1564 

1169.2015 

1174.2575 

1179.3244 

1181.4022 

1189.4910 

1194.5906 

1199.7011 

1204.8226 

1209.9550 

1215.0982 

1220.2524 

1225.4175 

1230.5935 

1235.7804 

1240.9782 

1246.1869 

1251.4065 

FtJ.ii. 

40 0 
1 

2 

3 

4 

5 

6 

7 

8 
9 

10 

11 

41 0 
1 
2 

3 

4 

5 

6 

7 

8 
9 

10 

11 

42 0 

1 

2 

3 

4 

5 

6 

7 

8 
9 

10 

11 

43 0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

44 0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

Feet. 

125.6637 

125.9255 

126.1873 

126.4491 

126.7109 

126.9727 

127.2345 

127.4963 

127.7581 

128.0199 

128.2817 

128.5135 

128.8053 

129.0671 

129.3289 

129.5907 

129.8525 

130.1143 

130.3761 

130.6379 

130.8997 

131.1615 

131.4233 

131.6851 

131.9469 

132.2087 

132.4705 

132.7323 

132.9941 

133.2559 

133.5177 

133.7795 

134.0413 

134.3031 

134.5649 

134.8267 

135.0885 

135.3503 

135.6121 

135.8739 

136.1357 

136.3975 

136.6593 

136.9211 

137.1829 

137.4447 

137.7065 

137.9683 

138.2301 

138.4919 

138.7537 

139.0155 

139.2773 

139.5391 

139.8009 

140.0627 

140.3245 

110.5863 

140.8481 

141.1099 

Sq. ft. 

1256.6371 

1261.8785 

1267.1309 

1272.3941 

1277.6683 

1282.9534 

1288.2493 

1293.5562 

1298.8740 

1304.2027 

1309.5424 

1314.8929 

1320.2543 

1325.6267 

1331.0099 

1336.4041 

1341.8091 

1347.2251 

1352.6520 

1358.0898 

1363.5385 

1368.9981 

1374.4686 

1379.9500 

1385.4424 

1390.9456 

1396.4598 

1401.9848 

1407.5208 

1413.0676 

1418.6254 

1424.1941 

1429,7737 

1435.3642 

1440.9656 

1446.5780 

1452.2012 

1457.8353 

1463.4804 

1469.1364 

1474.8032 

1480.4810 

1486.1697 

1491.8693 

1497.5798 

1503.3012 

1509.0335 

1514.7767 

1520.5308 

1526.2959 

1532.0718 

1537.8587 

1543.6565 

1549.4651 

1555.2847 

1561.1152 

1566.9566 

1572 8089 

1578.6721 

1584.5462 














































CIRCLES 


137 


TABLE 3 OF CIRCLES— (Continued). 


Diams in units and twelfths ; as in feet and inches. 


Dia. 

Icirciinif. 

Area. 

Dia. 

Circumf 

Area. 

Dia. 

Circumf 

Area. 

Ft.In 

Feet. 

Sq. ft. 

Ft.In 

Feet. 

Sq. ft. 

Ft.In 

Feet. 

Sq. It. 

45 0 

141.3717 

1590.4313 

50 0 

157.0796 

1963.4954 

56 0 

172.7876 

2375.8294 

1 

141.6335 

1596.3272 

1 

157.3414 

1970.0458 

1 

173.0494 

2383.0344 

2 

141.8953 

1602.2341 

2 

157.6032 

1976.6072 

2 

173.3112 

2390.2502 

3 

142.1571 

1608.1518 

3 

157.8650 

1983.1794 

3 

173.5730 

2397.4770 

4 

142 4189 

1614.0805 

4 

158.1268 

1989.7626 

4 

173.8348 

2404.7146 

5 

142.6807 

1620.0201 

5 

158.3886 

1996.3567 

5 

174.0966 

2411.9632 

6 

142.9425 

1625.9705 

6 

158.6504 

2002.9617 

6 

174.3584 

2419.2227 

7 

143.2043 

1631.9319 

7 

158.9122 

2009.5776 

7 

174.6202 

2426.4931 

8 

143.4661 

1637.9042 

8 

159.1740 

2016.2044 

8 

174.8820 

2433.7744 

9 

143.7279 

1643.8874 

9 

159.4358 

2022.842: 

9 

175.1438 

2441.0666 

10 

143.9897 

1649.8816 

10 

159.6976 

2029.4907 

10 

175.4056 

2448 3697 

11 

144.2515 

1655.8866 

11 

159.9594 

2036.1502 

11 

175.6674 

2455.6837 

40 0 

1 144.5133 

1661.9025 

51 0 

160.2212 

2042.8206 

56 0 

175.9292 

2463.0086 

1 

144.7751 

1667.9294 

1 

160.4830 

2049.5020 

1 

176.1910 

2470.3445 

2 

145.0369 

1673.9671 

2 

160.7448 

2056.1942 

2 

176.4528 

2477.6912 

3 

145.2987 

1680.0158 

3 

161.0066 

2062.8974 

3 

176.7146 

2485.0489 

4 

145.5605 

1686.0753 

4 

161.2684 

2069.6114 

4 

176.9764 

2492.4174 

5 

145.8223 

1692.1458 

5 

161.5302 

2076.3364 

5 

177.2:382 

2499.7969 

6 

146.0841 

1698.2272 

6 

161.7920 

2083.0723 

6 

177.5000 

2507.1873 

7 

146.3459 

1704.3195 

7 

162.0538 

2089.8191 

7 

177.7618 

2514.5886 

8 

146.6077 

1710.4227 

8 

162.3156 

2096.5768 

8 

178.0236 

2522.0008 

9 

146.8695 

1716.5368 

9 

162.5774 

2103.3454 

9 

178,2854 

2529.4239 

10 

147.1313 

1722.6618 

10 

162.8392 

2110.1249 

10 

178.5472 

2536.8579 

11 

147.3931 

1728.7977 

11 

163.1010 

2116.9153 

11 

178.8090 

2544.3028 

47 0 

147.6549 

1734.9445 

5*2 0 

163.3628 

2123.7166 

57 0 

179.0708 

2551.7586 

1 

147.9167 

1741.1023 

1 

163.6246 

2130.5289 

1 

179.3326 

2559.2254 

2 

148.1785 

1747.2709 

2 

163.8864 

2137.3520 

2 

179.5944 

2566.7030 

3 

148.4403 

1753.4505 

3 

164.1482 

2144.1861 

3 

179.8562 

2574.1916 

4 

148.7021 

1759.6410 

4 

164.4100 

2151.0310 

4 

180.1180 

2581.6910 

5 

148.9639 

1765.8423 

5 

164.6718 

2157.8869 

5 

180.3798 

2589.2014 

6 

149.2257 

1772.0546 

6 

164.9336 

2164.7537 

6 

180.6416 

2596.7227 

7 

149.4875 

1778.2778 

7 

165.1954 

2171.6314 

7 

180.9034 

2604.2549 

8 

149.7492 

1784.5119 

8 

165.4572 

2178.5200 

8 

181.1652 

2611.7980 

9 

150.0110 

1790.7569 

9 

165.7190 

2185.4195 

9 

181.4270 

2619.3520 

10 

150.2728 

1797.0128 

10 

165.9808 

2192.3299 

10 

181.6888 

2626.9169 

11 

150 5346 

1803.2796 

11 

166.2426 

2199.2512 

11 

181.9506 

2634.4927 

48 0 

150.7964 

1809.5574 

53 0 

166.5044 

2206.1834 

58 0 

182.2124 

2642.0794 

1 

151.0582 

1815.8460 

1 

166.7662 

2213.1266 

1 

182.4742 

2649.6771 

2 

151.3200 

1822.1456 

2 

167.0280 

2220.0806 

2 

182.7360 

2657.2856 

3 

151.5818 

1828.4560 

3 

167.2898 

2227.0456 

3 

182.9978 

2664.9051 

4 

151.8436 

1834.7774 

4 

167.5516 

2234.0214 

4 

183.2596 

2672.5354 

5 

152.1054 

1841.1096 

5 

167.8134 

2241.0082 

5 

183.5214 

2680.1767 

6 

152.3672 

1847.4528 

6 

168.0752 

2248.0059 

6 

183.7832 

2687.8289 

7 

152.6290 

1853.8069 

7 

168.3370 

2255.0145 

7 

184.0450 

2695.4920 

8 

152.8908 

1860.1719 

8 

168.5988 

2262.0340 

8 

184.3068 

2703.1659 

9 

153.1526 

1866.5478 

9 

168.8606 

2269.0644 

9 

184.5686 

2710.8508 

10 

153.4144 

1872.9346 

10 

169.1224 

2276.1057 

10 

184.8304 

2718.5467 

11 

153.6762 

1879.3324 

11 

169.3842 | 

2283.1579 

11 

185.0922 

2726.2534 

49 0 

153.9380 

1885.7410 

54 0 

169.6460 

2290.2210 

59 0 

185.3540 

2733.9710 

1 

154.1998 

1892.1605 

1 

169.9078 

2297.2951 

1 

185.6158 

2741.6995 

2 

154.4616 

1898.5910 

2 

170.1696 I 

2304.3800 

2 

185.8776 

2749.4390 

3 

154 7234 

1905.0323 

3 

170.4314 

2311.4759 

3 

186.1394 

2757.1893 

4 

154.9852 

1911.4846 

4 

170.6932 

2318.5826 

4 

186.4012 

2764.9506 

5 

155 2470 

1917.9478 

5 

170.9550 

2325.7003 

5 

186.6630 

2772.7228 

6 

155.5088 

1924.4218 

6 

171.2168 

2332.8289 

6 

186.9248 

2780.5058 

7 

155.7706 

1930.9068 

7 

171.4786 

2339.9681 

7 

187.1866 

2788.2998 

8 

156.0324 

1937.4027 

8 

171.7404 

2347.1188 

8 

187.4484 

2796.1047 

9 

156.2942 

1943.9095 

9 

172.0022 

2354.2801 

9 

187.7102 

2803.9205 

10 

156.5560 

1950.4273 

10 

172.2640 

2361.4523 

10 

187.9720 

2811.7472 

11 

156.8178 

1956.9559 

11 

172.5258 

2368.6354 

11 

188.2338 

2819.5849 














































I «I • 

Ill. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 


CIRCLES 


TABLE 3 OF CIRCL,ES-(Contimied). 


units and twelfths; as in feet and i 


Area. 

Ilia. 

C’ircunif. 

Area. 

Ilia. 

Circunif. 

Sq. ft. 

Ft.I d. 

Feet. 

Sq. ft. 

Ft.In. 

Feet. 

2827.4334 

65 

0 

204.2035 

3318.3072 

70 0 

2199115 

2835.2928 


1 

204.4653 

3326.8212 

1 

220.1733 

2843.1632 


2 

204.7271 

3335.3460 

2 

220.4351 

2851.0444 


3 

204.9889 

3343.8818 

3 

220.6969 

2858.9366 


4 

205.2507 

3352.4284 

4 

220.9587 

2866.8397 


5 

205.5125 

3360.9860 

5 

221.2205 

2874.7536 


6 

205.7743 

3369.5545 

6 

221.4823 

2882.6785 


7 

206.0361 

3378.1339 

7 

221.7441 

2890.6143 


8 

206.2979 

3386.7241 

8 

222.0059 

2898.5610 


9 

206.5597 

3395.3253 

9 

222.2677 

2906.5186 


10 

206.8215 

3403.9375 

10 

222.5295 

2914.4871 


11 

207.0833 

3412.5605 

11 

222.7913 

2922.4666 

66 

0 

207.3451 

3421.1944 

71 0 

223.0531 

2930.4569 


1 

207.6069 

3429.8392 

1 

223.3149 

2938.4581 


2 

207.8687 

3438.4950 

2 

223.5767 

2946.4703 


3 

208.1305 

3447.1616 

3 

223.8385 

2954.4934 


4 

208.3923 

3455.8392 

4 

224.1003 

2962.5273 


5 

208.6541 

3464.5277 

5 

224.3621 

2970.5722 


6 

208.9159 

3473.2270 

6 

224.6239 

2978.6280 


7 

209.1777 

3481.9373 

7 

224.8857 

2986.6947 


8 

209.4395 

3490.6585 

8 

225.1475 

2994.7723 


9 

209.7013 

3499.3906 

9 

225.4093 

3002.8608 


10 

209.9631 

3508.1336 

10 

225.6711 

3010.9602 


11 

210.2249 

3516.8875 

11 

225.9329 

3019.0705 

67 

0 

210.4867 

3525.6524 

72 0 

226.1947 

3027.1918 


1 

210.7485 

3534.4281 

1 

226.4565 

3035.3239 


2 

211.0103 

3543.2147 

2 

226.7183 

3043.4670 


3 

211.2721 

3552.0123 

3 

226.9801 

305] .6209 


4 

211.5339 

3560.8207 

4 

227.2419 

3059.7858 


5 

211.7957 

3569.6401 

5 

227.5037 

3067.9616 


6 

212.0575 

3578.4704 

6 

227.7655 

3076.1483 


7 

212.3193 

3587.3116 

7 

228.0273 

3084.3459 


8 

212.5811 

3596.1637 

8 

228.2891 

3092.5544 


9 

212.8429 

3605.0267 

9 

228.5509 

3100.7738 


10 

213.1047 

3613.9006 

10 

228.8127 

3109.0041 


11 

213.3665 

3622.7854 

11 

229.0745 

3117.2453 

68 

0 

213.6283 

3631.6811 

73 0 

229.3363 

3125.4974 


1 

213.8901 

3640.5877 

1 

229.5981 

3133.7605 


2 

214.1519 

3649.5053 

2 

229 8599 

3142.0344 


3 

214.4137 

3658.4337 

3 

230.1217 

3150.3193 


4 

214.6755 

3667.3731 

4 

230.3835 

3158.6151 


5 

214.9373 

3676.3234 

5 

230.6453 

3166.9217 


6 

215.1991 

3685.2845 

6 

230.9071 

3175.2393 


7 

215.4609 

3694.2566 

7 

231.1689 

3183.5678 


8 

215.7227 

3703.2396 

8 

231.4307 

3191.9072 


9 

215.9845 

3712.2335 

9 

231.6925 

3200.2575 


10 

216.2463 

3721.2383 

10 

231.9543 

3208.6188 


11 

216.5081 

3730.2540 

11 

232.2161 

3216.9909 

69 

0 

216.7699 

3739.2807 

o 

i'- 

232.4779 

3225.3739 


1 

217.0317 

3748.3182 

1 

232.7397 

3233.7679 


2 

217.2935 

3757.3666 

2 

233.0015 

3242.1727 


3 

217.5553 

3766.4260 

3 

233.2633 

3250.5885 


4 

217.8171 

3775.4962 

4 

233.5251 

3259.0151 


5 

218.0789 

3784.5774 

5 

233.7869 

3267.4527 


6 

218.3407 

3793.6695 

6 

234.0487 

3275.9012 


7 

218.6025 

3802.7725 

7 

234.3105 

3284.3606 


8 

218.8643 

3811.8864 

8 

234.5723 

3292.8309 


9 

219.1261 

3821.0112 

9 

234.8341 

3301.3121 


10 

219.3879 

3830.1469 

10 

235.0959 

3309.8042 


11 

219.6497 

3839.2935 

11 

235.3576 
































u 

a. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

!0 

1 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

0 

.1 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

.1 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

L0 

11 

0 

1 

2 

3 

4 

ft 

6 

7 

8 

9 


CIRCLES 


139 


TABLE 3 OF CIRCLES— (Continued), 
i units ami twelfths; as in feet and inches. 


Area. 

l)ia. 

Circuiiif. 

Area. 

Dia. 

Circumf. 

Area. 

Sq. ft. 

Ft.In. 

Feet. 

Sq. ft. 

Ft.In. 

Feet. 

Sq. ft. 

4417.8647 

80 0 

251.3274 

5026.5482 

85 0 

267.0354 

5674.5017 

4427.6876 

1 

251.5892 

5037.0257 

1 

267.2972 

5685.6337 

4437.5214 

2 

251.8510 

5047.5140 

2 

267.5590 

5696.7765 

4447.3662 

3 

252.1128 

5058.0133 

3 

267.8208 

5707.9302 

4457.2218 

4 

252.3746 

5068.5234 

4 

268.0826 

5719.0949 

4467.0884 

5 

252.6364 

5079.0445 

5 

268.3444 

5730.2705 

4476.9659 

6 

252.8982 

5089.5764 

6 

268.6062 

5741.4569 

4486.8543 

7 

253.1600 

5100.1193 

7 

268.8680 

5752.6543 

4496.7536 

8 

253.4218 

5110.6731 

8 

269.1298 

5763.8626 

4506.6637 

9 

253.6836 

5121.2378 

9 

269.3916 

5775.0818 

4516.5849 

10 

253.9454 

5131.8134 

10 

269.6534 

5786.3119 

4526.5169 

11 

254.2072 

5142.3999 

11 

269.9152 

5797.5529 

4536.4598 

81 0 

254.4690 

5152.9974 

86 0 

270.1770 

5808.8048 

4546.4136 

1 

254.7308 

5163.6057 

1 

270.4388 

5820.0676 

4556.3784 

2 

254.9926 

5174.2249 

2 

270.7006 

5831.3414 

4566.3510 

3 

255.2544 

5184.8551 

3 

270.9624 

5842.6260 

4576.3406 

4 

255.5162 

5195.4961 

4 

271.2242 

5853.9216 

4586.3380 

5 

255.7780 

5206.1481 

5 

271.4860 

5865.2280 

4596.3464 

6 

256.0398 

5216.8110 

6 

271.7478 

5876.5454 

4606.3657 

7 

256.3016 

5227.4847 

7 

272.0096 

5887.8737 

4616.3959 

8 

256.5634 

5238.1694 

8 

272.2714 

5899.2129 

4626.4370 

9 

256.8252 

5248.8650 

9 

272.5332 

5910.5630 

4636.4890 

10 

257.0870 

52595715 

10 

272.7950 

5921.9240 

4646.5519 

11 

257.3488 

5270.2889 

11 

273.0568 

5933.2959 

4656.6257 

82 0 

257.6106 

5281.0173 

87 0 

273.3186 

5944.6787 

4666.7104 

1 

257.8724 

5291.7565 

1 

273.5804 

5956.0724 

4676.8061 

2 

258.1342 

5302.5066 

2 

273.8422 

5967.4771 

4686.9126 

3 

258-3960 

5313.2677 

3 

274.1040 

5978.8926 

4697.0301 

4 

258.6578 

5324.0396 

4 

274.3658 

5990.3191 

4707.1584 

5 

258.9196 

5334.8225 

5 

274.6276 

6001.7564 

4717.2977 

6 

259.1814 

5345.6162 

6 

274.8894 

6013.2047 

4727.4479 

7 

259.4432 

5356.4209 

7 

275.1512 

6024.6639 

4737.6090 

8 

259.7050 

5367.2365 

8 

275.4130 

6036.1340 

4747.7810 

9 

259.9668 

5378.0630 

9 

275.6748 

6047.6149 

4757.9639 

10 

260.2286 

5388.9004 

10 

275.9366 

6059.1068 

4768.1577 

11 

260.4904 

5399.7487 

11 

276.1984 

6070.6097 

4778.3624 

83 0 

260.7522 

5410.6079 

88 0 

276.4602 

6082.1234 

4788.5781 

1 

261.0140 

5421.4781 

1 

276.7220 

6093.6480 

4798.8046 

2 

261.2758 

5432.3591 

2 

276.9838 

6105.1835 

4809.0420 

3 

261.5376 

5443.2511 

3 

277.2456 

6116.7300 

4819.2904 

4 

261.7994 

5454.1539 

4 

277.5074 

6128.2873 

4829.5497 

5 

262.0612 

5465.0677 

5 

277.7692 

6139.8556 

4839.8198 

6 

262.3230 

5475.9923 

6 

278.0309 

6151.4348 

4850.1009 

7 

262.5848 

5486.9279 

7 

278.2927 

6163.0248 

4860.3929 

8 

262.8466 

5497.8744 

8 

278.5545 

6174.6258 

4870 6958 

9 

263.1084 

5,508-8318 

9 

278.8163 

6186.2377 

4881.0096 

10 

263.3702 

5519-8001 

10 

279.0781 

6197.8605 

4891.3343 

11 

263.6320 

5530.7793 

11 

279.3399 

6209.4942 

4901.6699 

84 0 

263.8938 

5541.7694 

89 0 

279.6017 

6221.1389 

4912.0165 

1 

264.1556 

5552.7705 

1 

279.8635 

6232.7944 

4922.3739 

2 

264.4174 

5563.7824 

2 

280.1253 

6244.4608 

4932.7423 

3 

264.6792 

5574.8053 

3 

280.3871 

6256.1382 

4943.1215 

4 

264.9410 

5585.8390 

4 

280.6489 

6267.8264 

4953.5117 

5 

265.2028 

5596.8837 

5 

280.9107 

6279.5256 

4963.9127 

6 

265.4646 

5607.9392 

6 

281.1725 

6291.2356 

4974.3247 

7 

265.7264 

5619.0057 

7 

281.4343 

6302.9566 

4984.7476 

8 

265.9882 

5630.0831 

8 

281.6961 

6314.6885 

4995.1814 

9 

266.2500 

5641.1714 

9 

281.9579 

6326.4313 

5005.6261 

10 

266.5118 

5652.2706 

10 

282.2197 

63:18.1850 

5016.0817 

11 

266.7736 

5663.3807 

11 

282.4815 

6349.9496 



























140 


CIRCLES 




TABLE 3 OF CIRCLES— (Continued). 


Diams in units and twelfths; as in feet and inches. 


Dia. 

1 

Circumf. 

Area. 

Dia. 

Circumf. 

Area. 

Dia. 

Circumf. 

Area. 

Ft.In. 

Feet. 

Sq. ft. 

Ft.In. 

Feet. 

Sq. ft. 

Ft.In. 

Feet. 

Sq. ft. 

90 0 

282.7433 

6361.7251 

93 5 

293.4771 

6853.9134 

90 9 

303.9491 

7351.7686 

1 

283.0051 

6373.5116 

6 

293.7389 

6866.1471 

10 

304.2109 

7364.4386 

2 

283.2669 

6385.3089 

7 

294.0007 

6878.3917 

11 

304.4727 

73/ 7.1195 

3 

283.5287 

6397.1171 

8 

294.2625 

6890.6472 

97 0 

304.7345 

7389.8113 

4 

283.7905 

6108.9363 

9 

294.5243 

6902.9135 

1 

304.9963 

7402.5140 

5 

284.0523 

6120.7663 

10 

294.7861 

6915.1908 

2 

305.2581 

V41<).22 / 7 

6 

284.3141 

6432.6073 

11 

295.0479 

6927.4791 

3 

305.5199 

7427.9522 

7 

284.5759 

6444.4592 

94 0 

295.3097 

6939.7782 

4 

305.7817 

7440.687 / 

8 

284.8377 

6156.3220 

1 

295.5715 

6952.0882 

5 

306.0435 

7453.4340 

9 

285.0995 

6168.1957 

2 

295.8333 

6964.4091 

6 

306.3053 

7466.1913 

10 

285.3613 

6180.0803 

3 

296.0951 

6976.7410 

7 

306.5671 

7478.9595 

11 

285.6231 

6491.9758 

4 

296.3569 

6989.0837 

8 

306.8289 

7491.7385 

91 0 

285.8849 

6503.8822 

5 

296.6187 

7001.4374 

9 

307.0907 

7504.5285 

1 

286.1467 

6515.7995 

6 

296.8805 

7013.8019 

10 

307.3525 

7517.3294 

2 

286.4085 

6527.7278 

7 

297.1423 

7026.1774 

11 

307.6143 

7530.1412 

3 

286.6703 

6539.6669 

8 

297.4041 

7038.5638 

98 0 

307.8761 

7n42.9640 

4 

286.9321 

6551.6169 

9 

297.6659 

7050.9611 

1 

308.1379 

7555.7976 

5 

287.1939 

6563.5779 

10 

297.9277 

7063.3693 

2 

308.3997 

7568.6421 

6 

287.4557 

6575.5498 

11 

298.1895 

7075.7884 

o 

308.6615 

7581.4976 

7 

287.7175 

6587.5325 

95 0 

298.4513 

7088.2184 

4 

308.9233 

7594.3639 

8 

287.9793 

6599.5262 

1 

298.7131 

7100.6593 

5 

309.1851 

7607.2412 

9 

288.2411 

6611.5308 

2 

298.9749 

7113.1112 

6 

309.4469 

7620.1293 

10 

288.5029 

6623.5163 

3 

299.2367 

7125.5739 

7 

309.7087 

7633.0284 

11 

288.7647 

6635.5727 

4 

299.4985 

7138.0476 

8 

309.9705 

7645.9384 

92 0 

289.0265 

6647.6101 

5 

299.7603 

7150.5321 

9 

310.2323 

7658.8593 

1 

289.2883 

6659.6583 

6 

300.0221 

7163.0276 

10 

310.4941 

7671.7911 

2 

289.5501 

6671.7174 

7 

300.2839 

7175.5340 

11 

310.7559 

7684.7338 

3 

289.8119 

6683.7875 

8 

300.5457 

7188.0513 

99 0 

311.0177 

7697.6874 

4 

290.0737 

6695.8684 

9 

300.8075 

7200.5794 

1 

311.2795 

7710.6519 

5 

290.3355 

6707.9603 

10 

301.0693 

7213.1185 

2 

311.5413 

7723.6274 

6 

290.5973 

6720.0630 

11 

301.3311 

7225.6686 

3 

311.8031 

7736.6137 

7 

290.8591 

6732.1767 

96 0 

301.5929 

7238.2295 

4 

312.0649 

7749.6109 

8 

291.1209 

6744.3013 

1 

301.8547 

7250.8013 

5 

312.3267 

7762.6191 

9 

291.3827 

6756.4368 

2 

302.1165 

7263.3840 

6 

312.5885 

7775.6382 

10 

291.6445 

6768.5832 

3 

302.3783 

7275.9777 

7 

312.8503 

7788.6681 

11 

291.9063 

6780.7405 

4 

302.6401 

7288.5822 

8 

313.1121 

7801.7090 

93 0 

292.1681 

6792.9087 

5 

302.9019 

7301.1977 

9 

313.3739 

7814.7608 

1 

292.4299 

6805.0878 

6 

303.1637 

7313.8240 

10 

i 313.6357 

7827.8235 

2 

292.6917 

6817.2779 

7 

1 303.4255 

7326.4613 

11 

313.8975 

7840.8971 

3 

4 

292.9535 

293.2153 

6829.4788 

6841.6907 

8 

303.6873 

7339.1095 

100 0 

814.1593 

7853.9816 


Circumferences in feet, when the diam contains fractions 
of an inch. See similar process, p 133. 


Diam, 

Inch. 

Circumf, 

foot 

Diam, 

Inch. 

Circumf, 

foot 

Diam, 

Inch 

1 64 

.004091 

7-32 

.057269 

27-64 

1-32 

.008181 

15-64 

.061359 

7-16 

3-64 

.012272 

M 

.065450 

29-64 

1-16 

.016362 

17-64 

.069540 

15-32 

6-64 

.020453 

9-32 

.073631 

31-64 

3-32 

.024544 

19-64 

.077722 

K 

7-64 

.028634 

5-16 

.081812 

33-64 

Vs 

.032725 

21-64 

.085903 

17-32 

d-64 

.036816 

11-32 

.089994 

35-64 

5-32 

.040906 

23-64 

.094084 

9-16 

11-64 

.044997 

% 

.098175 

37-64 

3-16 

.049087 

25-64 

.102265 

19-32 

13-64 

.053178 

13-32 

.106356 

39-64 


Circumf, 

Diam, 

Circumf, 

Diam, 

Circumf, 

foot. 

Inch. 

foot. 

Inch. 

foot. 

.110447 

5-8 

.163625 

53-64 

.216803 

.114537 

41-64 

.167715 

27-32 

.220!'93 

.118628 

21-82 

.171806 

55-64 

.224984 

.122718 

43-64 

.175896 

7-8 

.229074 

.126809 

11-16 

.179987 

57-64 

.233165 

.130900 

45-64 

.184078 

29-32 

.237256 

.134990 

23-32 

.188168 

59-64 

.241346 

.139081 

47-64 

.192259 

15-16 

.245437 

.143172 

% 

.196350 

61-64 

.249528 

.147262 

49-64 

.200440 

31-32 

.253618 

.151353 

25-32 

.204531 

63-64 

.257709 

.155443 

51-64 

.208621 

1 

.261799 

.159534 

13-16 

.212712 




























































ii ikii; ci min 


CIRCULAR ARCS. 

CIRCULAR ARCS. 


141 



Fig.l 



Rules for Fig. 1 apply to all arcs equal to, or less than, a semi-circle. 

Fig. 2 “ “ “ or greater than, a semi-circle. 

Chord, a b, of whole arc, a d b, 

= (8'*X d ft§) —(3 X lgth of arc a d b) IT Fig 1. 
•— 2 X rad X sine of M a c b. Figs 1 and 2. 

= 2 X rise -r tang of a b d .* Figs 1 and 2. 

— 2 X d ft§ X cosine of a b d.* Figs 1 and 2. 


— 2 X V rad 2 — (rad — rise) 2 
~2X j/rad 2 — (rise — rad )2 
= 2 X V^rise X [(2 X rad) — rise]. 


Fig 1. 

Fig 2. 

Figs 1 and 2. 


— 2 X V d 62 — rise2. Figs 1 and 2. § 

For table of chords to radius 1, see p 105. 

Length, a d b, 

rad X 6.2832 X number of degrees subtended by arc a d b -r 360. Figs 1, 2. If the arc contains 
fractions of a degree, see table, p. 57 . 

chord X number in column of lengths, pp 143,144, opp “ rise -r chord.” Figl. For Fig 2, see p 144. 
rad X sum of numbers in columns of lengths, p 145, opp degs, mins, &c subtended by arc. Fig 1 
(radX6.2832) — lgth of small arc subtending ang acb. Fig 2. For said lgth see preceding formula 
approximately [(8 X d ft§) — chord a 5] -r 3. Fig 1 .** / 

Radius, c a, c d, c i, or c b, 


MM a ft) 2 -F rise2] -j- (2 X rise). Figs 1, 2. 
d ft2§ -j- (2 X rise). Figs 1, 2. 

M <* ft T sine of % a c ft. Figs 1, 2. 


= rise v (1 — cosine of Mac ft). Fig 1. 
= rise -r (1 -j- cosine of a c ftt). Fig 2. 
— M d b -j- sine of b c dll. Figs 1 and 2.§ 


For tables of radii to chords of 100 ft, and of 20 metres, see pp 726, 728. 

Rise, or middle ordinate, e d, 

~ rad — frad® — a ft) 2. Figl. 


= rad -(- ]/rad2 — (M a ft) 2 - Fig 2. 

= rad X (1 — cosine of 6 c dll). Fig 1. 
= rad X (1 + cosine of ft c dll)t. Fig 2. 
= db 2§ -s- (2 X rad). Figs 1, 2. 


— M a b X tang of a ft d.* Figs 1. 2 
= approxO^ a 6)2(2 X rad). Figl. 
If rad — a b, add .067 of the result 
If rad = 3 a 6, add .007 “ 

For tables of mid ords, see pp 726-730. 




= Urad2 — e n 2 — (rad — rise). 
— J^rad 2 — e»2-j- (rise — rad). 


Side ordinate, n i, 

i i . a n X n b 

"S 1, = approximately ————— . Figl. 

2 X radius 6 

* ig 2. | For tables of side ordinates, see p 730. 


Angle, acb, subtended by are, a d b, 

An angle and its supplement (as 6 c e and bed, Fig 2) have the same sine, cosine, and tangent. 

Caution. The following sines, &c, are those of only half a c b. 

Sine of Macb~%ab-r rad. Figs 1, 2. 

Cosine “ $ = < rad — rise -> f rad - Fi S 1 - 

uosme J = (rise _ rad) g. rad pig 2 . 

For areas of segments a d b e, see p 146. For areas of sectors a d ft c, see p 146. For 
cens of grav of arcs, segments, and sectors, see p. 


Tane of M acb\~^ ah ~ (rad ~ rise) - F, S *• 
1 ang or Macb^ =1Aabi _ (rise _ rad) pig 2 

Versed sine “ = rise -r rad. Figs 1, 2. 


348. 


* a 6 d is = % of the angle, acb, subtended bv the arc. In Fig 2 the latter angle exceeds 180°. 
t Strictly, this should read 1 minus cosine, but the cosines of angles between 90° and 270° must 
then be regarded as minus or negative. Our rule, therefore, amounts practically to the same thing. 

" bed — half the angle acb subtended by the arc. 


f If rise = .5 ch’d, add .036 of the result. 

** =.4 “ “ .0196 

“ =.333 “ “ .0114 

“ =r .3 “ “ .0083 


If rise = .25 ch’d,add .0044 of the result. 

= .2 “ “ .0021 “ 

= .125 “ “ .00036 “ 

= .1 “ “ .00015 “ 


§ d ft = chord of half -arc dib = V'rise 2 -|- (M a 6 - 2 . Figs 1, 2- 

* # If rise =.5 ch'd, add .012 of the result. 1 If rise = .25 ch'd, add .0015 of the result. 
“ rr .4 “ “ .0065 *• “ =.2 “ “ .0007 

“ = .333 “ “ .0038 “ 

“ = .3 “ “ .0028 “ 


.2 

= .125 
= .1 


.00012 

.00005 
































142 


CIRCULAR ARCS 


To describe the arc of a circle too large for the dividers. 


Let ac be the chord; and o b the height of 
the reqd arc, as laid down on the. drawing. On 
a separate strip of paper, s e. m n, dra w a c. o b, 
and ah: also h e, parallel to the chord ac. It 
is well to make b s, and b e, each a little longer 
than a b. Then cut off the paper carefully 
along the lines s h and 6 e. so as to leave remain¬ 
ing only the strip sahemn. Now, if the 
straight sides s b and 6 e be applied to the draw¬ 
ing, so that any parts of them shall touch at 
the same time the points a and b, or b and 
of the arc, and mav be pricked off. Thus, any 


Or thus : draw the span a b; 
theriserc; andac, 4c. From 
c with rad c r describe a circle. 
Make each of the arcs o t and 
i l equal to r o or ri\ aud draw 
c t, cl. Div ct, cl, cr, each iut« 
half as many equal parts as tha a 
curve is to be divided into. I 
Draw the lines b 1, b 2, b3; and 
a 4, a 5, a 6, extended to meet 
the first ouesat e, s, h. Then 
e.s, ft, are pointsiu one half tha 
curve. Then for the other half, 
draw similar lines from a to 7, 

8,9; and others from b to meet 
them, as before. Trace the 
curve by hand. 




ward united to form the curve. 


Remark.— It may frequently be of use to remember, that in any 
arc bos, not exceeding 29°, or in other words, whose chord b s is at 
least sixteen times its rise, the middle Ordinate a o, will be one 
half of a c, quite near enough for many purposes ; 6 c and s c being 
tangents to the arc.tAnd vice versa,if in such an arc we make o c equal 
a o, then will c be. very nearly, the point at which tangents from the 
ends of the arc will meet. Also the middle ordinate n, of the half arc 
o 6. or o s, will be approximately of a o, the mid ord of the whole 
arc. Indeed, this last observation will apply near enough for mauy 
approximate uses even if the arc be as great as 45°; for if in that case we take *4 of o a for the ord n, 
n will then be but 1 part in 103 too small; and therefore the principle may often be used in draw ings, 
for finding points in a curve of too great rad to be drawn by the dividers; for in the same manner, 34 
of n will be the mid ord for the arc n b or no; and so on to any extent. Below will be found 

a table by which the rise or middle ord of a half arc cau be 

obtained with greater accuracy when required for more exact drawings. 

CIRCULAR ARCS IN FREQUENT USE. 

The fifth column is of use for finding points for drawing aros too large for the beam-compass, on 
the principle given above. In even the largest office drawings it will not be necessary to 

use more than the first three decimals of the fifth column : and after the aro is subdivided into parts 
smaller than about 35° each, the first two decimals .25 will generally suffice. Original. 


Rise 

in 

parts 

of 

chord. 

Deg in 
whole 
arc. 

For rad 
mult rise 
by 

For 

length of 
arc mult 
chord 
by 

For rise 
of half 
aro 

mult rise 
by 

Rise 

in 

parts 

of 

chord. 

Deg in 
whole 
arc. 

For rad 
mult rise 
by 

For | 
lengt h of j 
arc mult 
chord 
by 

For 
rise of 
half arc 
mult 
rise by 

1-50 

O ' 

9 9.75 

313. 

1.00107 

.2501 


O ' 

56 8.70 

8.5 

1.04116 

.2538 

1-45 

10 10.75 

253.625 

1.00132 

.2501 

1-7 

63 46.90 

6.625 

1.05356 

.2549 

1 40 

11 26.98 

200.5 

1.00167 

.2502 

.155 

68 53.63 

5.70291 

1.06288 

.2557 

1-35 

13 4.92 

153.625 

1.00219 

.2502 

1-6 

73 44.39 

5. 

1.07250 

.2566 

1-30 

15 15.38 

113. 

1.00296 

.2503 

.18 

79 11.73 

4.35803 

1.08428 

.2576 

1-25 

18 17.74 

78.625 

1.00426 

.2504 

15 

87 12.34 

3.625 

1.10347 

.2593 

1-20 

22 50.54 

50 5 

1.00665 

.2506 

.207107 

90 

3.41422 

1.11072 

.2599 

1 19 

24 2.16 

45.625 

1.00737 

.2507 

.225 

96 54.67 

2.96913 

1.12997 

.2615 

1-18 

25 21.65 

41. 

1.00821 

.2508 

Ya 

106 15.61 

2.5 

1.15912 

.2639 

1-17 

26 50.36 

86.625 

1.00920 

.2509 

.275 

115 14.59 

2 15289 

1.19082 

.2665 

1-16 

28 30.00 

32.5 

1.01038 

.2510 

.3 

123 51.30 

1.88889 

1.22495 

.2692 

1-15 

30 22.71 

28.625 

1.01181 

.2511 

H 

134 45.62 

1.625 

1.27401 

.2729 

1-14 

.32 31.22 

25. 

1.01355 

.2513 

.365 

144 30 98 

1.43827 

1.32413 

.2766 

1-13 

34 59.08 

21.625 

1.01571 

.2515 

.4 

154 38.35 

1.28125 

1.38322 

.2808 

1-12 

37 50.96 

18.5 

1.01842 

.2517 

.425 

161 27.52 

1 19204 

1.42764 

.2838 

1-11 

41 13.16 

15.625 

1.02189 

.2520 

-45 

167 56.93 

1.11728 

1.47377 

.2868 

1-10 

45 14.38 

13. 

1.02646 

.2525 

.475 

174 7.49 

1.05402 

1.52152 

.2899 

19 

50 6.91 

10.625 

1.03260 

.2530 

-5 

180 

1 . 

1.57080 

.2929 


t At 29° o c thus found will be but about _Lpart too short. 

3 3 














































MENSURATION 


143 


IiC‘ii^ths of circular arcs. If arc exceeds a semicircle, seep 144. 

Knowing its chord and height, divide the height by the chord. Find in the column of heights the 
number equal to this quotient. Take out the corresponding number from the column of lengths. 
Multiply this last number by the leugtb of the given chord. 

TAI5RE OF CIRCULAR ARCS. Noerrors 


H'ghts. 

Lengths. 

H’ghts. 

Lengths. 

H gbts 

.001 

1.00002 

.076 

1.01533 

.151 

002 

1.00002 

.077 

1.01573 

.152 

.003 

1.00003 

.078 

1.01614 

.153 

001 

1.00001 

.079 

1.01656 

.154 

.005 

1.00007 

.080 

1.01698 

.155 

.000 

1.00010 

.081 

1.01711 

.156 

.007 

1.00013 

.082 

1.01784 

.157 

.000 

1.00017 

.083 

1.01828 

.158 

.000 

1.00022 

.084 

1.01872 

.159 

010 

1.00027 

.085 

1.01916 

.160 

.011 

1.00032 

.086 

1.01961 

.161 

012 

1.00038 

.087 

1.02006 

.162 

.010 

1.00015 

.088 

1.02052 

.163 

■OH 

1.00053 

.089 

1.02098 

.164 

.015 

1.00061 

.090 

1.02116 

.165 

.010 

1.00069 

.091 

1.02192 

.166 

017 

1.00078 

.092 

1.02210 

.167 

.018 

1.00087 

.093 

1.02289 

.168 

.010 

1.00097 

.091 

1.02339 

.169 

.020 

1.00107 

.095 

1.02389 

.170 

.021 

1.00117 

.096 

1.02410 

.171 

.022 

1.00128 

097 

1.02191 

.172 

.023 

1.00110 

.098 

1.02512 

.173 

.024 

1.00153 

.099 

1.02593 

.171 

.025 

1 00167 

.100 

1.02616 

.175 

.028 

1.C0182 

.101 

1.02698 

.176 

.027 

1.00196 

.102 

1.02752 

.177 

.028 

1.00210 

.103 

1.02806 

.178 

.020 

1.00225 

.101 

1.02860 

.179 

.030 

1.00210 

.105 

1.02911 

.180 

.031 

1.00256 

.106 

1.02970 

.181 

.032 

1.00272 

.107 

1.03026 

.182 

.033 

1.00289 

.108 

1.03082 

.183 

.031 

1.00307 

.109 

1.03139 

.184 

.035 

1.00327 

.110 

1.03196 

.185 

.030 

1.00315 

.111 

1.03251 

.186 

.037 

1.00361 

.112 

1.03312 

.187 

.038 

1.C0381 

.113 

1.03371 

.188 

.030 

1.00105 

.111 

1.03130 

.189 

.010 

1.00126 

.115 

1.03190 

.190 

.011 

1.00417 

.116 

1.03551 

.191 

.012 

1.00169 

.117 

1.03611 

.192 

.013 

1.00492 

.118 

1.03672 

.193 

.041 

1.00515 

.119 

1.03731 

.191 

.015 

1.00539 

.120 

1.03797 

.195 

.010 

1.00563 

.121 

1.03860 

196 

.017 

1.00587 

.122 

1.03923 

.197 

018 

1.00612 

.123 

1.03987 

.198 

.010 

1 00638 

.121 

1.04051 

.199 

.050 

1.00665 

.125 

1 01116 

.200 

.051 

1.00692 

.126 

1.01181 

.201 

.052 

1 00720 

.127 

1.01217 

.202 

053 

1 00718 

.128 

1.01313 

.203 

.051 

' 1.00776 

.129 

1.01380 

.201 

.055 

1.00805 

.130 

1 01117 

.205 

.050 

1.00831 

.131 

1 01515 

.206 

.057 

1 00861 

.132 

1.01584 

.207 

.058 

1.00895 

.133 

1.01652 

.208 

.059 

1.00926 

131 

1.01722 

.209 

.000 

1.00957 

.135 

1.01792 

.210 

.001 

1.00989 

.136 

1.01862 

.211 

.002 

1.01021 

.137 

1.01932 

.212 

.003 

1.01051 

.138 

1.05003 

.213 

.001 

1.01088 

.139 

1.05075 

.211 

.005 

1.01123 

.110 

1.05117 

.215 

.060 

1.01158 

.111 

1.05220 

.216 

.067 

1.01193 

.112 

1.05293 

.217 

.068 

1.01228 

.113 

1.05367 

.218 

.009 

1 01261 

• 111 

1.05111 

.219 

.070 

1.01302 

.115 

1 05516 

.220 

.071 

1.01338 

.116 

1.05591 

.221 

.072 

1.01376 

.117 

1.05667 

.222 

.073 

1.01111 

.118 

1.05713 

.223 

.071 

1.01153 

.119 

1.05819 

.224 

.075 

1.01193 

.150 

1.05896 

.225 


lengths. 

H'ghts. 

Lengths. 

H’ghts. 

Lengths. 

1.05973 

• 226 

1.13108 

.301 

1.22636 

1.06051 

• 227 

1.13219 

.302 

1.22778 

1.06130 

• 228 

1.13331 

.303 

1.22920 

1.06209 

•229 

1.13114 

.304 

1.23063 

1.06288 

• 230 

1.13557 

•305 

1.23206 

1.06368 

• 231 

1.13671 

.306 

1.23349 

1.06119 

• 232 

1.13785 

.307 

1.23492 

1.06530 

• 233 

1.13900 

.308 

1.23636 

1.06611 

.231 

1.14015 

.309 

1.23781 

1.06693 

• 235 

1.14131 

.310 

1.23926 

1.06775 

• 236 

1.11217 

• 311 

1.21070 

1.06858 

•237 

1.14363 

.312 

1.24216 

1.06911 

• 238 

1.11180 

• 313 

1.24361 

1.07025 

• 239 

1.11597 

.311 

1.24507 

1.07109 

•210 

1.14711 

.315 

1.24654 

1.07191 

•211 

1.11832 

•316 

1.24801 

1.07279 

• 212 

1.11951 

.317 

1.24948 

1.07365 

• 213 

1.15070 

• 318 

1.25095 

1.07151 

.241 

1.15189 

.319 

1.25243 

1.07537 

• 215 

1.15308 

.320 

1.25391 

1.07621 

.216 

1.15428 

.321 

1.25540 

1.07711 

.217 

1.15519 

.322 

1.25689 

1.07799 

.218 

1.15670 

.323 

1.25838 

1.07888 

.219 

1.15791 

.321 

1.25988 

1.07977 

.250 

1.15912 

.325 

1.26138 

1.08066 

.251 

1.16031 

.326 

1.26288 

1.08156 

.252 

1.16156 

.327 

1.26437 

1.08216 

.*253 

1.16279 

.328 

1.26588 

1.08337 

.254 

1.16402 

.329 

1.26740 

1.08128 

.255 

1.16526 

.330 

1.26892 

1.08519 

.256 

1.1HH50 

.331 

1.27041 

1.08611 

.257 

1.16771 

.332 

1.27196 

1.08701 

.258 

1.16899 

.333 

1.27349 

1.08797 

.259 

1.17024 

.334 

1.27502 

1.08890 

.260 

1.17150 

.335 

1.27656 

1.08984 

.261 

1.17276 

.336 

1.27810 

1.09079 

.262 

1.17103 

.337 

1.27964 

1.09171 

.263 

1.17530 

.338 

1.28118 

1.09269 

.261 

1.17657 

.339 

1.28273 

1.09365 

.265 

1.17781 

.340 

1.28428 

1.09161 

.266 

1.17912 

.341 

1.28583 

1.09557 

.267 

1.18010 

.312 

1.28739 

1.09651 

.268 

1.18169 

.313 

1.28895 

1.09752 

.269 

1.18299 

.344 

1.29052 

1.09850 

‘ .270 

1.18129 

.345 

1.29209 

1.09919 

.271 

1.18559 

.346 

1.29366 

1.10018 

.272 

1.18689 

.317 

1.29523 

1.10117 

.273 

1.18820 

.318 

1.29681 

1.10217 

.271 

1.18951 

.349 

1.29839 

1.10317 

.275 

1.19082 

.350 

1.29997 

1.10117 

.276 

1.19211 

.351 

1.30156 

1.10518 

.277 

1.19346 

.352 

1.30315 

1 10650 

.278 

1.19179 

.353 

1.30174 

1.10752 

.279 

1.19612 

.351 

1 30631 

1.10855 

.280 

1.19746 

.355 

1.30791 

1.10958 

.281 

1.19880 

.356 

1.30954 

1.11062 

.282 

1.20011 

.357 

1.31115 

1.11165 

.283 

1.20149 

.358 

1.31276 

1.11269 

.281 

1.20284 

.359 

1.31437 

1.11371 

.285 

1.20119 

.360 

1.31599 

1.11479 

.286 

1.20555 

.361 

1.31761 

1.11581 

.287 

1.20691 

.362 

1.31923 

1.11690 

.288 

1.20827 

.363 

1.32086 

1.11796 

.289 

1.20961 

.361 

1 32249 

1.11901 

.290 

1.21102 

.365 

1.32113 

1.12011 

.291 

1.21239 

.366 

1.32577 

1.12118 

.292 

1.21377 

.367 

1.32741 

1.12225 

.293 

1.21515 

.368 

1.32905 

1.12334 

.294 

1.21651 

.369 

1.33069 

1.12111 

.295 

1.21794 

.370 

1.33234 

1.12554 

.296 

1.21933 

.371 

1.33399 

1.12661 

.297 

1.22073 

.372 

1.33561 

1.12771 

.298 

1.22213 

373 

1.33730 

1.12885 

.299 

1.22354 

.371 

1.33896 

1 12997 

.300 

1.22195 

.375 

1.34063 
































144 


MENSURATION, 


TABLE OF CIRCULAR A RCS —(Continued.) 


H’ghts. 

Lengths. 

H'ghts. 

Lengths. 

H’ghts. 

Lengths. 

H’ghts. 

Lengths. 

H’ghts. 

Lengths. 

.376 

1.34229 

.401 

1.38496 

.426 

1.42945 

.451 

1.47565 

.476 

1.52346 

.377 

1.34393 

.402 

1.38671 

.427 

1.43127 

.452 

1.47753 

.477 

1.52541 

.378 

1.34563 

.403 

1.38846 

.428 

1.43309 

.453 

1.47642 

.478 

1.52736 

.379 

1.34731 

.404 

1.39021 

.429 

1.43491 

.454 

1.46131 

.479 

1.525 31 

.380 

1.34899 

.405 

1.39186 

.430 

1.43673 

.455 

1.453.0 

.480 

1.53126 

.331 

1.35068 

.406 

1.39372 

.431 

1.43856 

.4.56 

1.485(9 

.481 

1.53322 

.382 

1.35237 

.407 

1.39548 

.432 

1.44039 

.457 

1.48659 

.482 

1.53518 

.383 

1.35408 

.408 

1.33724 

.433 

1.44222 

.458 

1.48819 

.483 

1.53714 

.384 

1.35 57.5 

.409 

1.3S900 

.134 

1.44405 

.459 

1.45079 

.484 

1.53610 

.385 

1.35744 

.410 

1.40077 

.435 

1.44589 

.460 

1.49569 

.485 

1.54106 

.386 

1.35914 

.411 

1.40254 

.436 

1.44773 

.461 

1.46460 

.486 

1.54302 

.387 

1.36084 

.412 

1.40432 

.437 

1.44957 

.462 

1.46651 

.487 

1.5445 9 

.338 

1.35254 

.413 

1.40610 

.438 

1.45142 

.463 

1.46842 

.488 

1.54656 

.389 

1.36425 

.414 

1.40788 

.439 

1.45327 

.464 

1.50033 

.489 

1.54853 

.390 

1.36596 

.415 

1.40966 

.410 

1.45512 

.465 

1.'0224 

.490 

1.55091 

.391 

1.36767 

.416 

1.41145 

.441 

1.45657 

.466 

1.50416 

.491 

1.55289 

.392 

1.36939 

.417 

1.41324 

.442 

1.45883 

.467 

1.50608 

.492 

1.55487 

.393 

1.37111 

.418 

1.41503 

.443 

1.46C69 

.468 

1.50800 

.493 

1.55685 

.394 

1.37283 

.419 

1.41682 

.444 

1.46255 

.469 

1.5065 2 

.494 

1.55884 

.395 

1.37455 

.420 

1.41861 

.445 

1.46441 

.470 

1.51185 

.495 

1.56083 

.396 

1.37628 

.421 

1.42041 

.446 

1.46628 

.471 

1.51378 

.466 

1.56282 

.397 

1.37801 

.422 

1.42221 

.447 

1.46815 

.472 

1.51571 

.497 

1.56481 

.398 

1.37974 

.423 

1.42402 

.448 

1.47002 

.473 

1.51764 

.498 

1.56681 

.399 

1.38148 

.424 

1.42583 

.419 

1.47189 

.474 

1.51958 

.469 

1.56881 

.400 

1.38322 

.425 

1.42764 

.450 

1.47377 

.475 

1.52152 

.500 

1.57080 


If the arc is greater than a semicircle, then, as directed at top of 

p 147, find the diain of the circle. Then find its circumf. From diaru take ht. of arc. Th** rem 
will be ht of the smaller arc of the circle. By rule at top of p 143 find the length of this smaller arc. 
Subtract it from circumf. 

The length of 1 degree of a circular arc is equal to .017+53 292 520 X its radius. 

“ “ “ 1 minute “ “ “ “ “ .000290 888 209 X “ 

“ “ “ 1 second “ “ “ “ “ .000004 848 137 X “ “ 

An arc of 1° of the earth's gfreat circle is but 4.6356 feet longer than its 

chord. Its leneth is 69.16 land or statute miles. Earth’s equatorial rad rr 3962.5705 miles. Polar 3949.67. 

An are of 1°, rad 1 mile, is 92.1534 feet; a minute is 1.5359 feet; a second is .0256 of a foot; 
or very nearly 5-sixteeuths of an inch. Arc of 1°, rad 100 ft = 1.74533 feet. 

































MENSURATION 


145 


To find the length of a circular arc by the following table. 


Knowing the rad of the circle, and the measure of the arc in deg, min, Ac. 

Rule. Add together the lengths in the table found respectively opposite to the deg, min &c, of 
the arc. Mult the sum by the rad of the circle. 

Ex. In a circle of 12.43 feet rad, is an arc of 13 deg, 27 min, 8 sec. How long is the arc 7 
Here, opposite 13 deg in the table, we find, .2268928 
“ 27 min “ “ “ .0078540 

“ 8 sec “ “ “ .0000388 


Sum — .2347856 

And .2347856 X 12.43 or rad = 2.918385 feet, the reqd length of arc. 

LENGTHS OF CIRCULAR ARCS TO RAO 1. 


No errors. 


Deg. 

Length. 

Deg. 

Length. 

Deg. 

Length. 

Min. 

Length. 

Sec. 

Length. 

1 

.0174533 

61 

1.0646508 

121 

2.1118484 

1 

.0002909 

1 

.0000048 

2 

.0349066 

62 

1.0821041 

122 

2.1293017 

2 

.0005818 

2 

.00C0O97 

3 

.0523599 

63 

1.0995574 

123 

2.1467550 

3 

.0008727 

3 

.0000145 

4 

.0698132 

64 

1.1170107 

124 

2.1642083 

4 

.0011636 

4 

.0000194 

5 

.0872665 

65 

1.1344640 

125 

2.1816616 

5 

.0014544 

5 

.0000242 

6 

.1047198 

66 

1.1519173 

126 

2.1991149 

6 

.0017453 

6 

.0000291 

7 

.1221730 

67 

1.1GJ3706 

127 

2.2165682 

7 

.0020362 

7 

.0000339 

8 

.1396263 

68 

1.1863239 

128 

2.2340214 

8 

.0023271 

8 

.0O0C388 

9 

.1570796 

69 

1.2042772 

129 

2.2514747 

9 

.0026180 

9 

.0000436 

10 

.1745329 

70 

1.2217305 

130 

2.2689280 

10 

.0029089 

10 

.0000485 

11 

.1919832 

71 

1.2391838 

131 

2.2863813 

11 

.0031998 

11 

.0000533 

12 

.2094395 

72 

1.2566371 

132 

2.3038346 

12 

.0034907 

12 

.0000582 

13 

.2268928 

73 

1.2740904 

133 

2.3212879 

13 

.0037815 

13 

.0000630 

14 

.2443461 

74 

1.2915436 

134 

2.3387412 

14 

.0040724 

14 

.0000679 

15 

.2817994 

75 

1.3089969 

135 

2.3561945 

15 

.0043633 

15 

.0000727 

16 

.2792527 

76 

1.3264502 

136 

2.3736478 

16 

.0046542 

16 

.0000776 

17 

.2967060 

77 

1.3439035 

137 

2.3911011 

17 

.0049451 

17 

.0000824 

IS 

.3141593 

73 

1.3613568 

138 

2.4083544 

18 

.0052360 

18 

.0000873 

19 

.3316126 

79 

1.3788101 

139 

2.4260077 

19 

.0055269 

19 

.0000921 

20 

.3490659 

80 

1.3962834 

140 

2.4434610 

20 

.0058178 

20 

.0000970 

21 

.3665191 

81 

1.4137167 

141 

2.4609142 

21 

.0061087 

21 

.0001018 

22 

.3839724 

82 

1.4311700 

142 

2.4783675 

22 

.0063995 

22 

.0001067 

23 

.4014257 

83 

1.4483233 

143 

2.4958208 

23 

.0066904 

23 

.0001115 

24 

.4188790 

84 

1.4660766 

144 

2.5132741 

24 

.0069813 

24 

.0001164 

25 

.4363323 

85 

1.4835299 

145 

2.5307274 

25 

.0072722 

V5 

.0001212 

26 

.4537856 

86 

1.5009832 

146 

2.5481807 

26 

.0075631 

26 

.0001261 

27 

.4712389 

87 

1.5184364 

147 

2.5656340 

27 

.0078540 

27 

.0001309 

28 

.4886922 

88 

1.5358897 

148 

2.5830873 

28 

.0081449 

28 

.0001357 

29 

.5061455 

89 

1.5533430 

149 

2.6005406 

29 

.0084358 

29 

.0001406 

30 

.5235938 

90 

1.5707963 

150 

2.6179939 

30 

.0087266 

30 

.0001454 

31 

.5410521 

91 

1.5882196 

151 

2.6354472 

31 

.0090175 

31 

.0001503 

32 

.5585054 

92 

1.6057029 

152 

2.6529005 

32 

.0093084 

32 

.0001551 

33 

.5759587 

93 

1.6231562 

153 

2.6703538 

33 

.0095993 

33 

.0001600 

34 

.5934119 

94 

1.6406095 

154 

2.6878070 

34 

.0098902 

34 

.0001648 

35 

.6108652 

95 

1.6580628 

155 

2.7052603 

35 

.0101811 

35 

.0001697 

36 

.6283185 

96 

1.6755161 

156 

2.7227136 

36 

.0104720 

36 

.0001745 

37 

.6457718 

97 

1.692969! 

157 

2.7401669 

37 

.0107629 

37 

.0001794 

38 

.6632251 

98 

1.7104227 

158 

2.7576202 

38 

.0110538 

38 

.0001842 

39 

.6806784 

99 

1.7278760 

159 

2.7750735 

39 

.0113446 

39 

.0001891 

40 

.6981317 

100 

1.7453293 

160 

2.7925268 

40 

.0116355 

40 

.00,01939 

41 

.7155850 

101 

1.7627825 

161 

2.8099801 

41 

.0119264 

41 

.000 '.988 

42 

.7330383 

102 

1.7802358 

162 

2.8274334 

42 

.0122173 

42 

.01002036 

43 

.7504916 

103 

1.7976S91 

163 

2.8448867 

43 

.0125082 

43 

.0002085 

41 

.7679449 

104 

1.8151424 

164 

2.8623400 

44 

.0127991 

44 

.0002133 

45 

.7853982 

105 

1.8325957 

165 

2,8797933 

45 

.0,130900 

45 

.0002182 

46 

.8028515 

106 

1.8500490 

166 

2.8972166 

46 

.0133809 

46 

.0002230 

47 

.8203017 

107 

1.8675023 

167 

2.9146999 

47 

.0136717 

47 

.0002279 

48 

.8377580 

108 

1.8849556 

168 

2.9321531 

48 

.0139626 

48 

.0002327 

49 

.8552113 

109 

1.9024089 

169 

2.9496061 

49 

.0142535 

49 

.0002376 

50 

.8726646 

110 

1.9198622 

170 

2.9670597 

50 

.0145444 

50 

.0002424 

51 

.8901179 

111 

1.9373155 

171 

2.9845130 

51 

.0148353 

51 

.0002473 

52 

.9075712 

112 

1.9547688 

172 

3.0019663 

52 

.0151262 

52 

.0002521 

53 

.9250245 

113 

1.9722221 

173 

3.0194196 

53 

.0154171 

53 

.0002570 

54 

.9424778 

1 14 

1.9896753 

174 

3.0368729 

54 

.0157080 

54 

.0002618 

55 

.9599311 

115 

2.0071286 

175 

3.0543262 

55 

.0159989 

55 

.0002666 

56 

.9773844 

116 

2.0245819 

176 

3.0717795 

56 

.0162897 

56 

.0002715 

57 

.9948377 

117 

2.0420352 

177 

3.0892328 

57 

0165806 

57 

.0002763 

58 

1.0122910 

1 18 

2.0594885 

178 

3.1066861 

58 

.0168715 

58 

.0002812 

59 

1.0297443 

119 

2.0769418 

179 

3.1241394 

59 

.0171624 

59 

.0002860 

6) 

1.0471976 

120 

2.0943951 

180 

3 1415927 

60 

.0174533 

60 

.0002909 


10 






























146 


MENSURATION, 


(TRUITEAR SECTORS, RINGS, SEGMENTS, ETC. 


Fig. A. 
d 




To find the area of a circular sector a db c. 

Rule 1. Mult the length of the arc a d b, by the rad c a or c b ; and 
divide prod by 2. 

Rem. Knowing the chord a b, and the heights d; or the rad. and 
the number of degs in the arc, the length of the arc itself can be found 
from table, p 143 or 145. 

Rule 2. As 360° is to the number of degs in the angle a c b, so is 
the area of the entire circle to the area of the sector. 

To find the area of a circular ring 1 . 

Find the areas of the two circles, and take the least from the greatest. Or 
mult the sum of the two diams, a b, c d, by their diff; mult the prod by .7854. 
Or mult together the thickness c a; the mean diam 11 ; and 3.1416. 

To find the rad of a circle which shall have the 
same area as a given circular ring cs da b. 

Draw any rad nr of the outer circle; and from where said rad cuts the 
inner circle at t, draw Uat right angles to it. Then will ts be the reqd rad. 


To find the breadth of a circular ring. 

Having its area, and the diam of its outer circle. Find the area of the whole circle. 

From it take the area of the ring. Mult the rem by 1.2732. Take the sq rt of the 

prod. This sq rt will be the diam of the inner circle. Take it from the diam of the outer one; and 

div the rem by 2, for the reqd breadth. 


To find the area of a circular zone abed. 

Knowing the diam of the circle; the chords a b, and c d ; 
and the heights om, and sn, of the segments a mb, and end. 
First find the area of the entire circle; then by means of the 
following table of circular segments, find the areas of the 
two segments a mb, and end) and subtract their sum from 
the area of the circle. 



To find the area of a circular lime a bco. 

A lune is a crescent-shaped fig, comprised between two 
arcs of circles of diff diams. Having the chord a c, and the 
heights of the two segments a o c, and ab c, find the areas of 
those segments; take the least of these areas from the great¬ 
est, ; the rem is evidently the area of the lune. 



v. 


Fig. D. 


To find the area of a circular segment, abed, Figs C, 

table, pp 147,148. Also, 

Area of Segment 

a d b n Fig A = Area of 
Sector a d b c — Area of 
Triangle a b c. 


Having the area of a segment reqd to be cut off from a given 
circle, to find its chord and height. 

Div the area by the square or the diam of the circle; look for the quot in the col of areas in the 
table of areas, pp 147, 148; take out from the table the corresponding height: and mult it bv the 
d.am The prod will be the reqd height. Then from the diam, take the actual height thus found • 
mult the rem by the actual height; take the sq rt of the prod ; mult it by 2, for the reqd chord. 















MENSURATION 


147 


TABLE OF AREAS OF CIRCULAR SEGMENTS, Figs C, D. 

If tile fluent exceeds a semicircle, its area is = area of circle— area 
of a segment whose rise is =r (diam of circle — rise of given segment). l>iam of circle — (square 
of half chotd f rise) -J- rise, whether the segment exceeds a semicircle or not. 


Rise 1 
div by I 
diam of 
circle. 

Area = 

(square 
of diam) 
mult by 

Rise J 
div by | 
diam of 
circle. 

Area = 

(square 
of diam) 
mult by 

Rise 
div by 
diam of 
circle. 

Area = 

(square 
of diam) 
mult by 

Rise 
div by 
diam of 
circle. 

Area = 

(square 
of diam) 
mult by 

Rise 
div by 
diam of 
circle. 

Area = 

(square 
of diam 
mult by 

.001 

.000042 

.064 

.021168 

.127 

.057991 

.190 

.103900 

.253 

.156149 

.002 

.000119 

.065 

.921660 

.128 

.058658 

.191 

.104686 

.254 

.157019 

.003 

.000219 

.066 

.022155 

.129 

.059328 

.192 

.105472 

.255 

.157891 

.004 

.000337 

067 

.022653 

.130 

.059999 

.193 

.106261 

.256 

.158763 

.005 

.000471 

.068 

.023155 

.131 

.060673 

.194 

.107051 

.257 

.159636 

.000 

.('00619 

.069 

.023660 

.132 

.061349 

.195 

.107843 

.258 

.160511 

.007 

.000779 

.070 

.024168 

.133 

.062027 

.196 

.108636 

.259 

.161386 

.008 

.000952 

.071 

.024680 

.134 

.062707 

.197 

.109431 

.260 

.162263 

.009 

.0011 35 

.072 

.025196 

.135 

.063389 

.198 

.110227 

.261 

.163141 

.010 

.001329 

.073 

.025714 

.136 

.064074 

.199 

.111025 

.262 

.164020 

.011 

.001533 

.074 

.026236 

.137 

.064761 

.200 

.111824 

.263 

.164900 

.012 

.001746 

.075 

.026761 

.138 

.065449 

.201 

.112625 

.264 

.165781 

.013 

.001969 

.076 

.027290 

.139 

.066140 

.202 

.113427 

.265 

.166663 

.014 

.002199 

.077 

.027821 

.140 

.066833 

.203 

.114231 

.266 

.167546 

.015 

.002438 

078 

.028356 

.141 

.067 528 

.204 

.115036 

.267 

.168431 

.016 

.002685 

.079 

.028894 

.142 

.068225 

.205 

.115842 

.268 

.169316 

.017 

.002940 

.080 

.029435 

.143 

.068924 

.206 

.116651 

.268 

.170202 

.018 

.003202 

.081 

.029979 

.144 

.069626 

.207 

.117460 

.270 

.171090 

.019 

.003472 

.082 

.030526 

.145 

.070329 

.208 

.118271 

.271 

.171978 

•020 

.003749 

.0<3 

.031077 

.146 

.071934 

.209 

.119084 

.272 

.172868 

.021 

.004032 

.084 

.031630 

.147 

.071741 

.210 

.119898 

.273 

.173758 

.022 

.004322 

.085 

.032186 

.148 

.072450 

.211 

.120713 

.274 

.174650 

.023 

.004019 

.086 

.032746 

. i 49 

.073162 

.212 

.121530 

.275 

.175542 

.024 

.004922 

.087 

.033308 

.150 

.073875 

.213 

.122348 

.276 

.176436 

.025 

.005231 

.088 

.033873 

.151 

.074599 

.214 

.123167 

.277 

.177330 

.026 

.005546 

.089 

.934441 

.152 

.075307 

.215 

.12398* 

.278 

.178226 

.027 

.005867 

.090 

.035012 

.153 

.07 6026 

.216 

.124811 

.279 

.179122 

.028 

.006194 

.091 

.035586 

.154 

.076747 

.217 

.125634 

.280 

.180020 

.029 

.006527 

.092 

.036162 

155 

.077470 

.218 

.126459 

.281 

.180918 

.030 

.006S66 

.093 

.036742 

.156 

.078194 

.219 

.127286 

.282 

.181818 

.031 

.007 20.-t 

.094 

.037324 

.157 

.078921 

.220 

.128114 

.283 

.182718 

.032 

.007 559 

.095 

.037909 

.158 

.079650 

.221 

.128943 

.284 

.183619 

.033 

.907913 

.096 

.038497 

.159 

.080380 

.222 

.129773 

.285 

.184522 

.034 

.008273 

.097 

.039087 

.160 

.081112 

.223 

.130605 

.286 

.185425 

.035 

.008638 

.098 

.939681 

.161 

.081847 

.224 

.131438 

.287 

.186329 

.036 

.009008 

.099 

.040277 

.162 

.082582 

.225 

.132273 

.288 

.187235 

.037 

.009383 

.100 

.040875 

.163 

.083320 

.226 

.133109 

.289 

.188141 

.0-»8 

.009764 

.101 

.041477 

.164 

.084060 

.227 

.133946 

.290 

.189048 

.039 

.010148 

.102 

.042081 

.165 

.084801 

.228 

.134784 

.291 

.189956 

.040 

.010538 

.103 

.042687 

.166 

.085545 

.229 

.135624 

.292 

.190865 

041 

.010932 

.104 

.043296 

.167 

.086290 

.230 

.136465 

.293 

.191774 

.042 

.011331 

.105 

.043908 

.168 

.087037 

.231 

.137307 

.294 

.192685 

.043 

.011734 

.106 

.044523 

.169 

.087785 

.232 

.138151 

.295 

.193597 

.044 

.012142 

.107 

.045140 

.170 

.088536 

.233 

.138996 

.296 

.194509 

.045 

.012555 

.108 

.045759 

.171 

.089288 

.234 

.139842 

.297 

.195423 

.046 

.012971 

.109 

.046381 

.172 

.090042 

.235 

.140689 

.298 

.196337 

.047 

.013393 

.110 

.947006 

.173 

.090797 

.236 

.141538 

.299 

.197252 

.048 

.013818 

.111 

.047633 

.174 

.091555 

.237 

.142388 

.300 

.198168 

.049 

.014248 

.112 

.048262 

.175 

.092314 

.238 

.143239 

.301 

.199085 

.050 

.014681 

.113 

.048894 

.176 

.093074 

.239 

.144091 

.302 

.200003 

.051 

.015119 

.114 

.049529 

.177 

.093837 

.240 

.144945 

.303 

.260922 

.052 

.015561 

.115 

.050165 

.178 

.094601 

.241 

.145800 

.304 

.201841 

.053 

.016908 

.116 

.050805 

.179 

.095367 

.242 

.146656 

.305 

.202762 

.054 

.016458 

.117 

.051446 

.180 

.096135 

.243 

.147513 

.306 

.203683 

.055 

.016912 

.118 

.052090 

.181 

.096904 

.244 

.148371 

.307 

.204605 

.056 

.017369 

.119 

.052737 

.182 

' .097675 

.245 

.149231 

.308 

.205528 

.057 

.017831 

.120 

.053385 

.183 

.098447 

.246 

.150091 

.309 

.206452 

.058 

.018297 

.121 

.0540: 7 

.184 

.099221 

.247 

.150953 

.310 

.207376 

.059 

.018706 

.122 

.054690 

.185 

.099997 

.248 

.151816 

.311 

.208302 

.060 

.019239 

.123 

.055346 

.186 

.100774 

.249 

.152681 

.312 

.209228 

.061 

.019716 

.124 

.056004 

.187 

.101553 

.250 

.153546 

.313 

.210155 

.062 

.020197 

.125 

.056664 

.188 

.102334 

.251 

.154413 

.314 

.211083 

.063 

.020681 

.126 

.057327 

.189 

.103116 

.252 

.155281 

.315 

.212011 

























148 


MENSURATION 


TABLE OF AREAS OF CIRCULAR SEGMENTS— (Continued.) 


Rise 

Area = 

Rise 

Area — 

Rise 

Area = 

Rise 

Area = 

Rise 

Area = 

div bv 

(square 

div by 

(square 

div by 

(square 

div by 

(square 

div by 

(square 

diam of 

of (iiam) 

diam of 

of diam) 

diam of 

of diam) 

iiam of 

of diam) 

diam of 

of diam 

circle. 

mult by 

circle. 

mult by 

circle. 

mult bv 

circle. 

mult by 

circle. 

mult by 

.316 

.212941 

.353 

.247845 

•390 

.283593 

.427 

.319959 

.464 

.356730 

.317 

.213871 

.354 

.248801 

•391 

.284569 

.428 

.320949 

.465 

.357728 

.318 

.214802 

.355 

.249758 

•392 

.285545 

.429 

.321938 

.466 

.358725 

.319 

.215734 

.356 

.250715 

•393 

.286521 

.430 

.322928 

.467 

.359723 

.3-JO 

.216666 

.357 

.251673 

•394 

.287499 

.431 

.323919 

.46" 

.360721 

.3JI 

.217600 

.358 

.252632 

•395 

.288476 

.432 

.324909 

.469 

.361719 

.3JJ 

.218534 

.359 

.253591 

•396 

.289454 

.433 

.325900 

.470 

.362717 

.323 

.219469 

.360 

.254551 

■397 

.290432 

.434 

.326891 

.471 

.363715 

.324 

.220404 

.361 

.255511 

•398 

.291411 

.435 

.327883 

.472 

.364714 

.325 

.221341 

.362 

.256472 

•399 

.292390 

.436 

.328874 

.473 

.365712 

.326 

.222278 

.363 

.257433 

•400 

.293370 

.437 

.329866 

.474 

.366711 

.327 

.223216 

.364 

.258395 

•401 

.294350 

.438 

.330858 

.475 

.367710 

.328 

.224154 

.365 

.259358 

•402 

.295330 

.439 

.331851 

.476 

.368708 

.329 

.225094 

.366 

.260321 

•403 

.296311 

.■440 

.332843 

.477 

.369707 

•3»30 

.226034 

.367 

.261285 

•404 

.297292 

441 

.333836 

.478 

.370706 

.331 

.226974 

.368 

.262249 

•405 

.298274 

.442 

.334829 

.479 

.371705 

.33-2 

.227916 

.369 

.263214 

•406 

.299256 

.443 

.335823 

.480 

.372704 

.333 

.228858 

.370 

.264179 

•407 

.300238 

.444 

.336816 

.481 

.373704 

.334 

.229801 

.371 

.265145 

•4(i8 

.301221 

.445 

.337810 

.482 

.374703 

.335 

.2:30745 

.372 

.266111 

•409 

.302204 

.446 

.338804 

.483 

.375702 

.336 

.231689 

.373 

.26707.'- 

•410 

.303187 

.447 

.339799 

.484 

.376702 

.337 

.232634 

.374 

.26^04* 

•411 

.304171 

.448 

.340793 

.485 

.377701 

.338 

.233580 

.375 

.269014 

•412 

.305156 

.449 

.341788 

.486 

.378701 

.339 

.234526 

.376 

.269982 

•413 

.306140 

.450 

.342783 

.487 

.379701 

.340 

.235473 

.377 

.270951 

•414 

.307125 

.451 

.343778 

.488 

.380700 

.341 

.236421 

.378 

.271921 

•415 

.308110 

.452 

.344773 

.489 

.381700 

.342 

.237369 

.379 

.272891 

416 

.309096 

.453 

.345768 

.490 

.382700 

.343 

.238319 

.380 

.273861 

•417 

.310082 

.454 

.346764 

.491 

.383700 

.344 

.23926 s 

.381 

.274832 

•418 

.311068 

.455 

.347760 

.492 

384699 

.345 

.240219 

.382 

.275804 

•419 

.312065 

456 

.348756 

.493 

.385699 

.346 

.241170 

.383 

.2767 7 6 

•420 

.313042 

.457 

.349752 

.494 

.386699 

.347 

.242122 

.384 

.277748 

.421 

.314029 

.458 

.350749 

.495 

387699 

.348 

.243074 

.385 

.278721 

.422 

.315017 

.459 

.351745 

.496 

.388699 

.349 

.244027 

.386 

.279695 

.423 

.316005 

.460 

.352742 

.497 

.389699 

.350 

.244980 

.387 

.280669 

.424 

.316993 

.461 

.353739 

.49s 

.390699 

.351 

.245935 

.388 

.281643 

.425 

.317981 

.462 

.354736 

.499 

.391699 

.352 

.24689(1 

.389 

.282618 

.426 

.318970 

.463 

.355733 

.500 

.392699 


1 
































MENSURATION 


149 


THE ELLIPSE. 




An ellipse is a curve, e eee, Pig 1, formed bj an oblique section of either a cone or a cylinder, pass¬ 
ing through its curved surface, without cutting the base. Its nature is such that if two lines, as 
n / and n g, Fig. 2, be drawn from any point n in its periphery or circumf, to two certain points/ 
and g, in its loug diani c w, (and called the foci of the ellipse.) their sum will be equal to that of any 
other two lines, as b /, and b g drawn from any other point, as b, in the circumf, to the foci / and g; 
also the sum of any two such lines will be equal to the loug diam c w. The line c tv dividing the ellipse 
into two equal parts lengthwise, is balled its transverse, or major axis, or loug diam ; and a 6, which 
divides it equally at right-angles to c w, is called the conjugate, or minor axis, or short diam. To 
tind the position of the foci of an ellipse, from either end, as b, of the short diam, measure off the 
dists b f and b g, Fig 2, each equal to o c, or one half the long diam. 

The parameter of an ellipse is a certain length obtained thus ; as the long diam : short diam : : 
short diam : parameter. Any line r v, or s d, Fig 3, drawn from the circumf, to, and at right angles 
to, either diam, is called an ordinate; and the parts c v and v w, b s and a o, of that diam, between 
the ord and the circumf, are called abscissae, or abscisses. 

To find the length of any ordinate, r v or sd, drawn to either 

diam, C W Or b ll. Knowing the absciss, c t) or s a, and the two diams, cw, b a; 

c w 2 : b a 2 :: c v X v w : r v 2 . b a 2 : c w 2 :: b s X s a : s d 2 . 


To find the cireumf of an ellipse. 


Mathematicians have furnished practical men with no simple working rule for this purpose. The 
so-called approximate rules do not deserve the name. They are as follows, D being the long diam: 
and d the short one. 


Rule 1. Circumf—3.1416 ^ ~h d , R UL e 2. 3.1416 

2 > 


\/< 


Dj4-d2. \ . Rulb g 2.22151/ D2+d2; 

2 Jf 


this is the same as Rule 2, but in a diff shape. Rule 4. 2xj/ D*-}-1.4674 <Z 2 . Now, in an ellipse 


whose long and short diams are 10 and 2, the circumf is actually 21, very approximately; but rule 1 
gives it —18.85: rule 2, or 3, rr 22.65; and rule 4, 1 20.51. Again, if the diams be 10 and 6, the cir¬ 
cumf actually — 25.59: but rule 4 gives 24.72. These examples show that none of the rules usually 
given are reliable. The following one by the writer, is sufficiently exact for ordinary purposes; not 
being in error probably more than 1 part in 1000. When D is not more than 5 times as long as d, 


Circumf = 3.1416 N /D 2 +d 2 _ (D — d )2 

~ 2 8.8 


If D exceeds 5 times rl, then in¬ 
stead of dividing (D — d) 2 by 8.8, div it by 
the number in Unstable. 

The following rule originated with Mr. M. 
Arnold Pears, of New South Wales, Australia, 
and was by hint kindly communicated to the 
rate than our own, it is much neater. 

Circumf = 3.1416 d + 2(D — d) — 


05 ft 3) ® ® 53 ft 53 O 0 O O 0 

a_ 

,1 -s ”8 -etsts-s -8 ■« ’S’S-W-S •e-s-s'e -8 -tf 

(Cl-005)ON.tCXOiflOOCOOCO 

Q r- — — 

author. Although not more accu- 


d(D — d) 


v/(D + d) X (D + 2 d) 


Tile following: table of semi-elliptic arcs was prepared by our rule. 



To use this table, div the height or rise of thp arc, by its span or chord. The quofc 
will be the height of an arc whose span is 1. Find this quot in the column of 
heights ; and take out the corresponding number from the col. of leugths. Mult this 
number by the actual span. The prod will be the reqd length. 

When the height becomes .500 of the chord fas at the end of the table) the ellipse 
becomes a circle. When the height exceeds .500 of the chord, as in a 6 c, then take 
a o. or half the chord, as the rise; and div this rise by the long diam b d. for the 
quot to be looked lor in the col of heights; and to be mult by long diam. We thus 
get the arc bad, which is evidently equal to a be. 




















150 


MENSURATION 


TABLE OF LENGTHS OF SEMI-ELLIPTIC ARCS. (Original.) 


Height 
+ span. 

Length = 
spanx by 

Height 
•3-span. 

Length = 
spanx by 

Height 
' ■£ span. 

Length = 
spanx by 

Height 

•$-span. 

Length = 
spanx by 

.005 

1.000 

.130 

1.079 

.255 

1.219 

.380 

1.390 

.01 

1.001 

.135 

1.084 

.260 

1.226 

.385 

1.397 

.015 

1.002 

.140 

1.089 

.265 

1.233 

.390 

1.404 

.02 

1.003 

.145 

1.094 

.270 

1.239 

.395 

1.412 

025 

1.004 

.150 

1.099 

.275 

1.245 

.400 

1 419 

.03 

1.006 

.155 

1.104 

.280 

1.252 

.405 

1.426 

.0 ’5 

1.008 

.160 

1.109 

.285 

1.259 

.410 

1.434 

.01 

1.011 

.165 

1.115 

.290 

1.265 

.415 

1.441 

.045 

1.014 

.170 

1.120 

.295 

1.272 

.420 

1.449 

.05 

1.017 

.175 

1.125 

.300 

1.279 

.425 

1.4/6 

.055 

1.020 

.180 

1.131 

.305 

1.285 

.430 

1.464 

.06 

1.023 

.185 

1.137 

.310 

1.292 

.435 

1.471 

.065 

1.026 

.190 

1.142 

.315 

1.298 

.440 

1.479 

.07 

1.029 

.195 

1147 

.320 

1.305 

.445 

1.486 

.075 

1.032 

.200 

1.153 

.325 

1.312 

.450 

1.494 

.08 

1.036 

.205 

1.159 

.330 

1.319 

.455 

1.501 

.085 

1.039 

.210 

1.165 

.335 

1.325 

.460 

1.509 

.09 

1.043 

.215 

1.171 

.340 

1.332 

.465 

1.517 

.095 

1.046 

.220 

1.177 

•345 

1.339 

.470 

1.524 

.100 

1.051 

.225 

1.183 

.350 

1.346 

.475 

1.532 

.105 

1.055 

.230 

1.189 

.355 

1.353 

,480 

1.540 

.110 

1.059 

.235 

1.196 

.360 

1.361 

.485 

1.547 

.115 

1.064 

.240 

1.202 

.365 

1.368 

.490 

1.555 

.120 

1.069 

.245 

1.207 

.370 

1 375 

.495 

1.563 

.125 

1.074 

.250 

1.213 

.375 

1.382 

.500 

1.571 


Area of ail ellipse = prod of diams X .7854. Ex. D= 10; d — 6. Then 10 X 6X .7854 
= 47.124 area. The area of an ellipse is a mean proportional between the areas of two circles, de¬ 
scribed on its two diams; therefore it may be found by mult together the areas of those two circles ; 
and taking the sq rt of ihe prod. The area of an ellipse is therefore always greater than that of the 
circular section of the cylinder from which it may be supposed to be derived. 

Diam of circ of same area as a given ellipse = |/ Long diam * sllort diana- 

To find the area of an elliptic segment whose base is paral> 
lei to either diam. Div the height of the segment, by that diam of which said height 
is a part. From the table of circular segments take out the tabuiar area opposite the quot Mult 
together this area, the long diam, and the short diam. 


To draw an ellipse. Having its long and short diams a b and c d, Fig. 4 



Rule 1. From either end of the short 
diam, as c, lay off the dists c /, c/’, each equal 
to e a, or to one-half of the long diam. The 
points /, /’ are the foci of the ellipse. Pre¬ 
pare a string, /’ «/, or /’ g /, with a loop at 
each end ; the total length of string from end 
to end of loop, being equal to the long diam. 

Place pins at /and /’; and placing the loops 
ever them, trace the curve by a pencil, which 
in every position, as at n. or g, keeps the string 
/' n /, or/’ gf, equally stretched all the time. 

Note. Owing to the difficulty of keeping 
the string equally stretched, this method is 
not as satisfactory as the following. 

Rule 2. On the edge of a strip of paper 
w s, mark to l equal to half the short diam; 
and ws equal half the long diam. Then in 
whatever position this strip be placed, keep¬ 
ing l on the long diam, and s on the short Fig. 4. 

diam, w will mark a point in the circumf of 

the ellipse. We may thus obtain as many such points as we please; and then draw the curve through 
them by hand. 

Rule 3. Front the two foci / and /', Fig 4, with a rad equal to any part whatever of the long diam, 
describe 4 short arcs, o o o o : also with a rad equal to the remaining part of the long diam, describe 
4 other arcs, i » ( f. The intersections of these four pairs of arcs, will give four points in the circumf. 
In this manner any number of such points may be found, and the curve be drawn by hand. 


To draw a lang-ent 11, at any point » of an ellipse. Draw nf and 

»/\ to the foci; bisect the angle/n/' by the line x p ; draw t n t at right angles to xp. 

To draw a joint «i),of an elliptic arch, from any point n, in 

the arch. Proceed as in the foregoing rule for a taugeut, only omitting ft; np will be the 
reqd joint. 






































MENSURATION, 


151 


e 



To draw an oval, or false ellipse. 

When only the long diam a b is given, the following 
will give agreeable curves, of which the span a b will 
not exceed about three times the rise c o. On a b de¬ 
scribe two intersecting circles of any rad; through 
their intersections*, v, draw eg; make s g and v e 
each equal to the diam of one of the circles. Through 
the centers of the circles, draw e y, eft, g d, g t. From 
e describe ft i y; and from g describe dot. 


d 



y 


When the span, tn n, and the 
rise, s t, are both given. 

Make any t w and m r, equal to each other, 
but each less than t s. Draw r to; aud through 
its center o draw the perp io y. Draw y >■ z. 
Make n x equal m r, and draw yxb. From x and 
r describe nc and m a; and from y describe 
ate. By making s d equal to s y, we obtain 
the center for the other side of the oval. 

The beauty of the curve will depend upon 
what portion of t s is taken for m r and t w. 
When an oval is very flat, more than three cen¬ 
ters are required for drawing a graceful curve; 
but the finding of these centers is quite as trou¬ 
blesome as to draw the correct ellipse. 


On the given line, a ft, to draw a 
eynia recta, a c s. 

Find the center c. of as. From a, o, ands, with one-half 
of a s as rad, draw the four small arcs at o, o. The inter¬ 
sections o, o, are the centers for drawing the cyma, with 
the same rad. By reversing the position of the arcs, we 
obtain the cyma reversa, or ogee, d ef. 















152 


MENSURATION, 


THE PARABOLA. 




Fig. 2. 

The common or conic parabola. 


« 6 c, Fig 1, is a curve formed by cutting a cone in a direction b a, parallel to its side. The 
curved liue o b c itself is called the perimeter of the parabola; the line o c is called its bane; b a its 
height or axi»; b its apex or vertex; any line e s, or o a, Fig 2, drawn from the curve, to, and at right 
angles to, the axis, is an ordinate ; and the part s b, or a b, of the axis, between the ordinate and the 
apex b, is an abscissa. The focus of a parabola is that point in the axis, where the abscissa b s, is 
equal to oue-half of the ord e s. The dist from the apex to the focus, is called the focal dist. The 
focus may be entirely beyond or outside of the curve itself. Its dist from the apex is found thus : 
square any ord, as oa; div this square by the abscissa ha of that ord; div the quot by 4. The 
nature of the parabola is such that its abscissas, as b s, b a. &c, are to each other as, or in proportion 
to. the squares of their respective ords e s, o a, &c; that is, as b * ; b a ; ; ea 2 : o a 2 ; or b s : e »2 :: b a : 
o a 2 . If the square of any ord be divided by its abscissa, the quot will be a constant quantity ; that 
is, it will be equal to the square of any other ord divided by its abscissa. This quot or constant quan¬ 
tity is also equal to a certain quantity called the parameter of the parabola. Therefore the paiameter 
may be found by squariug e s, or o a, (one-half of the base.) and dividing said square by the height 
b s, or b a, as the case may be. If the square of any ord be divided by the parameter, the quot will 
be the abscissa of that ord. 


To fimi tlie length of a parabolic curve. 

The approximate rule given by various pocket-books, is as follows: 

Length = 2 X V base) 2 4- \ X / A times the (Height 2 ) 


a 




Fig. 4. 


w 



Fig. 3. 


Where the height does not exceed l-10th of the base, this rule may, for practical 
purposes, be called exact. With ht = ^ base, it gives about per cent too 
much: ht = % base, about 3H percent; ht = base, about 8)^ per cent; ht = 
twice the base, about 1'2% percent; ht= 10 X base, or more, about 15^ per cent. 

The following by the writer Is correct 
within perhaps 1 part in 800, in all cases ; uud will 
therefore auswer for many purposes. 

Let « d b, Fig 3, or n u d. Fig 4, be the parabola, 
in which are given the base a 6 or n d; and the 
height cdorc a. Imagine the complete fig a d b s, 
or n a d b. to be drawn : aud in either case, assume 
its long diam a b to be the chord or base; aud one- 
half the short diam, or c d, to be the height, of a 
circular are. Find the length of this circular arc, 
by means of the rule and table given for that pur¬ 
pose. Then div the chord or base a b, or n d of 
the parabola, by its height c d or c a. Look for 
the quot in the columu of bases in the following 
table, and take from the table the corresponding 
multiplier. Mult the length of the circular are by 
this; the prod will be the length of arc a d b, or 
n a d, as the case may be. For bases of parabolas 
less than .05 of the height, or greater than lOtimcs 
the height, the multiplier is 1, and is very approx¬ 
imate; or in other words, the parabola will be 
of almost exactly the same length as the circular 
arc. 

To find (he area of a parabola m a n It. 

Mult its base m n, Fig 5, by its height a b ; and take %ds of the prod. 
The area of any segment, as u b v, whose base u v is parallel to n in, is 
found in the same way, using u v aud s b, instead of m n and a b. 

To find the area of a parabolic zone, or frns- 

t nin, as m n u v. 

Rui.k 1. First find by the preceding rule the area of the whole parabola 
mbn; then that of the segment ubv; and subtract the last from the 
first. 

Ritlk 2. From the cube of m n. take the cube of u v; call the difT c. 
From the square of m n, take the square of u v ; call the diff s. Div c bv 
«. Mult the quot by %ds of the height a s. 


✓ 















MENSURATION 


153 


Table for Ijeiigfhs of Parabolic Curves. See opp page. (Original.) 


Base. 


Mult. 


.05 

.10 

.15 

.•20 

.*25 

.30 

.35 

.40 

.45 

.50 

.55 

.60 

.65 

.70 

.75 

.80 

.85 

.90 

.95 

1.00 

1.05 


1.000 
1.001 
1.002 
i .004 
1.006 
1.007 
1.007 
1.008 
1.009 
1010 
1.010 
1.010 
1.011 
1.011 
1.010 
1.009 
1.008 
1.006 
1.004 
1.002 
1.001 


Base. 


1.10 

1.15 

1.20 

1.25 

1.30 

1.35 

1.40 

1.45 

1.50 

1.55 

1.60 

1.65 

1.70 

1.75 

1.80 

1.85 

1.90 

1.95 

2.00 

2.05 

2.10 


Mult. 


.999 

.997 

.995 

.993 

.990 

.987 

.984 

.980 

.977 

.974 

.970 

.966 

.963 

.960 

.957 

.953 

.950 

.946 

.942 

.944 

.946 


Base. 


2 15 
2.20 
2.25 
2.30 
2.35 
2.40 
2.45 
2.50 
2.55 
2.60 
2.65 
2.70 
2.75 
2.80 
2.85 
2.90 
2.95 
3.00 

3 05 
3.10 
3.15 


Mult. 


.949 

.951 

.954 

.956 

.958 

.960 

.962 

.963 

.965 

.967 

.969 

.970 

.972 

.973 

.975 

.976 

.978 

.979 

.980 

.981 

.982 


Base. 


3.20 

3.30 

3.40 

3.50 
3.60 
3.70 
3.80 
3.90 
4.00 

4.25 

4.50 

4.75 
5.00 

5.25 

5.50 

5.75 
6.00 
7.00 
8.00 

10.00 


Mult. 


.983 
.9S4 
.985 
.9S6 
.987 
.988 
.989 
.990 
.991 
.992 
.993 
.994 
.995 
.996 
.997 
.998 
.998 
.999 
1.000 
1.000 


To draw a parabola, having base cs and height e o. 

eos, Fig 6. Make o t equal to the height e o. Draw ct and f 

; and divide each of them into any number of equal parts ; 
umbering them as in the Fig. join 1,1; 2,2; 3, 3, &c; 
then draw the curve by hand. It will be observed that the 
intersections of the lines 1,1; 2, 2, &c, do not give points in 
the curve; but a portion of each of those lines forms a tan¬ 
gent to the curve. By increasing the number of divisions 
on c t and s t, an almost perfect curve is formed, scarcely 
requiring to be touched up by hand. In practice it is best 
Jrst to draw only the center portions of the two lines which 
cross each other just above o ; and from them to work down¬ 
ward; actually drawing only that small portion of each 
successive lower line, which is necessary to indicate the 
curve. 

Or file parabola may be drawn 
tEsus: 

Let 5 c. Fig 7, be the base ; and a d the height. Draw the 
rectangle b nm c; div each half of the base into any num¬ 
ber of equal parts, and number them from the center each 
way. Div n b, and m c into the same number of equal parts ; 
and number them from the top, downward. From the points 
on 6 c draw vert lines; and from those at the sides draw lines 
to d. Then the intersections of lines 1,1; 2. 2, &c, 
will form points in the parabola. As in the pre¬ 
ceding case, it is not necessary to draw the entire 
lines; but merely portions of them, as shown be¬ 
tween d and c. 

Or a parabola may be drawn by first div the 
height a b, Fig 5, into any number of parts, either 
equal or unequal; and then calculating the ordi¬ 
nates ms, &c; thus, as the height a b : square of 
half base am:: any absciss b s : square of its 
oid m «. Take the sq rt for us. 

Rem.— When the height of a parabola is not 
greater than l-10th part its base, the curve coin¬ 
cides so very closely with that of a circular are, 
that in the preparation of drawings for suspen¬ 
sion bridges. &c., the circular arc may be em¬ 
ployed; or if no great accuracy is reqd, the circle 
may be used even when the height is as great as 
on<>\eighth of the base. 

To draw a tang'ent «' v, Fig. 5, to a parabola, from any point v. 

Draw v s perp to axis a b ; prolong a b until b tv equals « 6. Join w v. 



© 




































154 


MENSURATION. 


The Cycloid, 

acb, is the curve described by a point a in the circumference of a circle, 
an , during one complete revolution of the circle, rolled along a straight line 

aft; which is called the base of the 
cycloid. 

The vertex of the cycloid is at e. 

Base, a ft, = circumference of generat¬ 
ing circle a u 

= diameter, c<l, of generat¬ 
ing circleX’r = 3.1416&L 

Axis, or height, cd—an. 
Eength. acb, = 4c d. 

Area, acbd — 3 X area of generating circle, an 
= 3^^ =* cd2 X Jjt = cd n - X 2.3562. 

Center of gravity at g. eg = §cd. 

To draw a tangent, en, from any point e in a cycloid; draw es at right 
angles to the axised; on cd describe the generating circle del ; join tc\ from 
e draw e o parallel to t c. The cycloid is the curve of quickest descent; 
so that a body would fall from ft to c along the curve bm c, in less time than 
along the inclined plane ft i c, or any other line. 



SOLIDS. 

THE REGULAR BODIES. 


A regular body, or regular polyhedron, is one which has all its 

sides, and its solid angles, respectively similar and equal to each other. There 
are but five such bodies, as follows : 


lame. 

Bounded by 

Surface 

(=sum of surfaces 
of all the faces). 

Multiply the square 
of the length of 
one edge by 

Volume. 

Multiply the 
cube of the 
length of one 
edge by 

Tetrahedron. 

4 equilateral triangles. 

1.7320 

.1178 

Hexahedron or cube 

6 squares. 

6. 

1 

Octahedron. 

8 equilateral triangles. 

3.4641 

.4714 

Dodecahedron. 

12 “ pentagons. 

20.6458 

7.6631 

Icosahedron. 

20 “ triangles. 

8.6602 

2.1817 


Guldinus' Theorem. To find 
Fig. A. Fig. B. 

I 

I 


X 



I 

I 

If the revolution is incomplete, 

complete . incomplete 
revolution ' revolution : * 


the volume of any body fas the 

irregular mass dftcro, Fig A, or the ring 
ah cm, Fig B), generated by a complete 
or partial revolution of anv figure (as 
ahea) around one of its sides (as ae. 
Fig A), or around any other axis (as 
X V± Fig B). 

Volume = surface abcaX length 
of arc described by iis center of grav¬ 
ity G. 

If the revolution is complete, the arc 
described is = circumference = radius 
o G* X 2tt = radius o G* X 6.283186; and 
Volume ^surface abca X radius 
o G * X 6.283186. 

circumference . arc 
found as above ' described 


* Measured perpendicularly to the axis of revolution. 






























MENSURATION. 


155 


PAKALLELOPIPEDS. 



Fig. 1. Fig. 2. Fig. 3. Fig. 4. 

A parallelopiped is any solid contained within six sides, all of which are 
parallelograms; and those of each opposite pair, parallel to each other. We 
show but four of them ; corresponding lo the four parallelograms; namely, the 
cube. Fig 1, which has all its sides equal squares, and all its angles right angles; 
the right rectangular prism, Fig 2, has all its angles right angles, each pair of 
I opposite faces equal, but not all of its faces equal; the Rhumbohedron , Fig 3, 
which has all its sides equal rhombuses, and which, like the rhombus, p 119, is 
sometimes called “rhomb ” ; the Rhombic prism, Fig 4; its faces, rhombuses, or 
rhomboids, each pair of opposite faces equal, but not all its faces equal. All 
parallelopipeds are prisms. 

Volume of any _ area of any face, w perpendicular distance, p, 
parallelopiped as a, A to the opposite face. 

Volume of a cube = cube of length of one edge, 

= 1.90985 X volume of inscribed sphere, 

= 1.27324 X “ “ cylinder, 

= 3.81972 X “ “ cone. 

Diagonal of a cube = diameter of circumscribing sphere, 

— 1.7320508 X length of one edge of cube. 

The diagonal of a rhomb, or of a rhombic prism, cannot be calculated by 
means of its sides and their angles. 


PRISMS 



form the ends are equal, and the angles included 


equal, the prism is said to be regular : otherwise, irregular. 


A prism is any solid whose 
twoe«d.vare parallel, similar, 
and equal; and whose sides 
are parallelograms , as Figs 5 
to 10. Consequently the fore¬ 
going parallelopineds are 
prisms. A right prism is une 
whose sides are perpendic¬ 
ular to its ends as 5, 6, 7 ; 
w r hen not so, the prism is 
oblique, as 8, 9,10. When all 
the sides of the fi gures which 
between those sides are also 


Volume of any prism (whether regular or irregular, right or oblique) 
— area of one end X perpendicular distance, p, to the other end, 

= area of cross section perpendicular to the sides X actual length, ah, Figs 
5 to 10, 

= 3 X volume of pyramid whose base and height are = those of the prism. 


To find tlie volume of any frustum* 
of any prism. 

Whose cross section, perpendicular to its sides, 
is either any triangle; any parallelogram; a 
square, (as in Fig 10 1 /£) or a regular polygon of 
any number of sides; no matter how the twrn 
ends of the frustum may be inclined with regard 
to each other; or whether one, or neither of 
them, is parallel to the base of the original 
prism. 



Volu me 
of frustum 


sum of lengths of parallel edges, 
n+22+33+il 

number of such edges 
(4 in Fig 10 l 4) 


Figs. 10 

area of cross section 
perpendicular 
to such edges. 





















































156 


MENSURATION. 



Fig. 10)4 



This rule may be us p d for ascertaining beforehand, the quantity of earth to 
be removed from a “ borrow pit.” The irregular surface of the ground is first | 
staked out in squares; (the tape-line being stretched horizontaUy,viheu me as- j 

uring off their sides). These squares should be of such 
a size that without material error each of them may be j 
considered to lie a plane surface, either horizontal or in¬ 
clined. The depth of the horizontal bottom of the pit 
being determined on, and the levels being taken at every 
corner of the squares, we are thereby furnished with the 
lengths of the four parallel vertical edges of each of the 
resulting frustums of earth. In Figs 10 l 4 y may be sup¬ 
posed to represent one of these frustums. 

If the frustum is that of an iiregular 4-sided, or polyg¬ 
onal prism, first divide its cross section perpendicular to its sides, into tri¬ 
angles, by lines drawn from any one of its angles, as a, Fig 10*4 Calculate the 
area of each of these triangles separately ; then consider the entire frustum to 
be made up of so many triangular ones; calculate the volume 
of each of these by the preceding rule for triangular frustums; j 
and add them together, for the volume of the entire frustum. 

Volume of any frustum of any prism. 

Or of a cylinder. Consider either end to be the base ; and find its 
area. Also find the center of gravity c of the other end, and the 
n ^^ perpendicular distance n c, from the base to said center of gravity. 

Fig. 10%. Then Volume of frustum = area of base Xnc, Fig 10%. 

The slant end, c, is an ellipse. Its area is greater than that of the circular end. 
Surface of any prism. Figs 5 to 10, whether right or oblique, regular 
or irregular 

( circumference measured w „ .\ . sum of the areas 

= (perpendicular to the sides X actual leD S th ’ a b ) + of the two ends. 

CYLINDERS. 

A cylinder is any solid whose ends are 
paraliel, similar, and equal curved figures; 
and whose sections parallel to the ends 
are everywhere the same as the ends. 
Hence there are circular cvlind rs, ellip¬ 
tic cylinders (or cylmdroids) and many 
others; but when not otherwise expressed, 
the circular one is understood. A right 
cylinder is one whose ends are perpen¬ 
dicular to its sides, as Fig. 11; when oti er- 
wise, it is oblique, as Fig 12. If the ends 
of a right circular cylinder be cut so as to 




Fig. 11. 



make it oblique, it becomes an elliptic one ; because then both its ends, and all 
sections parallel to them, are ellipses. An oblique circular cylinder seldom 
o curs; it maybe conceived of by imagining the two ends of Fig 12 to be circles, 
united by straight lines forming its curved sides 
A cylinder is a prism having an infinite number of sides. 

Volume of any cylinder ( whether circular or elliptic, Ac, right or oblique) 
= area of one end X perpendicular distance, p, to the other end, 

= { m e“ s Tm/p%Tto e the°?Me S x *«<“' <* »• «*» 11 ”" d «• 

— 3 x volume of a cone who*e base and height are -= those of the cylinder. 
Surface of any cylinder (whether circular or elliptic, Ac, right or oblioue) 

/circumference " \ sum of the areas 

= 1 measured perpendicularly X actual length, ab )+" 0 f t p e two ends 
\to the sides, as at c o, Fig 12, ’ 

lligiit circular cylinder whose height = diameter. 

Volume = X volume of inscribed sphere. 

Curved surface = surface of inscribed sphere. 

Area of one end == i surface of inscribed sphere = £ curved surface. 

Entire surface = X surface of inscribed sphere = H X curved surface. 

















CONTENTS OF CYLINDERS, OR PIPES, 


157 


ogi C ^ ,lto,, * s one foot length, in Cub Ft, and in U. S. Gallons of 
^™. U Si n f» 01 i« 0:> na « s o° a ''V A cul> ft of water weighs about 62^ lbs ; and a gallon 
about lbs. Dunns 2. 8. or 10 times as great, give 4, 9, or 100 times the content. 

tor the weight of water in pipes, see Table 2 page 246 


No errors. 




For 1 ft. in 



For X ft in 



For 1 ft. in 

Diam. 

Diam. 

length. 



length. 



length. 



Diam. 

Diam. 




Diam. 



in 

in deci- 

-w —> 

«•- CD 

in 

in deci- 

. £3 

V- CO 

Diam. 

iu deci- 

. _ 

f 

Ius. 

mals ot 

0> ^ 

o a 

Ins. 

mals of 

1/ M 

o a 

iu 

mals of 

O _» 

0 a 


a foot. 

fa £ 

. a . 

a S « 

P.O 
£ 3 


a foot. 

ub. F( 

so are 

sq. ft 

E U3 
£ 3 
•3® 

Ins. 

a foot. 

£ 

i ei 

or. 

c O 
£ 3 

•30 




OSs 

M 




C5£ 

CM 




i co 

CM 

x 

.0208 

.0003 

.0025 

% 

.5625 

.2485 

1 859 

19. 

1.5S3 

1.969 

14.73 

5-16 

.0260 

.0005 

.0040 

7.- 

.5833 

.2673 

1.999 

X 

1.625 

2.074 

15.51 

% 

.0313 

.0008 

.0057 

x 

U 

.6042 

.2867 

2.145 

20. 

1.667 

2.182 

16.32 

7-16 

.0365 

.0010 

.0078 

.6^50 

.3068 

2.295 

y 2 

1.708 

2 292 

17.15 

X 

0417 

.0014 

.0102 

% 

.6458 

.3276 

2.450 

2 . 

1.750 

2 405 

17.99 

9 16 

.0469 

.0017 

.0129 

8. 

.6667 

.3491 

2.611 

X 

1.792 

2.521 

18.86 

% 

.0521 

.0021 

.0159 

l 4 

.6875 

.3712 

2.777 

22. 

1.833 

2.640 

19 75 

11-10 

.0573 

.0016 

.0193 

i 

.7083 

.3941 

2.948 

X 

1.875 

2.761 

20.66 

% 

.0625 

.0031 

.0230 

.7292 

.4176 

3.125 

23 

1.917 

2.885 

21.58 

1-5-1 6 

.0677 

.0)36 

.0269 

9. 

.7500 

.4418 

3.305 

X 

1.958 

3 012 

22.53 

Vh 

.(1729 

.0042 

.0312 

X 

U 

.7708 

.4667 

3 491 

24. 

2.000 

3.142 

29.50 

15-16 

.0781 

.0048 

.0359 

.7917 

.4922 

3.682 

25. 

2.083 

3.409 

25.50 

1. 

.083 5 

.0055 

.0408 

% 

.8125 

.5185 

3.879 

26. 

2.167 

3.687 

27.5S 

X 

.1042 

.0085 

.0638 

10. 

.833.3 

.5454 

4.085 

27. 

2.250 

3.97 6 

29.74 


.1250 

.0123 

.0918 

X 

.85 42 

.5730 

4.286 

28. 

2.333 

4 276 

31.99 


.1458 

.0167 

.1249 

1 2 

.8750 

.6013 

4.498 

29. 

2.417 

4.587 

34 31 

2 

.1667 

.0218 

.1632 

% 

.8958 

.6303 

4.715 

30. 

2.500 

4.909 

36.72 


.1875 

.0276 

.2066 

m . 

.9167 

.6600 

4.937 

31. 

2.583 

5.241 

39.21 


.2083 

.0341 

.2550 

X\ 

.9375 

.6903 

5.164 

32. 

2.667 

5 585 

41.78 

% 

.2292 

.0412 

.3085 

x 

.9583 

.7213 

5.396 

33. 

2.750 

5.940 

44 43 

3. 

.2500 

.0491 

.3672 

% 

.9792 

.7530 

5.633 

34. 

2.833 ■ 

6.305 

47.15 

X 

.2708 

.0576 

.4:109 

1 2. i 

1 Foot. 

.7854 

5.875 

$5* 

2.917 

6.681 

49.98 


.2917 

.0668 

.4998 

M 1-042 

.8522 

6.375 

36. 

3 000 

7.069 

52.88 

% 

.3125 

.0767 

.5738 

13. i 

1.083 

.9218 

6.895 

37. 

3.083 

7.467 

55.86 

4. 

.3333 

.0873 

.6528 

X 1-125 

.9940 

7.436 

38, 

3.16? 

7.876 

58.92 

X 

.3542 

.0985 

.7369 

14. 

1.167 

1.069 

7.997 

39. 

3.250 

8.296 

62.06 


.3750 

.1104 

.8263 

lXl-208 

1.147 

8.578 

40. 

3.3.33 

8.727 

65.28 

% 

.3958 

.1231 

.9206 

15. | 

1.250 

1.227 

9.180 

41. 

3.417 

9.168 

6S.58 

5. 

.4167 

.1364 

1.020 

X 1-292 

1.310 

9.80 L 

42! 

3.500 

9.621 

71.97 

X 

.4375 

.1503 

1.125 

16. |1.333 

1.396 

10.44 

43. 

3.5^3 

10.085 

75.44 

x 

.4583 

.1650 

1.234 

X 1.375 

1.485 

11.11 

44. 

3.667 

10.559 

78.99 

% 

.4792 

.1808 

1.349 

17. jl.417 

1 576 

11.79 

45. 

3.750 

11.045 

82.62 

6 

.5000 

.1963 

1.469 

X'A58 

1.670 

12.49 

46. 

3.833 11.541 

86.33 

X 

.5-208 

.21.31 

1.594 

18. 11.500 

1.767 

13.22 

47. 

3917 

12.048 

90.13 

x 

.5417 

.2304 

1.724 

X 1.542 

I 

1.867 

1 

13.96 

48. 

1 

4.000 

12.566 

94.00 


Table continued, but with the diams in feet. 


Diam. 

Feet. 

Cub. 

Feet. 

u. s. 

Galls. 

Diam. 

Feet. 

Cub. 

Feet. 

u. s. 

Galls. 

Dia. 

Feet. 

Cub. 

Feet. 

u. s. 

Galls. 

Dia. 

Feet. 

Cub. 

Feet. 

U. S. 
Galls. 

4 

12.57 

94.0 

7 

38.49 

287.9 

12 

113.1 

846.1 

24 

452.4 

3384 

X 

14.19 

106.1 

X 

41.28 

308.8 

13 

132.7 

992.8 

25 

490.9 

3672 

u 

15.90 

119.0 

X 

44.18 

330.5 

14 

153.9 

1152. 

26 

530.9 

3971 

k 

17.72 

132.5 

k 

47.17 

352.9 

15 

176.7 

1322. 

27 

572.6 

4283 

5 

19.64 

146.9 

8 

50.27 

376.0 

16 

201.1 

1504. 

28 

615.8 

4606 

X 

21.65 

161.9 

X 

56.75 

424.5 

17 

227.0 

1698. 

29 

660.5 

49-11 

Yi 

23.76 

177.7 

9 

63.62 

475.9 

18 

254.5 

1904. 

30 

706.9 

5288 

k 

25.97 

194.3 

X 

70.88 

530.2 

19 

283.5 

2121. 

31 

754.8 

5646 

6 

28.27 

211.5 

10 

78.54 

587.6 

20 

314.2 

2350. 

32 

804.3 

6017 

X 

30.68 

229.5 

X 

86.59 

647.7 

21 

346.4 

2591. 

33 

855.3 

6398 

X 

33.18 

248.2 

11 

95.03 

710.9 

22 

380.1 

2844. 

34 

907.9 

6792 

% 

35.79 

267.7 

X 

103.90 

777.0 

23 

415.5 

3108. 

35 

962.1 

7197 





































































158 


CONTENTS AND LININGS OF WELLS 


CONTEXTS AND EININCS OF WELLS. 


For diams twice as great as those in the table, for the cub yds of digging, take out those opposite 
one half of the greater diarn ; and mult them by 4. Thus, for the cub yds in each foot of depth of a 
well 31 feet in diam, tirst take out from the table those opposite the diam of 15>*> feet; namely, 6.989. 
Then 6.989 X 4 — 27.956 cub yds reqd for the 31 ft diam. But for the stoue lining or walling, bricks 
or plasteriug, mult the tabular quantity opposite half the greater diam, by 2. Thus, the perches of 
stoue walling for each foot of depth of a well of 31 ft diam, will be 2.073 X 2 — 4.146. If the wall is 
more or less than one foot thick, within usual moderate limits, it will generally be near euougli for 
practice to assume that the number of perches, or of bricks, will increase or decrease in the same pro¬ 
portion. 

The size of the bricks is taken at 8)4 X 4 X 2 inches: and to be laid dry, or without mortar. In 
practice an addition of about 5 per ceut should be made for waste. The brick lining is supposed to 
be 1 brick thick, or 8)4 ins. 

CACTION. — Be careful to observe that the diams to he used for the digging, 

are greater than those for the walling, bricks, or plastering. No errors. 


Diam. 

in 

Feet. 


1 . 


2 . 


3. 


4. 


5. 


6 . 


7. 


8 . 


10 . 


11 . 


12 . 


13. 


X 

X 

X 

X 

X 

X 

X 

X 

X 

X 

Vi 

X 

X 

Vi 

X 

X 

Vi 

X 

X 

Vi 

X 

X 

X 

X 

X 

X 

X 

X 

X 

X 

X 

X 

X 

X 

X 

X 


For each foot of depth. 

Diam. 

in 

Feet. 

For each foot of depth. 

For this 
col use the 
Diameter 
of the 
Digging. 

For these three cols use the 
diam in clear of the lining. 

For this 
col use the 
Diameter 
of the 
Digging. 

For these three cols use the 
diam in clear of the lining. 

Stone 

Dining 

1 ft thick. 
Perches of 
25 Cub Ft. 

No. of 
Bricks in 
a Lining 

1 Brick 
thick. 

Square 
Yards of 
Plaster¬ 
ing. 

Stone 

Lining 

1 ft thick. 
Perches of 
25 Cub Ft. 

No. of 
Bricks iu 
a Lining 

1 Brick 
thick. 

Square 
Y ards 
of Plas¬ 
tering. 

Cub Yds. 
of 

Digging. 

Cub Yds. 
of 

Digging. 

.0291 

.2513 

57 

.3491 

X 

5.107 

1.791 

750 

4.625 

.0455 

.2827 

71 

.4364 

X 

5.301 

1.822 

764 

4.713 

.0654 

.3142 

85 

.5236 

X 

5.500 

1.854 

778 

4.800 

.0891 

.3456 

99 

.6109 

14. 

5.701 

1.885 

792 

4.887 

.1164 

.3770 

114 

.6982 

X 

5.907 

1.916 

806 

4.974 

.1473 

.4084 

128 

.7855 

X 

6 116 

1.948 

820 

5.062 

.1818 

.4398 

142 

.8727 

X 

6.329 

1.979 

834 

5.149 

.2200 

.4712 

156 

.9000 

15. 

6.545 

2.011 

849 

5.236 

.2618 

.5027 

170 

1.047 

X 

6.765 

2.042 

863 

5.323 

3073 

.5341 

184 

1.135 

X 

6.989 

2.073 

877 

5.411 

.3563 

.5655 

198 

1.222 

X 

7.216 

2.105 

891 

5.498 

.4091 

.5969 

212 

1.309 

16. 

7.447 

2.136 

905 

5.585 

.4654 

.6283 

227 

1.396 

X 

7.681 

2.168 

919 

5 673 

.5254 

.6597 

241 

1.484 

X 

7.919 

2.199 

933 

5.760 

.5890 

.6912 

255 

1.571 

X 

8 161 

2.231 

948 

5.847 

.6563 

.7226 

269 

1.658 

17. 

8.407 

2.262 

962 

5.934 

.7272 

.7540 

283 

1.745 

X 

8.656 

2.293 

976 

6.022 

.8018 

.7854 

297 

1.833 

X 

8.908 

2.325 

990 

6.109 

.8799 

.8168 

311 

1.920 

X 

9.165 

2.356 

1004 

6.196 

.9617 

.8482 

326 

2.007 

18. 

9.425 

2.388 

1018 

6.283 

1.047 

.8796 

340 

2.095 

X 

9.688 

2.419 

1032 

6.371 

1.136 

.9111 

354 

2.182 

X 

9.956 

2.450 

1046 

6.458 

1 229 

.9425 

368 

2.269 

X 

10.23 

2.482 

1061 

6.545 

1.325 

.9739 

382 

2.356 

19. 

10.50 

2.513 

1075 

6.633 

1.425 

1.005 

396 

2.444 

X 

10.78 

2.545 

1089 

6.720 

1.529 

1.037 

410 

2.531 

X 

11.06 

2.576 

1103 

6 807 

1.636 

1.068 

425 

2.618 

X 

11.35 

2.608 

1117 

6.>*94 

1.747 

1.100 

439 

2.705 

20. 

11.64 

2.639 

1131 

6.982 

1.862 

1.131 

453 

2.793 

X 

11.93 

2.670 

1145 

7 069 

1.980 

1.162 

467 

2.880 

X 

12.22 

2.702 

1160 

7.156 

2.102 

1.194 

481 

2.967 

X 

12.52 

2.733 

1174 

7.243 

2.227 

1.225 

495 

3.054 

21. 

12.83 

2.765 

1188 

7.331 

2.356 

1.257 

509 

3.142 

X 

13.14 

2.796 

1202 

7.418 

2.489 

1.288 

523 

3.229 

X 

13.45 

2.827 

1216 

7.505 

2.625 

1.319 

538 

3.316 

X 

13.76 

2.859 

1230 

7 593 

2.765 

1.351 

552 

3.404 

22. 

14.08 

2.890 

1244 

7.680 

2.909 

1.382 

566 

3.491 

X 

14.40 

2.922 

1259 

7.767 

3.056 

1.414 

580 

3.578 

X 

14.73 

2.953 

1273 

7.854 

3.207 

1.445 

594 

3. 665 

X 

15.06 

2.985 

1287 

7.942 

3.362 

1.477 

608 

3.753 

23. 

15.39 

8.016 

1301 

8.029 

3.520 

1.508 

622 

3.840 

X 

15.72 

3.047 

1315 

8.116 

3.682 

1 539 

637 

3.927 

X 

16.06 

3.079 

1329 

8.203 

3.847 

1.571 

651 

4.014 

X 

16.41 

3.110 

1343 

8.291 

4.016 

1.602 

665 

4.102 

24. 

16.76 

3.142 

1357 

8..I78 

4.189 

1.634 

679 

4.189 

X 

17.11 

3.173 

1372 

8.465 

4.365 

1.665 

693 

4.276 

X 

17.46 

3 204 

1386 

8.552 

4.545 

1.696 

707 

4.364 

X 

17.82 

3.236 

1400 

8.6+0 

4.729 

1.728 

721 

4.451 

25. 

18.18 

3.267 

1414 

8.727 

4.916 

1.759 

736 

4.538 










A cub y«I 

= 202 U. S. s a 

Is. 


If perches are named in a contract, it is necessary, in order to prevent fraud* 

to specify the number of cub feet contained in the perch; for stonequarriers have one perch, stone¬ 
masons another, <tc. Knciueers, on this accouut. contract by the cubic yard The perch should be 
done away with entirely ; Perches of 25 cub ft X .926 = cub yds ; and cub yds -r .926 ~ pers of 25 cub ft. 



































CYLINDRIC UNGULAS, ETC. 


159 


CIRCULAR CYLINDRIC UNGULAS. 

I. When the cutting plane does not cut the base. Figs 13,14. 




Volumel = areaof baseo^X %sum of greatest & least perp heights, on, cm, 
ot f area of cross sec measd v % sum of greatest and least, lengths, 

uugula ) \ perp to sides, as a:, Fig 14 A gm, ot, measd along the sides. 

f circumf measd perp .. half sum of greatest and least lengths, 
1 to sides, as at x, A gm,o t, measd along the sides. 

Add areas of ends if required. 

For areas of sections perpendicular to the sides, see Circles, pp 123, &c. 
For areas of sections oblique to the sides, see The Ellipse, pp 149, &c. 


Area of 

curved 

surface 


II. When the cutting plane touches the base. Figs A to D. 



Volume 


Fig A = (%ab s — a c X area ad mb* of base) 
Fig B = 

Fig C = (% a 6 3 + a c X area a drub* of base) 


Curved 

surface 


Fig I> = % area of circle ym\ X mn 
= % volume of cylinder xy mn. 

Fig A — (ab X my — ac X length of arc dmb\) 


Fig R — my X mn. 

Fig C = (ab X my + ac X length of arc dmb%) 

Fig 1>= % circumference of basef my X mn 
= 14 curved surface of cylinder xymn. 

For area of sloping plane, see p ISO. 


* For area of base (segment of circle), see pp 146 to 148. 
f For circles, see pp 123, etc. 

X For length of circular arc, see pp 141, etc. 


































































160 


PYRAMIDS AND CONES 


PYRAMIDS AND CONES. 




A pyramid, 

Figs 1,2,3, Is any solid which has for its base, a plane fig of any number of sides: and for its sides, 
plane triangles, ali terminating at one point d, called its apex, or top. When the base is a regular fig, 
the pyramid is regular ; otherwise irregular. For regular figs see Polygons, p. 110. 

A cone. 

Figs 1 and5,is a solid, of which the base is a curved fig; and which may be considered as made or 
generated by a line, of which one end is stationary at a certain point d, called the apex or top, while 
the line is being carried around the circumf of the base, which may be a circle, ellipse, or other curve. 

The axis of a pyramid, or cone, is a straight line do in Figs 1, 2,4: and d i in Figs 3 and 5. from the 
apex d, to the centre of the base. When the axis is perp to the base, as in Figs 1, 2,4, the solid is said 
to be a right one; when otherwise, as Figs 3. 5, an oblique one. When the word cone is used alone, 
the right circular cone. Fig 4, is understood. If such a cone be cut, as at t t, obliquely to its base, the 
new base 11 will be an ellipse ; and the cone d 1 1 becomes an oblique elliptic one. Fig 5 will represent 
either an oblique circular cone, or an oblique elliptic one, according as its base is a circle, or au ellipse. 

To find the solidity of any pyramid, or cone. 

Whether regular or irregular; right or oblique; mult the area of its base, by one-third of its perp 
height d o. Figs 1 to 5. Every pyramid, or cone, has one-third of the solidity of either a prism or a 
cylinder having the same area of base, and the same perp height; and one-half that of a hemisphere 
of the same base and height; in other words, a cone, hemisphere, and cylinder of the same base and 
height, have solidities as 1, 2, 3. 

To find the surface of any regular right pyramid, or right 

cone. 




r 

i 




f 


i; 


Mult the circumf or outline of its base, by the slant height; take half the prod. This will give the 
surf of the sides ; to which add that of the base if reqd. In the pyramid, this slaut height must he 
measd from d to the middle of one of the equal sides, and not alone one of the edees of the pyramid. 
Mathematicians have been unable to devise any measurement of the surf of an oblique cone. 

To find the surface of an irregular pyramid. 

Whether right or oblique, each side must be calculated as a separate triangle; and the several area* 
added together. Add the area of base if reqd. 


To find the solid* 
ity of any frus¬ 
tum of any pyr¬ 
amid, or cone, 
when the batte 
and top are par¬ 
allel. 


Rule 1. Whether regular 
or irregular, right or oblique, 
add together the areas of the 
base, and top, and the area of 
their mean proportional; mult 
the sum by one-third of the 




Fig. 6. 


Fig. 7. 


perp height o o, Figs 6 and 7. between the base and top. 

Rem. For the area of the mean proportional, (which is not the area halfway between, and parallel 
with the ends,) mult together the areas of base and top , and take the sq rtof the prod. 

Rule 2. This applies only to the right circular conical frust. Add together the squares of the two 
diams; and the prod of the two diams; mult together the sum, one-third the perp height oo, and .7854. 


To find tlie surface of any frustum of a regular right pyramid, 
or cone. Figs « ami 7. when the base and top are parallel. 




Add toeether the clrcumfs of the two ends; mult the sum by the slant height s t ; take half the prod. 
This gives the surf of the sides: to which add that of the ends when reqd. In the frustum of the 
pyramid, the slant height must be measd between the middles s and t of two corresponding sides o* 
the base, and top. Fig. 6. 

If the pyramid is not regular: or if it is oblique, then the surf of the sides must be obtaiued for 
each side separately as a trapezoid. 


























PIUSMOIDS 


161 




PR IS MO IDS. 




A prismoid is 

Any solid bounded by six plane surfaces, of which but two, as a 5 c d and e /g h, Figs 1 and 2, are 
necessarily parallel; aud at least two other opposite ones not parallel. 

To find the solidity of any prisnioid. 

Add together the areas of the two parallel surfaces ; and four times the area of the section taken 
halfway between them, and parallel to them : mult the sum by the pern dist between the two parallel 
sides; div the prod by 6. To find the areas referred to, see Trapezoids, Trapeziums. 

p 120. 

The foregoing rule is the well-known “ prismoidal formula the very extended application of 
which to other solids than those which fall strictly wvthin the definition of the prismoid, was first 
discovered aud made known by Ellwood Morris, civ eng of Philadelphia, in 1840. It embraces all 
parallelopipeds, prisms, pyramids, cones, wedges, &c, whether regular or irregular, right or oblique; 
together with their frustums when cut parallel to their bases; indeed all solids whatever having two 
parallel faces, or sides, provided these two faces are united by surfaces, whether plaue or curved, upoa 
which, and through every point of w hich, a straight line may be drawn from one of the parallel faces 
to the other. The follow ing six Figs represent a few such solids ; they may be regarded as one-chain 
lengths of railroad cuttings; a o being the perp dist between the two parallel ends. 





The prismoidal formula applies also to the sphere, 
hemisphere, aud other spherical segments; also to 
any sections such as abed, and o n i d b c, of the 
cone, in which the sides ad, a c, or od, ic , are straight f 
as they are only when the cutting plane a d c passes 
through the apex, or top a. Also to the cylinder 
when a plane parallel to the sides passes th rough 
both ends, but not if the plaue w x is oblique, as 
in the fig, though never erring more than I in 
142. In this last ease we must imagine the 
plane to be extended until it cuts the side of the 
cylinder likewise extended; and then by page 159 
find the solidity of the ungula thus formed. Then 
find the solidity of the small ungula above to, also 
thus formed, and subtract it from the large one. 


11 
















162 


WEDGES, SPHERES. 


WEDGES. 



A wedge 

Js usually defined to be a solid, Figs 8 aDd 9, generated by a plane triangle, one, moving parallel to 
itself, in a straight line. This definition requires that the two triangular ends of the wedge should be 
parallel; but a wedge may be shaped as in Fig 10 or 11. We would therefore propose the following 
definition, which embraces all the figs; besides various modifications of them. A solid of five plane 
faces; one of which is a parallelogram abed, two opposite sides of which, as a c and b d. are united 
by means of two triangular faces a cn, and A dm, to an edge or line n m. parallel to the other opposite 
sides a b and c d. The parallelogram abed may be either rectangular, or not; the two triangular 
faces may be similar, or not; and the same with regard to the other two faces. The following rule 
applies equally to all. 

To find the solidity of any wedge. 

Add together the length of the edge m n. and twice the length a b or c d of the back ; mult the sum 
by the perp height p. from the edge to the back; mult this prod by the breadth of the back, measd 
perp to Us two sides a b and c d ; div this last prod by 6. 


SPHERES OR GLOBES. 


A sphere 

Is a solid generated by the revolution of a semicircle around its diam. Any line passing entirely 
through a sphere, and through its center, is called its axis, or diam. Any circle described on the 
surface of a sphere, from the center of the sphere as the center of the circle, is called a great circle of 
that sphere; in other words, any entire circumf of a sphere is a great circle. A sphere has a greater 
content or solidity than any other solid with the same amount of surface; so that if the shape of a 
sphere be any way changed, its content will be reduced. 


To find the solidity of a sphere.* 

Cube its diam; mult said cube by .5236. Or cube its circumf; and mult by .01689. Or mult its 
surface by its diam; and div by 6. Or refer to the following table of spheres. The solidity of a sphere 
is % that of its circumscribing cylinder; or .5236 that of its circumscribing cube; or = 4 great 
circles X diam; = cube of rad X 4.1888; = surface X % diam. 8 


To find the surface of a sphere.* 

Square its diam; mult said square by 3.1416. Or mult the diam by the circumf. Or sonare its 
circumf: and mult by .3183. The surf of a sphere is equal to 4 times the area of its great clrcTe or 
to the area of a circle whose diam is twice as great as that of the sphere; or to the curved surf of' its 
circumscribing cylinder; or to sq of rad X 12.5664; or to solidity -r % diam. 1 1 


toTS 1 * tab,es of snrfaces «n<l solidities of spheres, see pp 163 


* If the diam is measured in inches, divide the surfaces in the table by 144 if it is rennirort 

them to square feet; and divide the solidities by 1728, if required in cubic feet. q d 10 reduoe 























MENSURATION 


163 


SPHERES. (Original.) 

Some errors of 1 in'thelast figure only. 


a 

3 

GO 

o 

GO 

a 

a 

GO 

O 

GG 

a 

a 

GO 

CG 

a 

3 

O 

G fl 

1 64 

.00077 


13-32 

18.190 

7.2949 

h 

170.87 

210.03 

X 

921.33 

2629.6 

1 32 

.00307 

.00002 

7-16 

18.6Mi 

7.5829 

k 

176.71 

220.89 

X 

934.83 

2687. H 

3-64 

.00690 

.00005 

16-32 

19.147 

7.8783 

X 

182.66 

232.13 

h 

948.43 

2746.5 

1 16 

.01227 

.00013 


19.635 

8.1813 

X 

188.69 

243.73 

X 

962.12 

2806.2 

3-32 

.02761 

.00043 

17-32 

20.129 

8.4919 

X 

194.83 

255.72 

% 

975.91 

2866.8 

X 

.04909 

.00102 

9-16 

20.629 

8 8103 

8. 

201.06 

268.08 

X 

989.80 

2928.2 

5-32 

.07670 

.00200 

19-32 

21.135 

9.1366 

X 

207.39 

280.85 

X 

1003.8 

2990.5 

3-16 

.11045 

.00345 

H 

21.648 

9.4708 

% 

213.82 

294.01 

18. 

1017.9 

3053.6 

7-32 

.15033 

.00548 

21-32 

22.166 

9.8131 

^8 

220.36 

307.58 

X 

1032.1 

3117.7 

X 

.19635 

.00818 

11-16 

22.691 

10.164 

X 

226.98 

321.56 

X 

1046.4 

3182.6 

9-32 

.24851 

.01165 

23-32 

23.222 

10.522 

X 

233.71 

335.95 

% 

1060.8 

3248.5 

5-16 

.30680 

.01598 

3 4 

23.758 

10.889 

X 

240.53 

350.77 

X 

1075.2 

3315.3 

11-32 

.37123 

.02127 

25-32 

24.302 

11.265 

X 

247.45 

366.02 

X 

1089 8 

3382.9 

X 

.44179 

.02761 

13 16 

24.850 

11.649 

9. 

254.47 

381.70 

X 

1104.5 

3451.5 

13-32 

.51848 

.03511 

27-32 

25.405 

12.041 

H 

261.59 

397.83 

X 

1119.3 

3521.0 

7 16 

.60132 

.04385 

X 

25.967 

12.443 

X 

268.81 

414.41 

19. 

1134.1 

3591.4 

15-32 

.69028 

.05393 

29 32 

26.535 

12.853 

h 

276.12 

431.44 

X 

1149.1 

3662.8 

X 

.78540 

06545 

15-16 

27.109 

13.272 

X 

283.53 

448.92 

X 

1164.2 

3735.0 

17-32 

.88664 

.07850 

31-32 

27.688 

13.700 

% 

291.04 

466 87 

X 

1179.3 

8808.2 

9-16 

.99403 

.09319 

a. 

28.274 

14.137 

X 

298.65 

485.31 

X 

1194.6 

3882.5 

19-32 

1.1075 

.10960 

1-16 

29.465 

15.039 

X 

306.36 

504.21 

% 

1210.0 

3957 6 

% 

1.2272 

.12783 

X 

30.680 

15.979 

10. 

314.16 

523.60 

X 

1225.4 

4033.7 

21-32 

1.3530 

.14798 

3-16 

31.919 

16.957 

X 

322.06 

543 48 

X 

1241.0 

4110.8 

11-16 

1.4849 

.17014 

X 

33.183 

17.974 

X 

330.06 

563.86 

20. 

1256.7 

4188.8 

23-32 

1.6230 

.19442 

5-16 

34.472 

19.031 

X 

338.16 

581.74 

X 

1272.4 

4267.8 

X 

1.7671 

.22089 

X 

35.784 

20.129 

X 

346.36 

606.13 

X 

1288 3 

4347.8 

25-32 

1.9175 

.24967 

7-16 

37.122 

21.268 

X 

354.66 

628.04 

% 

1304.2 

4428.8 

13-16 

2.0739 

.28084 

X 

38.484 

22.449 

X 

363.05 

650.46 

X 

1320.3 

4510.9 

27-32 

2.2365 

.31451 

9 16 

39.872 

23.674 

X 

371.54 

673.42 

% 

1336.4 

4593 9 

X 

2.4053 

.35077 

% 

41.283 

24.942 

11. 

380.13 

696 91 

X 

1352.7 

4677 9 

29 32 

2.5802 

.38971 

11-16 

42 719 

26.254 

X 

388.83 

720.95 

X 

1369.0 

4763.0 

15-16 

2.7611 

.43143 

h 

44.179 

27.611 

X 

397.61 

745.51 

21. 

1385.5 

4849.1 

31-32 

2.9483 

.47603 

13-16 

45 664 

29.016 

% 

406.49 

770.64 

X 

1402.0 

4936.2 

1. 

3.1416 

.52360 

x 

47.173 

30.466 

X 

415.48 

796.33 

X 

1418.6 

5024.3 

1-32 

3.3410 

.57424 

15-16 

48.708 

31.965 

X 

424.56 

822.58 

X 

1435.4 

5113.5 

1-16 

3.5466 

.62804 

4. 

50.265 

33.510 

X 

433.73 

849.40 

X 

1452.2 

5203.7 

3-32 

3.7583 

.68511 

1 16 

51.848 

35.106 

X 

443.01 

876.79 

% 

1469.2 

5295.1 

X 

3.9761 

.74551 

X 

53.456 

36.751 

12. 

452.39 

904.78 

X 

1486.2 

5387.4 

5-32 

4.201X1 

.80939 

3 16 

55.089 

38.448 

X 

461.87 

933.34 

X 

1503.3 

5480.8 

3-16 

4.4301 

.87681 

X 

56.745 

40.195 

X 

471.44 

962.52 

22. 

1520.5 

5575.3 

7-32 

4.6664 

.94786 

5-16 

58.427 

41.994 

' % 

481.11 

992.28 

X 

1537 9 

5670.8 

X 

4.9088 

1.0227 

% 

60.133 

43.847 

X 

490.87 

1022.7 

X 

1555.3 

5767.6 

9-32 

5.1573 

1.1013 

7-16 

61.863 

45.752 

X 

500.73 

1053.6 

h 

1572.8 

5865.2 

5-16 

5.4119 

1.1839 

X 

63.617 

47.713 

X 

510.71 

1085.3 

X 

1590.4 

5964.1 

11-32 

5.6728 

1.2704 

9 16 

65.397 

49.729 

X 

520.77 

1117.5 

% 

1608.2 

6064.1 

% 

5.9396 

1.3611 

% 

67.201 

51.801 

13. 

530.93 

1150.3 

X 

1626.0 

6165.2 

13-32 

6.2126 

1.4561 

11-16 

69.030 

53.929 

X 

641.19 

1183.8 

X 

1643.9 

6267.3 

7-16 

6.4919 

1.5553 

% 

70.883 

56.116 

X 

651.55 

1218.0 

23. 

1661.9 

6370.6 

15-32 

6 7771 

1.6590 

13-16 

72.759 

58.359 

% 

562 00 

1252.7 

X 

1680.0 

6475.0 

X 

7.0686 

1.7671 

% 

74.663 

60.663 

X 

572.55 

1288.3 

X 

1698.2 

6580.6 

17-32 

7.3663 

1.8799 

15-16 

76.589 

63.026 

X 

583.20 

1324.4 

% 

1716.5 

6687.3 

9-16 

7.6699 

1.9974 

5. 

78.540 

65.450 

X 

593.95 

1361.2 

X 

1735.0 

6795.2 

19-32 

7.9798 

2.1196 

1 16 

80.516 

67.935 

X 

604.80 

1398.6 

% 

1753.5 

6904 2 

% 

8.2957 

2.2468 

X 

82.516 

70.482 

14. 

615.75 

1436.8 

X 

1772.1 

7014.3 

21-32 

8.6180 

2.3789 

3-16 

84.541 

73.092 

X 

626.80 

1475.6 

X 

1790.8 

7125.6 

11-16 

8.9461 

2.5161 

X 

86.591 

75 767 

X 

637.95 

1515.1 

24. 

1809.6 

7238.2 

23 32 

9.2805 

2.6586 

6-16 

88.664 

78.505 

X 

649.17 

1555 3 

X 

1828.5 

7351.9 

H 

9.6211 

2.8062 

% 

90.763 

81.308 

X 

660 52 

1596.3 

X 

1847.5 

7466.7 

25-32 

9.9678 

2.9592 

7-16 

92.887 

84.178 

X 

671.95 

1637.9 

% 

1866.6 

7583.0 

13-16 

10.321 

3.1177 

X 

95.033 

87.113 

X 

683.49 

1680.3 

X 

1885.8 

7700.1 

27-32 

10.680 

3.2818 

9-16 

97.205 

90.118 

X 

695.13 

1723.3 

% 

1905.1 

7818.6 

X 

11.044 

3.4514 

% 

99.401 

93.189 

15. 

706.85 

1767.2 

X 

1924.4 

7938.3 

29-32 

11.416 

3.6270 

11-16 

101.62 

96.331 

X 

718.69 

1811.7 

X 

1943.9 

8059.2 

15 16 

11.793 

3.8083 

X 

103.87 

99.541 

X 

730.63 

1857.0 

25. 

1963.5 

8181.3 

31 32 

12.177 

3.9956 

13-16 

106.14 

102.82 

X 

742.65 

1903.0 

X 

1983.2 

8304.7 

». 

12.566 

4.1888 

X 

108.44 

106.18 

X 

754.77 

1949.8 

X 

2002.9 

8429.2 

1-32 

12.962 

4.3882 

15-16 

110.75 

109.60 

X 

767.00 

1997.4 

% 

2022.9 

8554.9 

1 16 

13.364 

4.5939 

6. 

113.10 

113.10 

X 

779.32 

2045.7 

X 

2042.8 

8682.0 

3 32 

13.772 

4.8060 

X 

117.87 

120.31 

X 

791.73 

2094.8 

% 

2062.9 

8810.3 

% 

14.186 

5.0243 

X 

122.72 

127.83 

16. 

804.25 

2144.7 

X 

2083.0 

8939.9 

5-32 

14.607 

5.2493 

X 

127 68 

135.66 

X 

816.85 

2195.3 

X 

2103.4 

9070.6 

3 16 

15.033 

5.4809 

X 

132.73 

143.79 

X 

829.57 

2246.8 

26. 

2123.7 

9202.8 

7-32 

15.466 

5.7190 

% 

137.89 

152.25 

% 

842.40 

2299.1 

X 

2144.2 

9336.2 

yA 

15.904 

5.9641 

X 

143.14 

161.03 

X 

855.29 

2352.1 

X 

2164.7 

9470.8 

9 32 

16.349 

6 2161 

X 

148.49 

170.14 

X 

868.31 

2406.0 

% 

2185.5 

9606.7 

5-16 

16.800 

6.4751 

7 . 

153.94 

179.59 

X 

881.42 

2460.6 

X 

2206.2 

9744.0 

11-32 

17.258 

6 7412 

X 

159 49 

189.39 

X 

894.63 

2516.1 

% 

2227.1 

9882.5 

a l 

17 721 

7.0144 


165.13 

199.53 

17. 

907.93 

2572.4 

X 

2248.0 

10022 




































164 


MENSURATION 


SPHERES — (Continued.) 


s' 

G 

Surface. 

Solidity. 

Diam. 

0 

3 

h 

3 

C/2 

•3 

© 

C/2 

Diam. 

Surface. 

y 

VI 

Diam. 

i _ 

Surface. 

1 

Solidity. 

X 

2269.1 

10164 

X 

4214.1 

2.7724 

X 

6756.5 

52222 

X 

9896.0 

92570 

37. 

22 o 0.2 

1030b 

X 

4243.0 

25988 

X 

6792.9 

52645 

X 

9940.2 

931! 0 

X 

2311.5 

10450 

X 

4271.8 

26254 

X 

6829.5 

53071 

X 

9984.4 

93812 1 

X 

2332.8 

10595 

37. 

4300.9 

26522 

X 

6866.1 

53199 

X 

10029 

94438 

X 

2354.3 

10741 

X 

4330.0 

26762 

X 

6902.9 

63929 

X 

10073 

95066 

X 

2375.8 

10886 

X 

4359.2 

27063 

47. 

6939.9 

54362 

X 

1(118 

95697 

X 

2397.5 

11038 

X 

4388.5 

27337 

X 

6976.8 

54797 

X 

10163 

9(330 

% 

2419.2 

11189 

X 

4417.9 

27612 

X 

7013.9 

55234 

57. 

10207 

96967 I 

y» 

2441.1 

11341 

X 

4447.5 

27889 

X 

7050.9 

55674 

X 

1(252 

97606 

28. 

2463.0 

11494 

X 

4477.1 

2S168 

X 

7088.3 

56115 

X 

10297 

98248 

X 

2485.1 

11649 

X 

4506.8 

28449 

X 

7127.6 

56559 

X 

10342 

98893 

V\ 

2507.2 

11805 

38. 

4536.5 

28731 

X 

7163.1 

570C6 

X 

1(387 

99541 

% 

2529.5 

11962 

X 

4566.5 

29016 

X 

72C0.7 

57455 

X 

10432 

100191 

X 

2551.8 

12121 

X 

4596.4 

29302 

48. 

7238.3 

57 {106 

X 

10478 

100845 

% 

2574.3 

12281 

% 

4626.5 

29:790 

X 

7276.0 

58360 

X 

10523 

101501 

X 

2596.7 

12143 

X 

4656.7 

29980 

X, 

7313.9 

58815 

58. 

10568 

102161 

X 

2619.4 

126(6 

X 

4686.9 

30173 

X 

7351.9 

59274 

X 

10614 

102823 

29. 

2642.1 

12770 

X 

4717.3 

30466 

X 

7389.9 

59734 

X 

10660 

103488 

X 

2665.0 

12636 

X 

4747.9 

30762 

X 

7428.0 

60197 

X 

10706 

104155 

X 

2687.8 

13103 

39. 

4778.4 

31059 

X 

7466.3 

6C6C3 

X 

10751 

104826 

% 

2710.9 

13272 

X 

4809.0 

31359 

X 

7504.5 

61131 

X 

10798 

105499 

X 

2734.0 

13442 

X 

4839.9 

31661 

49. 

7543.1 

61601 

X 

10844 

106175 

% 

2757.3 

13614 

X 

4870.8 

31964 

X 

7581.6 

62074 

X 

10990 

106854 

% 

2780.5 

13787 

X 

4901.7 

32270 

X 

7620.1 

62549 

59. 

109 36 

107536 

y» 

2804.0 

13961 

X 

4932.7 

32.777 

X 

7658.9 

63026 

X 

10! 83 

108221 

ao. 

2827.4 

14137 

X 

4964.0 

32686 

X 

76S7.7 

63506 

X 

11029 

108909 

X 

2851.1 

14315 

X 

4995.3 

33197 

X 

7736.7 

63989 

X 

11076 

109600 

x 

2874.8 

14494 

40. 

5026.5 

33510 

X 

7775.7 

64474 

X 

11122 

110294 

X 

2898.7 

14674 

X 

£058.1 

33826 

X 

7814.8 

64961 

X 

11169 

110990 

X 

2922.5 

14856 

X 

5089.6 

34143 

50. 

7854.0 

65450 

X 

11216 

111690 

X 

2946.6 

15039 

X 

5121.3 

3 462 

X 

7893.3 

65941 

X 

11263 

112392 

x 

2970.6 

15224 

X 

6153.1 

34783 

X 

7932.8 

66 s 36 

60. 

11310 

113098 

X 

2994.9 

15411 

X 

5184.9 

371C6 

X 

7972.2 

66934 

X 

11357 

113806 

31. 

3019.1 

15599 

X 

5216.8 

35431 

X 

8011.8 

67433 

x 

11404 

114518 

X 

3043.6 

15788 

X 

5248.9 

35758 

X 

8051.6 

67935 

X 

11452 

115232 

X 

3068.0 

15979 

41. 

5281.1 

36087 

X 

8091.4 

68439 

X 

11499 

115949 

% 

3062.7 

16172 

X 

5313.3 

36418 

X 

8131.3 

68946 

X 

11547 

116669 

X 

3117.3 

16366 

X 

5345.6 

36751 

51. 

8171.2 

69456 

X 

11595 

117392 

% 

3142.1 

16561 

X 

5378.1 

37086 

X 

8211.4 

69967 

X 

11612 

118118 

X 

3166.9 

167.58 

X 

5410.7 

37423 

X 

8251.6 

70482 

61. 

11690 

118847 

. % 

3192.0 

16957 

X 

5443.3 

37763 

X 

8292.0 

70999 

X 

11738 

119579 

32. 

3217.0 

17157 

X 

5476.0 

38104 

X 

8332.3 

71519 

X 

11786 

120315 

% 

3242.2 

17359 

X 

5508.9 

38448 

X 

8372.8 

720-.0 

X 

11834 

121053 } 

X 

3267.4 

17563 

42. 

5541.9 

38792 

X 

8413.4 

7256' 

X 

11882 

121794 

% 

3292.9 

. 17768 

X 

6574.9 

39140 

X 

8454.1 

73092 

X 

11931 

122538 

X 

3318.3 

17974 

X 

5608.0 

39490 

52. 

8494.8 

73622 

X 

11980 

123286 

% 

3343.9 

18182 

X 

5641.3 

39841 

X 

8535.8 

74154 

X 

12028 

124036 

X 

3369.6 

18392 

X 

5674.5 

40194 

X 

8576.8 

7*168! 

62. 

12076 

124789 

o, % 

3395.4 

18604 

X 

570S.0 

405.71 

X 

8617.8 

75226 

X 

12126 


33. 

3421.2 

18817 

X 

5741.5 

40! 08 


8658.9 

75767 

X 

12174 

126305 

X 

3447.3 

19032 

, K 

5775.2 

41268 

X 

8700.4 

763C6 

X 

12223 

127067 

X 

3473.3 

10218 

43. 

.6808.8 

41630 

X 

8741.7 

76854 

X 

12272 

127832 

% 

3499.5 

19466 

X 

5842.7 

41994 

X 

8783.2 

77401 

X 

12322 

128601 

% 

3525.7 

19685 

X 

5876.5 

42360 

53. 

8824.8 

77952 

X 

12371 

lv< 373 

% 

3552.1 

19907 

X 

5910.7 

42729 

X 

8866.4 

78505 

X 

12420 

130147 

X 

3578.5 

20129 

X 

5944.7 

430919 

X 

8908.2 

79060 

63. 

12-469 

130925 

% 

3605.1 

20354 

X 

5978.9 

43172 

X 

8950.1 

79617 

X 

12519 

131706 

34. 

3631.7 

20580 

X] 

6013.2 i 

43846 

X 

8992.0 

80178 

X 

12568 

132490 

H 

3658.5 

20808 

X 

6047.7 ) 

44224 

X 

9034.1 

80741 

X: 

12618 

133277 

X 

368.5.3 

21037 

44. 

6082 1 

44602 

X 

9076.4 

81308 

X 

11668 

134067 

% 

3712.3 

21268 

X 

6116.8 | 

44984 

X 

9118.5 

81876 

X 

12718 

134860 

X 

3739.3 

21501 

X 

6151.5 I 

47367 

54. 

9160.8 

82448 

X 

12768 

135657 

X 

3766.5 

21736 

X 

6186.3 

47773 

X 

9203.3 

83021 

X 

12818 

136456 

X 

3793.7 

21972 

X 

6221.2 

46141 

X 

9246.0 

83598 

64. 

12868 

137259 

„ X 

3821.1 

22210 

X 

6276 1 

46730 

X 

9288.5 

84177 

X 

12918 

138065 

35. 

3848.5 

22449 

X 

6291.2 

46922 

X 

9331.2 

84760 

X 

12969 

J 388*74 

X 

3876.1 

22691 

X 

6326.5 

47317 

X 

9374.1 

85344 

X 

13019 

1306Kg 

X 

3903.7 

22934 

45. 

6361.7 

47713 

X 

9417.2 

85931 

X 

13070 

l4n*inj[ 

X 

3931.5 

23179 

X 

6397.2 

48112 

X 

9460.2 

86521 

X 

13121 

14X320 

X 

3959.2 

23*25 

X 

6432.7 

48.713 

55. 

9503.2 

87114 

X 

13172 


X 

3987.2 

23674 

X 

6 168.3 

48916 

X 

9546.5 

87709 

X 

13222 


X 

4015.2 

23921 

X 

6703.9 

49321 

X 

9590.0 

88307 

65. 

13273 


X 

4043.3 

24176 

X 

6739.7 

49729 

X 

9633.3 

88908 

X 

13324 


8fi. 

4071.5 

24429 

X 

6775.5 

50139 

X 

9676.8 

89511 

X 

1.9376 


X 

40P9.9 

21685 

X 

6611 6 

50551 

X 

9720.6 

90117 

% 

13427 

i4n4o0 

X 

4128.3 

24942 

46. 

6617.6 

50967 

X 

9764.4 

90726 

X 

] 347 ft 

146297 

X 

4156.9 

25201 

X 

6683.7 

51382 

X 

9808.1 

9i338 

by 


14/13ft 

X 

4185.5 

25461 

X 

6720.0 

51801 

56. 

9852.0 

91953 

X 

13582 

14(982 

148828 













































MENSURATION 


165 


SPH EKES —(Continued.) 


a 

.2 

5 

Surface. 

Solidity. 

Diam. 

Surface. 

^ Solidity. 

Diam. 

Surface. 

j Solidity. 

Diam. 

Surface. 

Solidity. 

% 

13633 

149680 

X 

17437 

216505 

X 

21708 

300743 

X 

26446 

404406 

66. 

13685 

150533 

X 

17496 

217597 

X 

21773 

302100 

X 

26518 

40(060 

X 

13737 

151390 

X 

17554 

218693 

X 

21839 

303463 

92. 

26590 

407721 

X 

13789 

152251 

X 

17613 

219792 

X 

21904 

304831 

X 

26663 

409384 

x 

13841 

153114 

75. 

17672 

220894 

X 

21970 

306201 

X 

26735 

411054 

X 

13893 

153980 

X 

17731 

222001 

X 

22036 

307576 

X 

26808 

412726 

% 

13946 

154850 

X 

17790 

223111 

X 

22102 

308957 

% 

26880 

414405 

X 

13998 

155724 

X 

17849 

224224 

84. 

22167 

310340 

% 

26953 

416086 

X 

14050 

156600 

X 

17908 

225341 

X 

22234 

311726 

X 

27026 

417774 

67. 

14103 

157480 

X 

17968 

226463 

X 

22300 

313118 

X 

27099 

419464 

X 

14156 

158363 

X 

18027 

227588 

X 

22366 

314514 

93. 

27172 

421161 

X 

14208 

159250 

X 

18087 

228716 

X 

22432 

315915 

X 

27245 

422862 

x 

14261 

160139 

76. 

18146 

229848 

X 

22499 

317318 

X 

27318 

424567 

X 

14314 

161032 

X 

18206 

230984 

X 

22565 

318726 

X 

27391 

426277 

X 

14367 

161927 

X 

18266 

232124 

X 

22632 

320140 

X 

27464 

427991 

X 

14420 

162827 

X 

18326 

233267 

85. 

22698 

321556 

X 

27538 

429710 

x 

14474 

163731 

X 

18386 

234414 

X 

22765 

322977 

X 

27612 

431433 

68 . 

14527 

164637 

X 

18446 

235566 

X 

22832 

324402 

X 

27686 

433160 

X 

14580 

165547 

X 

18506 

236719 

X 

22899 

325831 

94. 

27759 

434894 

X 

14634 

166460 

X 


237879 

X 

22966 

327264 

X 

27833 

436630 

X 

14688 

167376 

77. 

18626 

239041 

X 

23034 

328702 

V* 

27907 

438373 

X 

14741 

168295 

X 

18687 

240206 

X 

23101 

330142 

X 

27981 

440118 

% 

14795 

169218 

X 

18748 

241376 

X 

23168 

331588 

X 

28055 

441871 

X 

14849 

170145 

X 

18809 

242551 

86. 

23235 

333039 

X 

28130 

443625 

% 

14903 

171074 

X 

18869 

243728 

X 

23303 

334492 

X 

28204 

445387 

69 

14957 

172007 

X 

18930 

244908 

X 

23371 

335951 

X 

28278 

447151 

X 

15012 

172944 

X 

18992 

246093 

X 

23439 

337414 

95. 

28353 

448920 

X 

15066 

173883 

X 

19053 

247283 

X 

23506 

338882 

X 

28428 

450695 

X 

15120 

174828 

78. 

19414 

248475 

X 

23575 

340352 

X 

28503 

452475 

X 

15175 

175774 

X 

19175 

249672 

X 

23643 

341829 

X 

28577 

454259 

% 

15230 

176723 

X 

19237 

250873 

X 

23711 

343307 

X 

28652 

456047 

X 

15284 

177677 

X 

19298 

252077 

87. 

23779 

344792 

X 

28727 

457839 

y» 

15339 

1786,35 

X 

19360 

253284 

X 

23847 

346281 

X 

28802 

459638 

70. 

15394 

179595 

X 

19422 

254496 

X 

23916 

347772 

X 

28878 

461439 

X 

15449 

180559 

X 

19483 

255713 

X 

23984 

349269 

96. 

28953 

463248 

X 

15504 

181525 

X 

19545 

256932 

X 

24053 

350771 

X 

29028 

465059 

X 

15560 

182497 

79. 

19607 

258155 

X 

24122 

352277 

X 

29104 

466875 

X 

15615 

183471 

X 

19669 

259383 

X 

24191 

353785 

X 

29180 

468697 

X 

15670 

184449 

X 

19732 

260613 

X 

24260 

355301 

X 

29255 

470524 

X 

15726 

185430 

X 

19794 

261848 

88. 

24328 

356819 

% 

29331 

472354 

x 

15782 

186414 

X 

19856 

263088 

X 

24398 

358342 

X 

29407 

474189 

71 . 

15837 

187402 

X 

19919 

264330 

X 

24467 

359869 

X 

29483 

476029 

X 

15893 

188394 

X 

19981 

265577 

X 

24536 

361400 

97. 

29559 

477874 

X 

15949 

189389 

X 

20044 

266829 

X 

24606 

362935 

X 

29636 

479725 

X 

16005 

190387 

80. 

20106 

268083 

X 

24676 

364476 

X 

29712 

481579 

X 

16061 

191389 

X 

20170 

269342 

X 

24745 

366019 

X 

29788 

483438 

% 

16117 

192395 

X 

20232 

270604 

X 

24815 

367568 

X 

29865 

485302 

X 

16174 

193404 

X 

20296 

271871 

89. 

24885 

369122 

X 

29942 

487171 

x 

16230 

194417 

X 

20358 

273141 

X 

24955 

370678 

X 

30018 

489045 

72. 

16288 

195433 

X 

20422 

274416 

X 

25025 

372240 

X 

30095 

490924 

X 

16343 

196453 

X 

20485 

275694 

X 

25095 

373806 

98. 

30172 

492808 

X 

16400 

197476 

X 

20549 

276977 

X 

25165 

375378 

X 

30249 

494695 

X 

16156 

198502 

81. 

20612 

278263 

X 

25236 

376954 

Va 

30326 

496588 

X 

16513 

199532 

x 

20676 

279553 

X 

25306 

378531 

X 

30404 

498486 

X 

16570 

200566 

X 

20740 

280847 

X 

25376 

380115 

X 

30481 

500388 

X 

16628 

201604 

X 

20804 

282145 

90. 

25447 

381704 

% 

30558 

502296 

y» 

16685 

202645 

X 

20867 

28.3447 

X 

25518 

383297 

X 

30636 

504208 

73. 

16742 

203689 

X 

20932 

284754 

X 

25589 

384894 

X 

30713 

506125 

X 

16799 

204737 

X 

20996 

286064 

X 

25660 

386496 

99. 

30791 

508047 

X 

16857 

205789 

X 

21060 

287378 

X 

25730 

388102 

X 

30869 

509975 

X 

16914 

206844 

82. 

21124 

288696 

X 

25802 

389711 

X 

30947 

51lDOfi 

X 

16972 

207903 

X 

21189 

290019 

X 

25873 

391327 

X 

31025 

513843 

X\ 

17030 

208966 

X 

21253 

291345 

X 

25944 

392946 

X 

31103 

515785 

x 

17088 

210032 

X 

21318 

292674 

91. 

26016 

394570 

X 

31181 

517730 

y* 

17146 

211102 

X 

21382 

294010 

X 

26087 

396197 

X 

31259 

519682 

74. 

17204 

212175 

X 

21448 

295347 

X 

26159 

397831 

X 

31338 

521638 

X 

17262 

213252 

X 

21512 

296691 

X 

26230 

399468 

100 . 

31416 

523598 

X 

17320 

2143.33 

X 

21578 

298036 

X' 

26302 

401109 




%\ 

1 

17379 

215417 

83. 

21642 

299388 

X| 

26374 

402756 
















































166 


SEGMENTS, ETC., OF SPHERES, 


To find the solidity of a spherical segment. 


s 



Rule 1. Square the rad o n, ofits base: mult this square by 3; to 
the prod add the square of its height os ; mult the sum by the height 
o s : and mult this last prod by .5236. 

Rule 2. Mult the diam a b of the. sphere, by §•, from the prod 
take twice the height o s of the segment; mult the rem by the square 
of the height o »; and mult this prod by .5236. 

The solidity of a sphere being %ds that of its circumscribing cylin¬ 
der. if we add to any solidity in the table, its half, we obtain that 
of a cylinder of the same diam as the sphere, aud whose height 
equals its diam. 


To find tlie curved surface of a spherical segment. 

Rule 1. Mult the diam a b of the sphere from which the segment is cut, by 3.1416; 
mult the prod by the height o s of the seg. Add area of base if reqd. Rem. Having the diam n r 
of the seg, and its height o s, the diam a It of the sphere may be fouud thus: Div the square of half 
the diam n r, by its height o s ; to the quot add the height o s. Rule 2. The curved surf of either 
a segment, last Fig, or of a zone, (uext Fig,) bears the same proportion to the surf of the whole 
sphere, that the height of the seg or zone bears to the diam of the sphere. Therefore, first find the 
surf of the whole sphere, either by rule or from the preceding table ; mult it by the height of the seg 
or zoue; div the prod by diam of sphere. Rule 3. Mult the circumf of the 6phere by the height o * 

of the seg. 




To find tlie solidity of a spherical zone. 

Add together the square of the rad e d. the square of rad o h, 
and t£d of the square of the perp height eo; mult the sum by 
1.5708; aud mult this prod by the height eo. 

To find the curved surface of a spher¬ 
ical zone. 

Rule 1. Mult together the diam m iiof the sphere; the height 
e o of the zone, and the number 3.1416. Or see preceding Rule 2 
for surf of segments. Rule 2. Mult the circumf of the sphere, by 
the height of the zoue. 

To find the solidity of a hollow spher¬ 
ical shell. 

Take from the foregoing table the solidities of two spheres having 
the diams o b, and c d. Subtract the least from the greatest. Here 
a c or b d is the thickness of the shell. 


THE ELLIPSOID, OR SPHEROID, 

Is a solid generated by the revolution of an ellipse around either its loug or its short diam. When 

around the long (or transverse) diam, as at a Fig 1, it is an oblong; or pro¬ 
late spheroid; when around the short (or conjugate) one, as at m, in Fig 2, 

it m oblate. 



Fig. 1. Fig;. 2. 


For tlie solidity in either case, mult the fixed diam or axis bv the 
of the revolving one; and mult the prod by .5236. square 















PARABOLOIDS, ETC 


167 


the paraboloid, or parabolic conoid, 

lext Fig, is a solid generated by the revolution of a parabola a c b, arouud its axis, c r. 

For its solidity mult the area of its base, by half its height, r c. Or mult 

.ogetuer the square of the rad a r of the base; the height rc; and the number 1.5708. 



For the solidity of a frustum, 

' * b g h, the ends of which are perp to the axis r c; add together the 
squares of the two diams a b and y A; mult the sum by the height r l; 
mult the prod by the decimal .3927. 

To find the surface of a paraboloid. 

Mult the rad a r of its base, by 6.2832; div the prod by 12 times the 
square of the height rc; cali the quot p. Then add together the square 
of the rad a r, and 4 times the square of the height r c. Cube the 
sum; take the sq rt of this cube; from the sq rt subtract the cube of 
the rad a r. Mult the rem by p. 

Either the solidity, or the surface of a frustum, a b g ft, when gh is 
parallel to a b, may be found by calculating for the whole paraboloid, 
and for the upper portion c g ft, as two separate paraboloids, and taking their diffi. 

THE CIRCULAR SPINDLE, 

Is a solid ab ny generated by the revolution of a circular segment 
a b n e a, around its chord a n as an axis. 

To find its solidity. 

Rule 1. First find the area of ab e, or half the generating circular 
segment. Then to the square of a e, add the square of be; div the sum 
by 6 y ; from the quot take 6 e: mult the rem bv the area of a e 6; call 
theprodp. Cube a e; div the cube by 3; from the quot take p. Multthei 
rem by 12.5664. 

Rulb 2. When the dist o e is known, from the center of the circle to ' 
the cen of the spindle, then mult that disto e, by the area of ab e; call 
the prod p ; cube a e: div the cube by 3; from the quot take p; mult the 
rem by 12.5664. 

To find its surface. 

Rulb 1. First find the length of the circular arc abn; and mult it by 
the dist o e from the center of the circle to the center of the spindle. Call 
the prod p. Next mult the length an of the spindle, by the rad o b of the circle, 
take p; mult the rem by 6.2832. 

Rule 2. First find the length of the arc abn. Square a e; also square be; add these squares 
together; div their sum by 6 y; call the quot «; and mult it by a n; call the prod p. Next from s taka 
b e : mult the rem by the length of the arc abn. Subtract the prod from p ; mult the rem by 6.2832. 

To find the solidity of a middle zone of a circular spindle. 

As A skp 

^ —■ ^o e X area of g A l k*j ^ X 6.2832. 



From the prod 



CIRCULAR RINGS. 1 


Volume — 
Surface — 


area of cross section of bar v \ su ra of inner and outer y - 3141593 
of which ring is made diameters, a a and bb 

circumference of bar y 3 sum of inner and outer .. g 141593 
of which ring is made A diameters, a a and bb ^ 









168 


LAND SURVEYING, 




LAND SUKVEYING. 


In surveying a tract ot * 
ground, the sides which cow j 
pose its outline are desig- ' 
nated by numbers in the 
order iu which they occur. , * 1 
That end of each side which 
first presents itself in the 
course or the survey, may be 
called its near end : and the 
other its far end. The uuni- | 

ber of each side is placed at its | 

far end. Thus, iu Kig 1, the 
survey being supposed to 
commence at the corner 6 , i 
and to follow the direction 
of the arrows, the first side 
is 6 , 1 ; and its number is 
placed at its far end at 1 ; 
and so of the rest. Let N S 
be a meridian liue, that is, a 
north and south liue; and 
E W an east and west line. 
Then in any side which runs 
northwardly, whether due 
north, or northeast, as side 2: or northwest, as sides 5 and I, the dist in a due north direction 
between its near end and its far end, is called its northing; thus, o 1 is the northing of side 1 ; 1 ft 
the northing of side 2; 4 c of side 5. In like manner, if any side runs in a southwardly direction, 
whether due south, or soutbeastwardly, as side 3; or southwestwardly, as sides 4 and 6 , the corre- j 
sponding dist in a due south direction between its near end and its far end. is called its southing; thus, 1 
d3 is the southing of side 3; 3 eof side 4 ; / 6 of side 6 . Both northings and southings are included in the 
general term Difference of Latitude of a side; or more commonly, but erroneously, its latitude. The dist 
due east, or due west, between the near and the far end of any side, is in like manner called the easting, 
or westing of that side, as the case may be : thus. 6 a is the westing of side 1: 5/ of side 6 ; c5 of side 
5; e 4 of side 4; and ft 2 is the easting of side 2; 2 d of side 3. Both eastings and westings are included 
in the general term Departure of a side; implying that the side departs so far from a north or south 
direction. We may employ the directions (or courses, or bearings, as they are usually called) of sides, 
as verbs; and say that a side norths, wests, southeasts, &c. We shall call the northings, southings, 

&c, the Ns, Ss, Es. and Ws; the latitudes, lats; and the departures, deps. The Traverse 

Table consists of the lats, (or Ns and Ss;) and the deps, (or Es and Ws.) corresponding to dift' angles 
or courses, for a side whose length is 1 ; therefore, to obtain the actual lat aud dep of any given side, 
those taken from the table must be mult by the length of the side. Beyond 44°, the lats and deps 
of this table must be read upward from the bottom of the page. The angles in the table are those 
which the course or bearing of any side would make with a meridian line drawn through either end 
of said side; but it is self-evident that what would be N from one end, would be S from the other; and 
so of E and W; in other words, the angle is the same at both ends; but the direction is reversed. 

Perfect accuracy is unattainable in any operation involving the measurements of angles and dists. 
That work is accurate enough, which cannot be made more so without an expenditure more than com¬ 
mensurate with the object to be gained. The writer conceives that the accuracy essential to constitute 
practically fair surveying is purely a matter of dollars and cents. Iu the purchase and sale of tracts 
of land, such as farms, &c, an uncertainty of about 1 part in 200 respecting the content, and consequently 
respecting the price, probably never prevents a transfer; and on this principle we assume that a survey 
which proves itself within that limit, may ordinarily be regarded as accurate enough. There is no 
great difficulty in attaining this limit, which, if exceeded, is the result of bad work. Many circum¬ 
stances combine to render trilling errors absolutely unavoidable:* they alwavs become apparent 
when we come to work out the field notes; and since the map or plot of the survey, and the calcula¬ 
tions for ascertaining the content, should be consistent within themselves, we do what is usnallv called 
correcting the errors, but what in fact is simply humoring them in. no matter how scientific the pro¬ 
cess may appear. We distribute them all around the snrvev. Two methods are used for this purpose 
both based upon precisely the same principle; one of them mechanical, by means of drawing - the 
other more exact, but much more troublesome, by calculation. We shall describe both • but will state 
now, that by proportioning tbe scale of the plot in the following manner, the mechanical method 
becomes, in the hands of a correct draftsman, sufficiently exact for all ordinary purposes. Add all 
the sides in feet together; and div the sum by their number, for the average length. Piv this average 
by 8 ; the quot will be the proper scale in feet per inch. In other words, take about 8 ins to represent 
an average side.t W e shall take it for granted, that an engineer does not consider It accurate work to 

* A 100 ft measuring chain may vary its length 5 feet per mile, between winter and summer, by 
mere change of temperature; and by this alone we shall make a difference of about l-l acres in a lot 

1 mile square, which contains 640 acres. Even this error amounts to 1 acre in 533. Not one farmer 
in a hundred would dream of paying for a scrupulously correct sui wey and plot of his property. 

t It will seldom happen that precisely this quotient will be adopted for a scale. For instance if 
quot should be 738 feet, or 83 feet, per inch, we should adopt the most couveuieat near number if 
smaller ihe better, as 700, or 80; or rather more than 8 inches to a side. W ith a scale so pronortioneV 
and with good drawing instruments, the error in protracting (excluding of course errors of the field 
work) will rarely exceed about part of the periphery of tbe plot; and the area may be found 

mechanically by dividing the plot into triangles, within ^ ^ part of the truth. This remark applies 

particularly to such plots as may be protracted, and computed within a period so short as 
to allow tbe paper to contract or expand appreciably by atmospheric chnnges. a lareer 
will insure proportionally greater accuracy. The young assistant should practise*’nlotiiuw 
from perfectly accurate data; as, for instance, from the example given iu the table p* 1'5 ■ 



















1.AND SURVEYING 


169 


measure his angles to the nearest quarter of a degree, which is the usual practice among land-survey 
ors. They can, by menus of the engineer's transit, now in universal use on our public works, be readily 
measured within a minute or two; and being thus much more accurate than the compass courses, 
(which cannot be read ofl so closely, aud which are moreover subject to many sources of error,) they 
serve to correct the latter in the office. The noting of the courses, however, should not be confined to 
the nearest quarters ot a degree, but should be read as closely as the observer can guess at the minutes. 
The back courses also should be taken at every corner, as an additional check, and for the detection 
of local attraction. It is 
well in taking the com¬ 
pass bearings, to adopt 
as a rule, always to point 
the north of the compass- 
box toward the object 
whose bearing is to be 
takeu, and to read off 
from the north end of the 
needle. A person who 
uses indifferently the N 
and the S of the box, and 
of the needle, will be very 
liable to make mistakes. 

It is best to measure the 
least angle (shown by 
dotted arcs. Fig 2,) at the 
corners; whether it be 
exterior, as that at corner 
5; or interior, as all the 
others; because it is al¬ 
ways less than 180°; so 
that there is less danger 
of reading it off incor¬ 
rectly, than if it exceeded 
180°; taking it for grant¬ 
ed that the trausit instrument is graduated from the same zero to 180° each way ; if it is graduated 
from zero to 360 9 the precaution is useless. When the small angle is exterior, subtract it from 360° 
for the interior one. 

Supposing the field work to be finished, and that we require a plot from which the contents may 
be obtained mechanically, by dividing it into triangles, (the bases and heights of which may be 
measured by scale, and their areas calculated one by one,) a production of it may be made at once 
from the field notes, either by using the angles, or by first correcting the hearings by means of the 
angles, and then using them. The last is the best, because in the first the protractor must be moved 
to each angle ; whereas in the last it will remain stationary while all the bearings are being pricked 
off. Every mhvement of it increases the liability to errors'. The manner of correcting the bearings 
is explained on the next page. 

Iu either case the protracted plot will certainly not close precisely; not only in consequence of errors in 
the field work, but also in the protractiug itself. Thus the last side, No 6 , Fig 2, instead of closing in at 
corner 6 , will end somewhere else, say, for instance, at t', the dist t 6 being the closing error, which, 
however, as represented in Fig 2, is more than ten times as great, proportionally to the size of the 
survey, as would be allowable in practice. Now to humor-in this error, rule through every corner 
a short line parallel to t 6; and, in all cases, in the direction from, t (wherever it may be) to the 
starting point 6 . Add all the sides together; and measure t 6 by the scale of the plot. Then begin¬ 
ning at corner 1 , at the far end of side 1 , say, as the 

Sum of all . Total closing . . Q . j , 

• error (6 • • fclQe 1 



the sides • 

Lay off this error from 1 to a. 

Sum of all . 

the sides • 


Error 
for sidel. 


Then at corner 2, say, as the 

Total closing . . Sum of . Error 

error t 6 • • sides 1 and 2 • for side 2 . 


Which error lay off from 2 to b; and so at each of the corners; always using, as the third term, the 
sum of the sides between the starting point and the given corner. Finally, join the points a, b, c, 
d, e, 6 ; and the plot is finished. 

The correction has evidently changed the length of every side; lengthening some and shortening 
others. It has also changed the angles. The new lengths and angles may with tolerable accuracy 
be found by means of the scale and protractor; and be marked on the plot instead of the old ones. 


from those to be found in books on surveying. This is the only way in which he can learn what is 
meant by accurate work. His semicircular protractor should be about 9 to 12 ins in diam and gradu¬ 
ated to 10 min. His straight edge and triangle should be of metal: we prefer German silver, which 
does not rust as steel does; and they should be made with scrupulous accuracy by a skilful instru¬ 
ment-maker. A very fine needle, with a sealing-wax head, should be used for pricking off dists and 
angles; it must be held vertically ; and the eye of the draftsman must be directly over it. The lead 
pencil should be bard (Faber's No. 4 is good for protracting), and must be kept to a sharp point by 
rubbing on a fine file, after using a knife for removing the wood. The scale should be at least as long 
as the longest side of the plot, and should be made at the edge of a strip of the same paper as the plot 
is drawn on. This will obviate to a considerable extent, errors arising from contraction and expan¬ 
sion. Unfortunately, a sheet of paper does not contract and expand in the same proportion length¬ 
wise and crosswise, thus preventing the paper scale from being a perfect corrective. In plots of com¬ 
mon farm surveys, Ac, however, the errors from this source may be neglected. For such plots as may 
be protracted, divided, and computed within a time loo short to admit of appreciable change, the ordi¬ 
nary scales of wood, ivorv or metal may be used; but satisfactory accuracy cannot be obtained wilh 
them on plots requiring several days, if the air be meanwhile alternately moist and dry, or subject to 
considerable variations in temperature. What is called parchment paper is worse in this respect than 
good ordinarv drawing-paper. 

With the foregoing precautions we may work from a drawing, with as much accuracy as is usually 
attained in the field work. 





170 


LAND SURVEYING, 




When the plot has many sides, this calculating the error for each ot them becomes tedious; and 
since, iu a well-performed survey aud protraction, the entire error will be but a very small quantity, 
(it should not exceed about part of the periphery,) it may usually be divided among the sides by 
merely placing about and Ys of it at corners about %, and H way around the plot; and at 

^ ; ^ intermediate corners propor- 


a- 


I 



I 


tion it by eye. Or calculation 
may be avoided entirely by 
drawiug a line a 6 of a length 
equal to the united lengths 
of all the sides ; dividing it 


Into distances a, l; 1, 2 ; &c, equal to the respective sides. Make 6 c equal to the entire closing error; 
join a c ; and draw 1,1’; 2 , 2 ’, &c, which will give the error at each corner. 

When the plot is thus completed, it may be divided by fine pencil lines into triangles, whose 
bases and heights may be measured by the scale, in order to compute the contents. With care in 
both the survey and the drawing, the error should not exceed about I_ part of the true area. At 
least two distinct sets of triangles should be drawn and computed, as a guard against mistakes; and if 
the two sets differ in calculated contents more than about P art > they have not been as carefully 
prepared as they should have been. The closing error due to imperfect field-work, may be accurately 
calculated, as we shall show, and laid down on the paper before beginning the plot; thus furnishiug 
a perfect test of the accuracy of the protraction work, which, if correctly done, will not close at the 
poiut of beginning, but at the point which indicates the error. But this calculation of the error, by 
a little additional trouble, furnishes data also for dividing it by calculation among the diff sides ; 
besides the means of drawing the plot correctly at once, without the use of a protractor ; thus ena¬ 
bling us to make the subsequent measurements and computations of the triangles with more cer¬ 
tainty. 

We shall now describe this process, but would recommend that even when it is employed, and 
especially in complicated surveys, a rough plot should first be made and corrected, by the first of the 
two mechanical methods already alluded to. It will prove to be of great service iu using the method 
by calculation, inasmuch as it furnishes an eye check to vexatious mistakes which are otherwise apt 
to occur ; for, although the principles involved are extremely simple, and easily remembered when 
once understood, yet the continual changes in the directions of the sides will, without great care, 
cause us to use Ns instead of Ss; Es instead of Ws, Ac. 

We suppose, then, that such a rough plot has been prepared, and that the angles, bearings, and 
distances, as taken from the field book, are figured upon it in lead pencil. 

Add together the interior angles formed at all the corners : call their sum a. Mult the number of 
sides by 180°; from the prod subtract 360° : if the remainder is equal to the sum a, it is a proof that 
the angles have been correctly measured.* This, however, will rarely if ever occur; there will 
always be some discrepancy ; but if the field work has been performed with moderate care, this will 
not exceed about two min for each angle. In this case div it in equal parts among all the angles, 
adding or subtracting, as the case may be, unless it amounts to less than a min to each angle, when 
it may be entirely disregarded in common farm surveys. The corrected angles may then be marked 
on the plot in ink, and the pencilled figures erased. We will suppose the corrected ones to be as 
shown in Fig 3. 



Next, by means of these 
corrected angles, correct the 
bearings also, thus. Fig 3 ; 
Select some side (the longer 
the better) from the two ends 
of w hich the bearing and the 
reverse bearing agreed ; thus 
showing that that bearing 
was probably not influenced 
by local attraction. Let side 
2 be the one so selected : as¬ 
sume its bearing, N 75°32' E, 
as taken on the ground, to be 
correct; through either end 
of it, as at its far end 2 . draw 
the short meridian line: par¬ 
allel to which draw others 
through every corner. Now, 
having the bearing of side 2 , 
N 75° 32' E, and requiring 
that of side 3, it is plain that 
the reverse hearing from cor¬ 
ner 2 is S 75° 32' W ; and 
that therefore the angle 1 . 2 , 
to, is75°32'. Therefore, if we 
take 75° 32' from the entire 
corrected angle 1,2.3, or 144° 
57', the rent 69° 25' will be 
the angle m 23 ; consequently 
the bearing of side3 must he 
S 69° 25' E. For finding the bearing of side 4, we now have the angle 23 a of the reverse bearing of 
side 3, also equal to 69° 25' ; and if we add this to the entire corrected angle 234, or to69° 32', we have 
the augle a 34 = 690 25'+ 69° 32'= 138° 57'; wl ,i ch tu keu from 180°, leaves the angle 6 34 = 41°3' ; 
consequently the bearing of side 4 must be S 41° 3' W. For the bearing of side 5 we now have the 
angle 34 c = 41° 3', which taken from the corrected angle 345, or 120° 43', leaves the angle c 45 = 79° 
40'; consequently the bearing of side5 must be N 79° 40' W. At corner 5, for the bearing of side 6 , 
we have the angle 45 <1 = 79° 40', which taken from 133° 10', leaves the angle d 56 = 53°30' ; conse 
queutly the bearing of side 6 must be S 53° 30' W. And so with each of the sides, nothing but 




* Because in every straight-lined figure the sum of all its interior angles is equal to twice as many 
right augles as the figure has sides, minus 4 right angles, or 360°. 


















LAND SURVEYING 


171 


Careful observation is necessary to see how the several angles are to be employed at each corner. 
Kules are sometimes given for this purpose, but unless frequently used, they are soon forgotten. 
The plot mechanically prepared obviates the necessity for such rules, inasmuch as the principle of 
proceeding thereby becomes merely a matter of sight, and teuds greatly to prevent error from using 
the wrong bearings; while the protractor will at ouce detect any serious mistakes as to the angles, 
and thus prevent their being carried tarther along. After having obtained all the corrected bearings, 
they may be figured on the plot instead of those taken in the field. They will, however, require a 
still further correction after a while, since they will be affected by the adjustment of the closing error. 

We now proceed to calculate the closing error 1 6 of Fig 2, which is done on the principle that in a 
correct survey the northings will be equal to the southings, and the eastings to the westings. Pre¬ 
pare a table of 7 columns, as below, and in the first 3 cols place the numbers of the sides, and their-'or- 
rected courses; also the dists or lengths of the sides, as measured on the rough plot, if such a one 
has been prepared ; but if not, theu as measured on the ground. Let them be as follows : 


Side. 

Bearing. 

Dist. Ft. 

Latitudes. 

Departures. 

N. 

S. 

E. 

W 

1 

N 16° 40' W 

1060 

1015.5 



304. 

2 

N 75° 32'E 

1202 

300.3 


1163.9 


3 

S 69° 25' E 

1110 


390.2 

1039.2 


4 

S 41° 3' W 

850 


6 *1. 


558.2 

6 

N 79° 40'W 

802 

143.9 



789. 

6 

S 53° 30' W 

705 


419.3 


566.7 




1459.7 

1450.5 

2203.1 

. 

2217.9 




1450.5 



2203.1 




9.2 

Error in 

Error in 

14.8 





Lat. 

Dep. 



Now find the N, S, E, W, of the several sides, and place them in the corresponding four columns, 
thus : By means of the Traverse Table find out the lat and dep for the angle of each course. Mult each 
of them by’ the length of the side; and place the prod in the corresponding col of N, S, E. W. Thus, 
for side 1. which is 1060 feet long, the latitude from the traverse table for 16° 40' is .9580; and the 
departure is .2868: and .9580 X 1060 = 1015.5 lat; which, since the side norths, we put in the N 
col. Again, .2868 X 1060 = 304 dep; which, since the side wests, we put in the W col. Proceed 
thus with all. Add up the four cols; find the diflf between the N and S cols; and also between 
the E and W ones. In this instance we fiud that the Ns are 9.2 feet greater than the Ss; and that 
the Ws are 14.8 ft greater than the Es; in other words, there is a closing error which would cause a 
correct protraction of our first three cols, to terminate 9.2 feet too far north of the starting point; and 
14.8 feet too far west of it. So that by placing this error upon the paper before beginning to protract, 
we should have a test for the accuracy of the protracting work ; but, as before remarked, a little more 
trouble will now enable us to div the error proportionally among all the Ns, Ss, Es, and Ws, and thereby 
give as data for drawing the plot correctly at once, without using a protractor at all. 

To divide the errors, prepare a table precisely the same as the foregoing, except that the hor spaces 
are farther apart: and that the addiugs-up of the old N, S, E, W columns are omitted. The additions 
here noticed are made subsequently. 

The new table is on the next page. 


Remark. The bearing: and the reverse bearing* from the two ends 
of a line will not read precisely the same angle; and the difference varies with the 
latitude and with the length of the line, but not in the same proportion with either. 
It is, however, generally too small to be detected by the needle, being, according to 
Gummere, only three quarters of a minute in a line one mile long in lat 40°. In 
higher lats it ia more,and in lower ones less. It is caused by the fact that meridians 
or north and south lines are not truly parallel to each other; but would if extended 
meet at the poles. 

Hence the only bearing: that can be run in a straight line, 

with strict accuracy, is a true N and S one; except on the very equator, where alone a due E and W 

one will also be straight. But a true curved E ami W line may be found 

anvwhere with sufficient accuracy for the surveyor’s purposes thus. Having first by means of the N 
sta'r p 177, or otherwise got a true N and S bearing at the starting point, lay off from it 90°, for a true 
E and W bearing at that point. This E and W bearing will be tangent to the true E and W curve. 
Ruu this tangent carefully ; and at intervals (say at the end of each mile) layoff from it (towards 
the N if iu N lat, or vice versa) an offset whose length in feet is equal to the proper one from the 
following table, multiplied by the square of the distance iu miles from the starting point. These 
offsets will mark points iu the true E and W curve. 


Latitude N or S. 

50 io° 15° 20° 25° 30° 35° 40° 45° 50° 55° 60 J 65° 

Qflfcets in ft one mile from starting- point. 

.058 .118 .179 . 243 .311 .385 .467 .559 .667 .795 .952 1.15 1.43 

Or, any offset in ft = .6666 X Total Dist in miles® X Nat, Tang of Lat. 

A rhumb line is any one that crosses a meridian obliquely, that is, is 

ueiiher due N aud S, nor E aud W. 

































172 


LAND SURVEYING 


Side. 

Bearing. 

Dist. Ft. 

Latitudes. 

Departures. 

N. 

S. 

E. 

W. 




1 

N 10° 40' W 

1060 

1015.5 

1.7 



304.0 

2.7 

__/■ 




1013 8 



... 301.3 

2 

N 75° 32' E 

1202 

300.3 

1.9 


1163.9 

3.1 





298.4 

. .. 

... 1107.0 


3 

S 69° 25' E 

1110 


390.2 

1.8 

1039.2 

2.9 






392 ... 

... 1042.1 


4 

S 41° 3' W 

850 


641.0 

1.3 


558.2 

2.2 





642.3... 

•••••••••••• • • • 

... 556.0 

5 

N 79° 40' W 

802 

143.9 

1.3 



789.0 

2.1 




142.6... 

. 

• •••• •••••••• 

... 786.9 

6 

S 53° 30' W 

705 


419.3 

1.1 


566.7 

1.8 




• 

420.4... 


564 0 










5729 

1454.8 

1454.7 

2209.1 

2209.1 



Sum of 
Sides. 

Cor’d Ns. 

Cor’d Ss. 

Cor’d Es. 

Cor’d Ws. 


Now we have already found by the old table that the Ns and the ITs are too long; consequently 
they must be shortened ; while the Ss, and Es, must be lengthened; all in the following proportions: 
As the 

Sum of all . Any given .. Total err of . Err of lat, or dep, 
the sides • side • • lat or dep • of given side. 

Thus, commencing with the lat of side 1, we have, as 

Sum of all the sides. . Side 1. .. Total lat err. . Lat err of side 1. 

5729 • 1060 • • 9.2 • 1.7 

Now as the lat of side 1 is north, it must be shortened ; hence it becomes = 1015.5—1.7 = 1013.8, as 
figured out in the new table. Again we have for the departure of side 1, 

Sum of all the sides. . Side 1. .. Total dep err. . Dep err of side 1. 

5729 • 1060 • ♦ 14.8 • 2 .7 


Now as the dep of side 1 is west, it must be shortened; hence it beoomes 304 — 2.7 = 301 3 as figured 
out in the new table. 


N 



Proceeding thus with each 
side, we obtain all the corrected 
lats and deps as shown in the 
new table; where they are con¬ 
nected with their respective 
sides by dotted lines; but in 
practice it is better to cross out 
the original ones when the cal¬ 
culation is finished and proved. 
If we now add up the 4 cols of 
corrected N, S, E, W, we find that 
the Ns = the Ss ; and the Es = 
the \Ts; thus proving that the 
work is right. There is, it is 
true, a discrepancy of .1 of a ft 
between the Ns, and the Ss; but 
this is owing to our carrying 
out the corrections to only one 
decimal place; and is too small 
to be regarded. Discrepancies 
of .1 or 4 tenths of a foot will 
sometimes occur from this 
cause; but mav be neglected. 
The corrected lats and deps 
must evidently change the 
bearing and distance of every 
survey by means of the corrected 






















































LAND SURVEYING 


173 


lats and deps alone. The principle is self-evident, explaining itself. First draw a meridian line 
4 ’ upoii it fix oti a point 1, to represent the extreme west * corner of the survey. 

Ihen from the point 1, prick off by scale, northward, the dist 1, 2 = the corrected northing 298.4 
i 2 » taken from the last table ; from 2' southward prick off the dist 2', 3% the corrected south- 

lng 392 of side 3^; from 3' southward prick off 3', 4', = southing 642.3 of side 4; from 4' northward 
prick off 4', o' — northing 142.6 of side 5; from 5' prick off southward 5', 6'= southing of side 6.t 
Theu from the points 2', 3', 4', 5', 6', draw indefinite lines due eastward, or at right angles to the 
nienman line. Make by scale, 2% 2corrected departure of side 2 ; and join 1, 2. Make 3', 3 = dep 
of side 2-j-dep ot side 3 ; and join 2, 3; make 4', 4 = 3', 3 — dep of side 4; and join 3, 4; make 5', 5 
, * * .^P sl ? e ^ ’ anc * *j°* u ^» make 6', 6 = 5', 5 — dep of side 6 ; and join 5, 6 J Finally join 

b, 1; and the plot is complete. If scrupulous accuracy is not required, the contents may be found by 
the mechanical method of triangles; the bearings, by the protractor; and the lengths of the sides, 
by the scale ; all with an approximation sufficient for ordinary purposes; and perhaps quite as close 
as by the method by calculation, when, as is customary, the bearings are taken only to the nearest 

quarter of a degree. We have already said that with a scale of feet per inch = ~ Fage leng -- 0fsid( '‘ a ‘ 

the error of area need not exceed the 2 ijWth part. 

But if it is required to calculate the area of the corrected survey with rigorous exactness, it may 
be done on the following 
principle, (see Fig 5.) If a a, 
meridian line N S be sup- IN 
posed to be drawn through 
the extreme west corner 1 of 
a survey; and lines (called 
middle distances) drawn (as 
the dotted ones in the Fig) 
at right angles to said me¬ 
ridian, from the center of 
each side of the survey; 
then if each of the middle 
dists of such sides as have 
northings, be mult by the 
corrected northing of its cor¬ 
responding side; and if each 
of the middle dists of such 
sides as have southings, be 
mult by the corrected south¬ 
ing of its corresponding 
side; if we add all the north 
prods into one sum ; and all 
the south prods into another 
sum; and subtract the least 
o f these sums from the great¬ 
est, the rem will be the area 



* The extreme east corner would answer as well, with a slight change in the subsequent oper¬ 
ations, as will become evident. 

t Instead of pricking off these northings and southings in succession, from each other, it will be 
wore correct in practice to prepare first a table showing how far each of the points 2',3', &c, is north 
or south from 1. This being done, the points can be pricked off north or south from 1, without mov¬ 
ing the scale each time; and of course with greater accuracy. Such a table is readily formed. Rule 
it as below; and in the first three columns place the numbers of the sides (starting with side 2 from 
point 1 ;) and their respective corrected northings and southings. The formation of the 4th and 5th 
cols by means of the 3d and 4th ones, explains itself. Its accuracy is proved by the final result 
being 0. 






Dist N or S from Point 1. 

Side. 

N. lat. 

S. lat. 


N. 

S. 

2 

298.4 



298.4 


3 


392. 



93.6 

4 


642.3 



735.9 

5 

142.6 




593.3 

6 


420.4 



1013.7 

1 

1013.8 

t 


000.0 

000.0 


t A similar table should be prepared beforehand for the dists of the points 2, 3, 4, &c, east from the 
meridian line. It is done in the same manner, but requires one col less, as all the dists are on th* 
same side of the mer line. Thus, starting from point 1, with side 2: 


Side. 

E. dep. 

W. dep. 

2 

1167.0 


3 

1042.1 


4 


556.0 

5 


786.9 

6 


564.9 

1 


301.3 


Dist east from 
meridian line. 


1167.0 

2209.1 

1653.1 
866.2 
301.3 
000.0 


This work likewise proves Itself by the final result being 0. 


































174 


LAND SURVEYING, 


of the survey.* The corrected northings and southings we have already found ; as also the ea. g 
and westings. The middle dists are found by means of the latter, by employing their halves; *ddiug 
half eastings, and subtracting half westiugs. Thus it is evident that the middle dist l o s < e • 
equal to half the easting of side 2. To this add the other half easting of side 2, and a half easting 
of side 3; and the sum is plainly equal to the middle dist 3' or side 8. ro tins add the other halt 
easting of side 3, and subtract a half westing of side 4, for the middle dist 4 of side 4. From thl * 
subtract the other half westing of side 4, and a half westing of side 5, for the middle dist o of sid 
5; and so on. The actual calculation may be made thus : 


Half easting of side 2 = 

1167 _ 

583.5 E = mid dist of side 2. 

2 

583.5 E 


1042.1 

1167.0 E 

Half easting of side 3 = 

2 

521.0 E 


1688.0 E mid dist of side 3. 



521.0 E 


556 

2209.0 E 

Half westing of side 4 = 

2 

278.0 W 


1931.0 E — mid dist of side 4. 



278.0 W 


786.9 

J653.0 E 

Half westing of side 5 = 

2 

393.5 W 


1259.5 E = mid dist of side 5. 



393.5 W 


564.9 

866.0 E 


— 

282.4 W f 





583.6 E = mid dist of side 6. 
282.4 W 


301.3 

301.2 E 

Half westing of side 1 = 

2 • 

150.6 W 


150.6 E = mid dist of side 1. 


The work always proves itself by the last two results being equal. 

Next make a table like the following, iu the first 4 cols of which place the numbers of the side*, 
the middle dists, the northings, and southings. Mult each middle dist by its corresponding northing 
or southing, aud place the products in their proper col. Add up each col; subtract the least from the 


Side. 

Middle dist. 

Northing. 

Southing. 

North prod. 

South prod. 


1 

150.6 

1013.8 


152678 



2 

583.5 

298.4 


174116 



3 

1688 


392 


661696 

4 

4 

1931 


642.3 


1240281 


5 

1259.5 

142.6 


179605 



6 

583.6 


420.4 


245345 






506399 

2147322 







506399 






435 

60)1640923(37. 

67 Acres. 


* Proof. To illustrate the principle upon wlfich this 
rule is based, let ah, be, and c a, Fig 6, represent in 
order the 3 sides of the triangular plot of a survey, with 
a meridian line oi/drawu through the extreme wester¬ 
ner, a. Let lines b d and of be drawn from each corner, 
perp to the meridian line; also from the middle of each 
side draw lines we, mn, so, also perp to meridian ; and 
representing the middle dists of the sides. Then since 
the sides are regarded in the order a6, be, ca, it is 
plain that a d represents the northing of the side ab; 
fa the northing of co; and df the southing of be. 
Now if we mult the northing ad of the side ab, by its 
mid dist ew, the prod is the area of the triangle abd. 
In like manner the northing fa of the side co. mult by 
its mid dist so. gives the area of the triangle a cf. Again, 
the southing df of the side b c, mult by its mid dist mn, 
gives the area cf the entire fig dhcfd. If from this 
area we subtract the areas of the two triangles abd, 
and acf, the rent is evidently the area of the plot a be. 
So with auy other plot, however complicated. 










































LAND SURVEYING, 


175 


greatest. Tlie rem will be the area of the survey in sq ft; which, div by 43560, (the number of sq ft 
in an acre,) will he the area iu acres; in this iustauoe, 37.67 ac. 

It now remains only to calculate the corrected beariugs and lengths of the sides of the survey, all 
of which are necessarily changed by the adoption of the corrected lats aud deps. To fiud the bearing 
of any side, div its departure (K or W) by its lat (N or S); iu the table of nat tang, hud the quot; 


the angle opposite it is the reqd angle of bearing. 


Thus, for the course of side 1, we have 


301.3 W 
m3.8 N 


~ .2972 = nat tang; opposite which in the table is the reqd angle, 16° 33'; the bearing, therefore, is 
N 16 3 33' W. 


Again ; for the dist or length of any side, from the table of nat cosines take the cos opposite to 
the angle of the corrected bearing ; divide the corrected lat (N or S) of the side by the cos. Thus, 
for the dist of side 1, we fiud opposite 16° 33', the cos .9586. And 


Lat. Cos. 

1013.8 -v .9586 = 1057.6 the reqd dist. 

The following table contains all the corrections of the foregoing survey ; consequently, if the bear¬ 


Side. 

Bearing. 

Dist. Ft. 

1 

N 16° 33' W 

1057.6 

2 

N 75° 39' E 

1204.0 

3 

S 69° 23' E 

1113.3 

4 

S 40° 53' W 

849.6 

5 

N 79°44'W 

too.i 

6 

S 53° 21' W 

704.3 


ings and dists are correctly plotted, they will close perfectly. The young assistant is advised to 
practise doing this, as well as dividing the plot into triangles, and computing the couteut. In this 
manner he will soon learn what degree of care is necessary to insure accurate results. 

The following hints may often be of service. 

1st. Avoid taking bearings and 
dists along a circuitous bound¬ 
ary line like a b c, Fig 7 ; but run 
the straight line a c; and at 
right angles to it, measure off¬ 
sets to the crooked line. Sid. 

Wishing to survey a straight 
line from o to c, but being una¬ 
ble to direct the instrument 
precisely toward c, on accouut 
of intervening woods, or other 
obstacles; first run a trial line, 
as a m, as nearly in the proper 
direction as can be guessed at. 

Measure m c, and say, as a m is to m c, so is 100 ft to ? Lay off a o equal to 100 ft. and o s equal 
to ? ; and run the final line a 8 c. Or. if m c is quite small, calculate offsets like o s for every 100 ft 
along a m, and thus avoid the necessity for running a second line. 8d. When c is visible from a, but 
the intervening ground difficult to measure along, on account of marshes, &c. extend the sidejy a 
to good ground at t : then, making the angle y t cl equal to y a c, run the line t n to that point a at 
which the angle n d c is found by trial to be equal to the angle a t d. It will rarely be necessary to 
make more than one trial for this point d ; for, suppose it to be made at x , see where it strikes a cat 
t; measure i c, and continue from x, making xd~ic. 4th. In case of a very irregular piece of 
land, or a lake, Fig 8, surround it by straight lines. Survey these, and at right angles to them, 
measure offsets to the crooked boundary. 6th. Surveying a straight hue from w toward y, Hg9 



a 



Fig. 9. 


an obstacle, o, is met. To pass it, lay off a right angle wtu; measure any t u; make tuv — 
90° ; measure u v ; make u v i = 90° ; make v i = t u ; make v i y — 90°. Then is ti—uv; and 
iy is in the straight line. Or, with less trouble, at g make t g a=60°j measure any g a; mane 
gas — 60° ; #ndas = ja: make a s i = 60°. Then is g s = g a or a s ; and t s, continued toward 
y, is iu the straight line. 6th. Being between two objects, m and n, and wishing to place myself in 
range with them, I lay a straight rod c ft on the ground, and point it to one of the objects w» 1 
going to the end c. I find that it does not point to the other object. By successive trials, I find the 
position e d, in which it points to both objects, and consequently is in range with them. If no rod 























176 


LAND SURVEYING 


is at hand, two stones will answer, or two chain-pins. A plumb-line (a pebble tied to a piece of 
thread) will add to the accuracy of ranging the rod, or stones, &c. 


THE FOLLOWING TABLE 


gives deductions or additions to be made every 100 ft as actually chained along sloping 

ground, in order to reduce the sloping measurements to Horizontal ones. Keen when it is so nearly 
level that the eye cannot detect the slope, an over-measurement of an inch or two in 100 ft may 
readily occur, it is plain, that, if we measure all the undulations of the ground, we shall get 
greater totals than if we measure hor, as is supposed always to be done ; but since few surveyors 
pretend to measure hor until the slope becomes apparent to the eye, their Hues are usually too long 
bv from one to two ins in 100 feet. To counteract this to some extent, chains are frequently made 
from one to two ins longer than 100 feet; and for ordinary purposes the precaution is a pood one. 
When greater accuracy is required the chainmen should be attended by a third person, with a rod 
and slope-level, for taking the inclinations of the ground. These deductions being made, the remain¬ 
der will he the actual hor dist. 

For example, in Fig 10**;, each 100 ft a o measured up or down the 
slope ae plainly corresponds to the shorter horizontal distauce a c; the 
difference or deduction being c n. Taking a o as Rad. then a c is the 
cosine, and c n the versed sine of the angle e an of the slope. There¬ 
fore a o multiplied by the nat. cosine of the angle eon erives the reduced 
hor dist a e ; which taken from a o gives the deduction c n of our table. 

But if while chaining along the slope a e we wish to drive 
stakes that shall correspond with hor dlsts a n of 100 ft. it is evident 
that we must add c n to each 100 ft a o. as shown at x e; and the stake 

must be driven at e instead of at o. Observe that x e — c n must 

be measured horizontally. 

When the ground is very sloping, all this calculation may be avoided where great accuracy is not 
required, by actually holding the chain horizontal, as nearly as can be judged byeyt, aud finding, 
by means of a plumb-line, where its raised end would strike the ground. A whole chain at a time 

cannot be measured in this way; but shorter distances must be taken as the ground requires; at 

times, on very steep ground, not more than 5 or 10 feet. See note, p 113. 



Table of Deductions or Additions to be made per 100 feet, 
in chaining' over sloping ground. 


IN OHDEK TO REDUCE THE INCLINED MEASUREMENTS TO HORIZONTAL ONES 

See pp 354, 723, 724, and 725 for other tables. 


Slope 

in 

Deg. 

Deduct 

Feet. 

Rise in 
100 ft 
hor. 

Slope 

in 

Deg. 

Deduct 

Feet. 

Rise in 
100 ft 
hor. 

Slope 

in 

Deg. 

Deduct 

Feet. 

Rise in 
100 ft 
hor. 

Slope 

in 

Deg. 

Deduct 

Feet. 

Rise in 
100 ft 
hor. 

X 

X 

X 

1 

X 

X 

X 

2 

X 

X 

X 

3 

X 

X 

H 

4 

X 

X 

X 

5 

.001 

.004 

.009 

.015 

.024 

.034 

.047 

.061 

.077 

.095 

.115 

.137 

.161 

.187 

.214 

.244 

.275 

.308 

.343 

.381 

.436 

.873 

1.309 
1.746 

2.182 
2.619 
3.055 
3.492 
3.929 
4.366 
4.803 
5.241 
5.678 
6.116 
6.554 
6.993 
7.431 
7.870 

8.309 
8.749 

X 

X 

X 

6 

X 

X 

X 

7 

X 

X 

X 

8 

X 

X 

X 

9 

X 

X 

X 

10 

.420 

.460 

.503 

.548 

.594 

.643 

.693 

.745 

.800 

.856 

.913 

.973 

1.035 

1.098 

1.164 

1.231 

1.300 

1.371 

1.444 

1.519 

9.189 

9.629 

10.07 

10.51 

10.95 

11.39 

11.84 
12.28 

12.72 

13 17 
13.61 
14.05 
14.50 

I 4.93 

15.39 

15.84 
16.29 

16.73 
17.18 
17.63 

X 

X 

X 

11 

X 

X 

X 

12 

X 

X 

X 

13 

X 

X 

X 

14 

X 

X 

X 

15 

1.596 

1.675 

1.755 

1.837 

1.921 

2.008 

2.095 

2.185 
2.277 
2.370 
2.466 
2.563 
2.662 
2.763 
2.866 
2.970 
3.077 

3.185 
3.295 

3 407 

18 08 
18.53 
18.99 
19.44 
19.89 
20.35 
20.80 
21.26 
21.71 
22.17 
22.63 
23.09 
23 55 
24.01 
24.47 
24.93 
25.40 
25.86 
26.33 
26.79 

X 

X 

X 

16 

X 

X 

X 

17 

X 

X 

X 

18 

X 

X 

X 

19 

X 

X 

X 

20 

3.521 

3.637 

3.754 

3.874 

3.995 

4.118 

4 243 
4.370 
4.498 
4.628 
4.760 
4.894 
5.030 
5.168 
5.307 
5.448 
5.591 
5.736 
5.882 
6.0.11 

27.26 
27.73 
28.20 
28.67 
29.15 
29.62 
30.10 
30.57 
31.05 
31.53 
' 32.01 
32.49 
32.98 
33.46 
33 95 
34.43 
34.92 
35.41 
35.90 
36.40 


Chain and Pins. 


use one n oflOO h ft V Gunter ’ s ch , ai “ of 66 ft - div into 100 links of 7.92 ins in length ; am 

,' * ot 't' 0 *t.. with links 1 ft long ; and calculate areas in sq ft; which, div bv 43560 reduces then 

should benf^nn C t atr pai t , 8 ’ | ustead " r roods nnd perches. Both the chain and the chain pins. Fig. u 
good strong steel; and there should be a stout leather bag for carrying them. To bear ham 

- mering into hard ground, the'pins may be of this sham 

and size, 11 or 12 ins long, X inch thick, % wide heat 
2)4 wide, w ith a circular hole of 1)4 diarn. Each pii 
should have a strip of bright red flannel tied to its ton 
that it may be readily found among grass & c bv thi 
hind chainman. The length of the chain should hi 
... „ Jested ever J few days; and the target-rod may be uset 

While locating, it is well to have the chain one or two ins longer than' inn r , 
>r 12. is a vornt slip ibr „ inn r, This is scant one-eighth inch diam W 1661 



r 


Fig. 11. 

for this purpose.... ... .... „„ 

Steel wire, No 11 or 12, is a good size for a 100 ft chain 

































LAND SURVEYING 


177 


Nat Sines of Polar Dists of Polaris or N. Star. 


Year. 

Sine. 

Year. 

Sine. 

Year. 

Sine. 


Year. 

Sine. 

Year. 

Sine. 

Year. 

Sine. 

1880 

.0232 

1883 

.0229 

1886 

.0227 


1889 

.0224 

1892 

.0221 

1895 

.0218 

1881 

.0231 

1884 

.0229 

1887 

.0226 


1890 

.0223 

1893 

.0220 

1896 

.0217 

1882 

.0230 

1885 

.0228 

1888 

.0225 


1891 

.0222 

1894 

.0219 

1897 

.0216 


Nat Secants of North Latitudes. 


Lat . 

Sec. 

Lat . 

Sec. 

Lat . 

Sec. 

Lit . 

Sec. 

Lat . 

Sec. 

Lat . 

Sec. 

0° 

1.000 

24° 

1.095 


1.196 

yo 

1.296 

%° 

1.433 

52° 

1.624 

2 

1.001 

34 

1.099 

% 

1.199 

74 

1.301 

46 

1.440 

Y 

1 633 

4 

1.002 

25 

1.103 

k 

1.203 

40 

1.305 

Y 

1.446 

u 

1.643 

5 

1.004 

34 

1.108 

34 

1.206 

Y* 

1.310 


1.453 

k 

1.652 

6 

1.006 

26 

1.113 

34 

1.210 

H 

1.315 

% 

1.460 

53 

1.662 

7 

1.008 

K 

1.117 

72 

1.213 

% 

1.320 

47 

1.466 

Y 

1.671 

8 

1.010 

'll 

1.122 


1.217 

41 

1.325 

Y 

1.473 

u 

1.681 

9 

1.013 

34 

1.127 

35 

1.221 

Y 

1.330 


1.480 

k 

1.691 

10 

1.016 

28 

1.133 

34 

1.225 


1.335 

74 

1.487 

54 

1.701 

11 

1.019 

34 

1.138 

34 

1.228 

% 

1.340 

48 

1.495 

Y 

1.712 

12 

1.022 

29 

1.143 

% 

1.232 

42 

1.346 

Y 

1.502 

U 

1.722 

13 

1.026 

34 

1.149 

36 

1.236 

34 

1.351 

V, 

1.509 

k 

1.733 

14 

1.031 

30 

1.155 

34 

1.240 

72 

1.356 

74 

1.517 

55 

1.743 

15 

1.035 

K 

1.158 


1.244 

74 

1.362 

49 

1.524 

Y 

1.754 

16 

1.040 

A 

1.161 

•34 

1.248 

43 

1.367 

Y 

1.532 

u 

1.766 

17 

1.046 

% 

1.164 

37 

1.252 

34 

>4 

1.373 

u 

1.540 

It 

1.777 

18 

1.052 

31 

1.167 

l 4 

1.256 

1.379 

3 A 

1.548 

1.788 

19 

1.058 


1.170 


1.261 

% 

1.384 

50 

1.556 

A 

1.800 

20 

1.064 

ii 

1.173 


1.265 

44 

1.390 

Y 

1.564 


1.812 

21 

1.071 

% 

1.176 

38 

1.269 

k 

1.396 

1/ 

1.572 

k 

1.824 

34 

1.075 

32 

1.179 

34 

1.273 


1.402 


1.581 

57 

1.836 

22 

1.079 


1.182 


1.278 

74 

1.408 

51 

1.589 

Y 

1.849 

34 

23 

1.082 


1.186 

% 

1.282 

45 

1.414 

Y 

1.598 

v> 

1.861 

1.086 

1.189 

39 

1.287 

34 

1.420 


1.606 

74 

1.874 

34 

1.090 

33 

1.192 

34 

1.291 

34 

1.427 

?4 

1.615 

58 

1.887 


To find a Meridian Line (a true North and South line) by 
means of the North Star. (Polaris.) 


The north star appears to describe a small circle, rt n', &c, Pig 14, around the true north point, or 
north pole, as a center. The rad of this circle is estimated bv the angle between the star and the 
pole, as measured from the earth ; and is called the polar dist of the star. This polar dist be¬ 
comes 19 -^ seconds, or very nearly of a minute less every year. On Jan 1, 1885, it is, approx¬ 
imately, 1° 17' 58". On the first of 1890, it will be about 1° 16' 41", &c. When, in its revolution, the 
star is farthest east or west from the pole, as at n' or n", it is said to be at its greatest K or \V 
elongation. Then its apparent motion for several min is nearly vert, and consequently art ,i us 
the best opportunity for an observation in the simple manner here described. The arrows in Pig It 
show the direction in which the stars appear to move from east to west when the spectator faces the 


north. 

The latitude of the place must be known approximately. Taking it at the closest one in our fore¬ 
going table, the error in the position of the meridian will not exceed half a min of azimuth in lat 57°, 
or one quarter min in lat 40° ; and still less in lower lats. 

About 3 ft above ground fix firmly, perfectly level, and as nearly east and west as may be, a 
smooth narrow pieceof board, about.! ft long, to serve asakindof 
table. Also prepare another piece a a, about a foot long; and fasten 
to it. at right angles, a compass-sight, or a strip of thin metal, with 
a straight slit, (shown by a black line in the fig.l about 6 ins long 
and yL iuch wide. This piece of board is to be slid along the 
table! as the observer follows the motions of the star toward the 
east or west: looking at it through the vert slit. Plant a stout 
pole, about 20 ft long, firmly in the ground, with its top as nearly 
north as possible from the middle of the table. Its top should 
lean 2 or 3 a toward either the east or the west; and a plumb- 


CLf 


a 


Fig. 12 . 


Hue must be suspended from its top, with a bob weighing one or two lbs, which may swing in a 
bucket of water placed on the ground. This is to prevent the line from being so easily moved by 
slight currents of air ; and for further steadiness, the pole itself should be well braced from within, a 


12 











































































178 


LAND SURVEYING, 


P 



few feet below its top. The proper dist a o, of the pole p o. from the table t a. may toe Joand 

thus: Make an angle n m s, equal approximately to the lat of the place. Open 

to eaual bv any convenient scale, the height t a of the table; and draw t a. Then take ny tne 
to equal, oy any cu 8anie sca i e , the hejght p 0 o{ the pole above groU ud ; and p ace 

it upon the sketch, so that the top p shall be by scale a it ot 
two above to n. Then a o, by the same scale, will be about the 
dist reqd; probably from 3 to 5 yards. A deviation of a ft or so 
from this will not be important. 

The correct clock time at any place, for the elongation, may be 
found within a few min from the following table. 

Instead of a pole and plumb-line, the writer would suggest a 
planed, straight-edged board planted vert and braced; its sid» 
toward the observer. 

The observer should be at his station at least au hour in 

advance of the time. Placing the board a a. upon the table aud 
in range with the plumb-line and star, he will watch both of them 
through the vert slit; sliding the board along the table, so as to 
keep the slit in the range as long as the star continues to move 
toward the east or west, as the case may be. An assistant must 
hold a candle, or lantern, on a pole near the plumb-line, to euahle 
tbe observer to see the latter. As the star approaches its elonga¬ 
tion, it will appear to move nearly vert for several min, so that it can Ik; seen without moving the 
slit. When certain by this that the star has reached its elongation, confine the slidiug board to the 
table by sticking a few tacks around its edges. Then let a third person, with another cam e ’ 8 !' 
some dist, (a hundred yards or more if convenient.) in a direction toward the star; an t en 
a stake as directed by the observer, who will take care that it is exactly in range with the slit and 
plumb-line. Another stake must then be driven exactly under either the slit or the plumb-line. 
Having thu 9 placed the two stakes in the range of the elongation, defer the remainder of tbe ope a 1 n 
antil morning. From the tables given above take out the sine of the polar di9t, and also the secant 
of the lat. Molt these together. The prod will be the nat sine of an angle called the azimuth 
of the star. Find the sine in table, p 60, &c. aud the angle which corresponds to it. This azimuth 
angle will be between 1° 20' and 2° 30', according to lat. Place an instrument over the S stake, sight 
to the N one, and lav off this angle to the E if the eloug was W, or vice versa, and drive a stake to 
mark it. This last direction is true N and S. It might be supposed that after driving the first two 
stakes, a true meridian could be had by merely laviug off tbe polar dist, by means of a compass or 
transit; but this is not so. Place the compass over the south stake, and take sight to the north one. 
If, then, the north end of the needle points east of the line, the variation of the compass is east, and 
vice versa. 

Times by a correct clock of Elongations of the N. Star. 

Deduced from U. S. Coast Survey table, calculated for April 1, 1883, to April 1. 1884, and for lat 
38° N ; but will answer within about 5 minutes for any lat up to 60° N, and until 1890. 


Times of Eastern Elongations. 


Day of 
Month. 

Apr. 

May* 

June. 

July. 

Au*. 

8cp. 


H 

M. 

H. M. 

H. M. 

H. M. 

H. M. 

H. 

M. 

1 

6 

41 A M 

4 43 A M 

2 41 A XI 

12 43 A M 

10 37 P Xf 

8 

36 P M 

7 

6 

18 “ 

4 20 “ 

2 18 *• 

12 20 " 

10 14 “ 

8 

12 " 

13 

5 

54 *• 

3 56 “ 

1 54 “ 

11 52 P M 

9 50 " 

7 

48 “ 

19 

5 

30 “ 

3 32 “ 

1 30 “ 

11 29 “ 

9 27 “ 

7 

25 “ 

25 

5 

7 “ 

3 9" 

1 7 “ 

11 5 “ 

9 3“ 

7 

1 “ 


Times of Western Elongations. 


Day of 
Month. 

Oct. 

Nov. 

Dee. 

Jan. 

Feb. 

Mar. 


H. M. 

H. M. 

H. M. 

H. M. 

H. M. 

H XI. 

1 

6 31 A M 

4 29 A M 

2 32 A M 

12 30 A M 

10 24 P M 

8 30 P M 

7 

6 8“ 

4 6“ 

2 8“ 

12 6 “ 

10 00 “ 

8 6“ 

13 

5 44 “ 

3 42 “ 

1 44 “ 

11 39 P M 

9 36 “ 

7 43 •< 

19 

5 21 » 

3 19 “ 

1 21 “ 

11 15 “ 

9 13 " 

7 19 “ 

25 

4 57 “ 

2 55 “ 

12 57 “ 

10 51 “ 

8 49 “ 

6 55 “ 


For days of the month intermediate of those in the table, it will be near enough to make the time 
4 min earlier each succeeding day. 

During nearly all of the four months, March, April, September, and October, the elongations take 
place in daylight; so that this method cannot then be used. Nor can it be used at anv time in places 
south of about 4° north of the equator, because there the north star is not visible. But during that 
time a meridian may be found by recollecting that when the north star n is on the meridian, or. in 
other words, is truly north from ns. the star Alioth, a, is very nearly vertically above it, if the 
north star « is on the meridian below the pole: or below it, as at a'", if the north star, «"' is on 
the meridian above the pole. When the north star n' is at its east elongation, Alioth is horizon¬ 
tally west of it, as at a’; and when the north star. n". is at its west elongation, Alioth is horizontally 
east of it. as at a". All that is necessary, therefore, is to prepare an arrangement of table, pole 
plumb line, Ac. precisely as before; except that the plumb-line must be nearer the observer, as he 
will have to watch Alioth above the north star. Watch through the movable slit until Alioth is on 
the same vert line with the north star. Then put in two stakes as before, and they will be nearly 
in a Hue north aud south liue. But to be more exact, either lay off (to the E if Alioth is above 







































LAND SURVEYING, 


179 


Polaris, or to the W if below) an azimuth Angle of 11 minutes; or else do not drive the stakes when 
the two stars are in a vert line, but note the time, and then wait 24.5 minutes. Then take the 
range to Polaris alone. This last range will be true N and S within 2 or 3 minutes depending on lat. 
until 1890. A transit with illuminated cross-wires can plainly be used in¬ 
stead of the plumbliue, &c, but is more troublesome except to an expert. A very correct method 
adapted to both N and S lats, is to take the two ranges to any N or S circumpolar star on the same 

night; when it is at any two equal altitudes. Half way between them 

will be true N and S. 



There can be no 
difficulty in find¬ 
ing Alioth, as it 
is one of the 7 
bright stars in 
the fine constel- 
lation so well 
known as the 
Great Bear, or 
the Wagon and 
Horses. Alioth is 
the horse nearest 
tothe fore-wheels 
of the wagon. 
The two hind- 
wheels t t are 
known to every 
schoolboy as the 
“Pointers," be¬ 
cause they point 
nearly in the di¬ 
re c t i o n to the 
North Star. The 
relative positions 
of these 7 stars, 
as shown in Fig 
14, are tolerably 
oorreet. 








180 


TRAVERSE TABLE. 


Traverse Table for a Distance = 1. 



Lat. 

or 

N. S. 

Dep. 

or 

E. W. 



Lat. 

or 

N. S. 

Dep. 

or 

E. W. 



Lat. 

or 

N. S. 

Dep. 

or 

E. W. 


0°0’ 

2 

4 

6 

8 

10 

12 

14 

18 

18 

20 

22 

24 

28 

28 

30 

32 

34 

36 

38 

40 

42 

44 

46 

48 

50 

52 

54 

56 

58 

1°0 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

32 

34 

36 

38 

40 

42 

44 

46 

48 

50 

52 

54 

56 

58 

2°0' 

1.0000 

1.0000 

1.0000 

l.oooo 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1.0000 

1.00(H) 

1.0000 
1.0000 
1.0000 
.9909 
.9999 
.9999 
.9999 
.9999 
.9999 
.9999 
.9999 
.9999 
.9999 
.9999 
.99)9 
.9998 
.9998 
.9998 
.9998 
.9998 
.9998 
.9998 
.9998 
.9998 
.9997 
.9997 
.9997 
.9997 
.9997 
.9997 
.9997 
.9996 
.9996 
.9996 
.9996 
.9996 
•9996 
.9995 
.9995 
.9995 
.9995 
.9995 
.9995 
.9994 
.9994 
.9994 

.0000 

.0006 

.0012 

.0017 

.0023 

.0029 

.0035 

.0041 

.0047 

.0052 

.0058 

.0064 

.0070 

.0076 

.0081 

.0087 

.0093 

.0099 

.0105 

.0111 

.0116 

.0122 

.0128 

.0134 

.0140 

.0145 

.0151 

.0157 

.0163 

.0169 

.0175 

.0180 

.0186 

.0192 

.0198 

.0204 

.0209 

.0215 

.0221 

.0227 

.0233 

.0239 

.0244 

.0250 

.0256 

.0262 

.0268 

.0273 

.0279 

.0285 

.0291 

.0297 

.0302 

.0308 

.0314 

.0320 

.0326 

.0332 

.0337 

.0343 

.0349 

90°0’ 

58 

56 

54 

52 

50 

48 

46 

44 

42 

40 

38 

36 

34 

32 

30 

28 

26 

24 

22 

20 

18 

16 

14 

12 

10 

8 

6 

4 

2 

89°0 

58 

56 

54 

52 

50 

48 

46 

44 

42 

40 

38 

36 

34 

32 

30 

28 

26 

24 

22 

20 

18 

16 

14 

12 

10 

8 

6 

4 

2 

88°0' 

2°0' 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

32 

34 

36 

38 

40 

42 

44 

46 

48 

50 

52 

54 

56 

58 

3°0' 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

32 

34 

36 

38 

40 

42 

44 

46 

48 

50 

52 

54 

56 

58 

4°0' 

.9994 

.9994 

.9993 

.9993 

.9993 

.9993 

.9993 

.9992 

.9992 

.9992 

.9992 

.9991 

.9991 

.9991 

.9991 

.9990 

.9990 

.9990 

.9990 

.9989 

.9989 

.9989 

.9989 

.9988 

.9988 

.9988 

.9987 

.9987 

.9987 

.9987 

.9986 

.9986 

.9986 

.9985 

.9985 

.9985 

.9984 

.9984 

.9984 

.9983 

.9983 

.9983 

.9982 

.9982 

.9982 

.9981 

.9981 

.9981 

.9980 

.9980 

.9980 

.9979 

.9979 

.9978 

.9978 

.9978 

.9977 

.9977 

.9976 

.9976 

.9976 

.0349 

.0355 

.0361 

.0366 

.0372 

.0378 

.0384 

.0390 

.0396 

.0401 

.0407 

.0413 

.0419 

.0425 

.0430 

.0436 

.0442 

.0448 

.0454 

.0459 

.0465 

.0471 

.0477 

.0483 

.0488 

.0494 

.0500 

.0506 

.0512 

.0518 

.0523 

.0529 

.0535 

.0541 

.0547 

.0552 

.0558 

.0564 

.0570 

.0576 

.0581 

.0587 

.0593 

.0599 

.0605 

.0610 

.0616 

.0622 

.0628 

.0634 

.0640 

.0645 

.0651 

.0657 

.0663 

.0669 

.0674 

.0680 

.0686 

.0692 

.0698 

88°0' 

58 

56 

54 

52 

50 

48 

46 

44 

42 

40 

38 

36 

34 

32 

30 

28 

26 

24 

22 

20 

18 

16 

14 

12 

10 

8 

6 

4 

2 

87°0' 

58 

56 

54 

52 

50 

48 

46 

44 

42 

40 

38 

36 

34 

32 

30 

28 

26 

24 

22 

20 

18 

16 

14 

12 

10 

8 

6 

4 

2 

86°0’ 

4°0' 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

32 

34 

36 

38 

40 

42 

44 

46 

48 

50 

52 

54 

56 

58 

5°0' 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

32 

34 

36 

38 

40 

42 

44 

46 

48 

50 

52 

54 

56 

58 

6°0' 

.9976 

.9975 

.9975 

.9974 

.9974 

.9974 

.9973 

.9973 

.9972 

.9972 

.9971 

.9971 

.9971 

.9970 

.9970 

.9969 

.9969 

.9968 

.9968 

.9967 

.9967 

.9966 

.9966 

.9965 

.9965 

.9964 

.9964 

.9963 

.9963 

.9962 

.9962 

.9961 

.9961 

.9960 

.9960 

.9959 

.9959 

.9958 

.9958 

.9957 

.9957 

.9956 

.9956 

.9955 

.9955 

.9954 

.9953 

.9953 

.9952 

.9952 

.9951 

.9951 

.9950 

.9949 

.9949 

.9948 

.9948 

.9947 

.9946 

.9946 

.9945 

.0698 

.0703 

.0709 

.0715 

.0721 

.0727 

.0732 

.0738 

.0744 

.0750 

.0756 

.0761 

.0767 

.0773 

.0779 

.0785 

.0790 

.0796 

.0802 

.0808 

.0814 

.0819 

.0825 

.0831 

.0837 

.0843 

.0848 

.0854 

.0860 

0866 

.0872 

.0877 

.0883 

.0889 

.0895 

.0901 

.0906 

.0912 

.0918 

.0924 

.0929 

.0935 

.0911 

.0947 

.0953 

.0958 

.0961 

.0970 

.0976 

.0982 

.0987 

.0993 

.0999 

.1005 

.1011 

.1016 

.1022 

.1028 

.1034 

.1039 

.1045 

86°0' 

58 

56 

54 

52 

50 

*8 

46 

44 

42 

40 

38 

36 

34 

32 

TO 

28 

26 

24 

22 

20 

18 

16 

14 

12 

10 

8 

6 

4 

2 

85°0' 

58 

56 

54 

52 

50 

48 

46 

44 

42 

40 

38 

36 

34 

32 

30 

28 

s 

22 

20 

18 

16 

14 

12 

10 

8 

6 

4 

2 

84°0’ 


Dep. 

or 

E. W. 

r. at. 
or 

N. S. 



Dep. 

or 

E. W. 

I.at. 

or 

N. S. 

1 


Dep. 

or 

E. W. 

Lat. 

er 

N.S. 































































TRAVERSE TABLE 


181 


Traverse Table for a Distance = 1 . (Continued.) 



I.at. 
or 

N. S. 

Dep. 

or 

E. W. 



Lat. 

or 

N. S. 

Dep. 

or 

E. W. 



Lat. 

or 

N.S. 

Dep. 

or 

E. W. 


6°0 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

32 

34 

36 

38 

40 

42 

44 

46 

48 

50 

52 

54 

56 

58 

7°0 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

32 

34 

36 

38 

40 

42 

44 

46 

48 

50 

52 

54 

56 

58 

8°0' 

.9945 

.9945 

.9944 

.9943 

.9943 

.9942 

.9942 

.9941 

.9940 

.9940 

.9939 

.9938 

.9938 

.9937 

.9936 

.9936 

.9935 

.9934 

.9934 

.9933 

.9932 

.9932 

.9931 

.9930 

.9930 

.9929 

.9928 

.9928 

.9927 

.9926 

.9925 

.9925 

.9924 

.9923 

.9923 

.9922 

.9921 

.9920 

.9920 

.9919 

.9918 

.9917 

.9917 

.9916 

.9915 

.9914 

.9914 

.9913 

.9912 

.9911 

.9911 

.9910 

.9909 

.9908 

.9907 

.9907 

.9906 

.9905 

.9904 

.9903 

.9903 

.1045 
• 1.051 
.1057 
.1063 
.1068 
.1074 
.1080 
.1086 
.1092 
.1097 
.1103 
.1109 
.1115 
.1120 
.1126 
.1132 
.1138 
.1144 
.1149 
.1155 
.1161 
.1167 
.1172 
.1178 
.1184 
.1190 
.1196 
.1201 
.1207 
.1213 
.1219 
.1224 
.1230 
.1236 
.1242 
.1248 
.1253 
.1259 
.1265 
.1271 
.1276 
.1282 
.1288 
.1294 
.1299 
.1305 
.1311 
.1317 
.1323 
.1328 
.1334 
.1340 
.1346 
.1351 
.1357 
.1363 
.1369 
.1374 
.1380 
.1386 
.1392 

84°0' 

58 

56 

54 

52 

50 

48 

46 

44 

42 

40 

38 

36 

34 

32 

30 

28 

26 

24 

22 

20 

18 

16 

1 14 

12 
10 

8 

6 

4 

2 

83 °0' 
58 
56 
54 

52 

50 

48 

46 

44 

42 

40 

38 

36 

34 

32 

30 

28 

26 

24 

22 

20 

18 

16 

14 

12 

10 

8 

6 

4 

2 

82°0 

8°0' 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

32 

34 

36 

38 

40 

42 

44 

46 

48 

50 

52 

54 

56 

58 

9°0 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

32 

34 

36 

38 

40 

42 

44 

46 

48 

50 

52 

54 

56 

58 

10°0' 

.9903 

.9902 

.9901 

.9900 

.9899 

.9899 

.9898 

.9897 

.9896 

.9895 

.9894 

.9894 

.9893 

.9892 

.9891 

.9890 

.9889 

.9888 

.9888 

.9887 

.9886 

.9885 

.9884 

.9883 

.9882 

.9881 

.9880 

.9880 

.9879 

.9878 

.9877 

.9876 

.9875 

.9874 

.9873 

.9872 

.9871 

.9870 

.9869 

.9869 

.9868 

.9867 

.9866 

.9865 

.9864 

.9863 

.9862 

.9861 

.9860 

.9859 

.9858 

.9857 

.9856 

.9855 

.9854 

.9853 

.9852 

.9851 

.9850 

.9849 

.9848 

.1392 

.1397 

.1403 

.1409 

.1415 

.1421 

.1426 

.1432 

.1438 

.1444 

.1449 

.1455 

.1461 

.1467 

.1472 

.1478 

.1484 

.1490 

.1495 

.1501 

.1507 

.1513 

.1518 

.1524 

.1530 

.1536 

.1541 

.1547 

.1553 

.1559 

.1564 

.1570 

.1576 

.1582 

.1587 

.1593 

.1599 

.1605 

.1610 

.1616 

.1622 

.1628 

.1633 

.1639 

.1645 

.1650 

.1656 

.1662 

.1668 

.1673 

.1679 

.1685 

.1691 

.1696 

.1702 

.1708 

.1714 

.1719 

.1725 

.1731 

.1736 

82°0' 

58 

56 

54 

52 

50 

48 

46 

44 

42 

40 

38 

36 

34 

32 

30 

28 

26 

24 

22 

20 

18 

16 

14 

12 

10 

8 

6 

4 

2 

81 °0’ 
58 

56 

54 

52 

50 

48 

46 

44 

42 

40 

38 

36 

34 

32 

30 

28 

26 

24 

22 

20 

18 

16 

14 

12 

10 

8 

6 

4 

2 

80°0' 

10°0' 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

32 

34 

36 

38 

40 

42 

44 

46 

48 

50 

52 

54 

56 

58 

I1°0' 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

32 

34 

36 

38 

40 

42 

44 

46 

48 

50 

52 

54 

56 

58 

12°0' 

.9848 

.9847 

.9846 

.9845 

.9844 

.9843 

.9842 

.9841 

.9840 

.9839 

.9838 

.9837 

.9836 

.9835 

.9834 

.9833 

.9831 

.9X30 

.9829 

.9828 

.9827 

.9826 

.9825 

.9824 

.9823 

.9822 

.9821 

.9820 

.9818 

.9817 

.9816 

.9815 

.9814 

.9813 

.9812- 

.9811 

.9810 

.9808 

.9807 

.9806 

.9805 

.9804 

.9803 

.9802 

.9800 

.9799 

.9798 

.9797 

.9796 

.9795 

.9793 

.9792 

.9791 

.9790 

.9789 

.9787 

.9786 

.9785 

.9784 

.9783 

.9781 

.1736 
.1742 
.1748 
.1754 
.1759 
.1765 
.1771 
.1777 
.1782 
.1788 
.1794 
.1799 
.1805 
.1811 
.1817 
.1822 
.1828 
.1834 
.1840 
.1845 
.1851 
.1857 
.1862 
.1868 
.1874 
.1880 
.1885 
.1891 
.1897 
.1902 
.1908 
.1914 
.1920 
.1925 
.1931 
.1937 
.1942 
.1948 
1954 
.1959 
.1965 
.1971 
.1977 
.1982 
.1988 
.1994 
.1999 
.2005 
.2011 
.2016 
.2022 
.2028 
.2034 
.2039 
.2045 
.2051 
.2056 
.2062 
.2068 
.2073 
.2079 

80'0' 
58 
56 
54 
52 
50 
48 
46 
44 

! « 
40 

1 38 

36 

34 

32 

30 

28 

26 

24 

22 

20 

18 

16 

14 

12 

10 

8 

6 

4 

2 

79°0' 

58 

56 

54 

52 

50 

48 

46 

44 

42 

40 

38 

36 

34 

32 

30 

28 

26 

24 

22 

20 

18 

16 

14 

12 

10 

8 

6 

4 

2 

78°0' 


Dep. 

Lat. 



Dep. 

Lat. 



Dep. 

Lat. 



or 

or 



or 

or 



or 

or 



E. W. 

N. S. 



E. W. 

N. S. 



E. W. 

N.S. 


































































182 


TRAVERSE TABLE, 


Traverse Table for a Distance = 1. (Continued.) 



Lat. 

or 

N. S. 

Dep. 

or 

E. W. 



Lat. 

or 

N. S. 

Dep. 

or 

E. W. 



Lat. 

or 

N. S. 

Dep. 

or 

E. W. 


12°0' 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

32 

34 

36 

38 

40 

42 

44 

46 

48 

50 

52 

54 

56 

58 

13°0' 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

32 

34 

36 

38 

40 

42 

44 

46 

48 

50 

52 

54 

56 

58 

14°0' 

.9781 

.9780 

.9779 

.9778 

.9777 

.9775 

.9774 

•9773 

.9772 

.9770 

.9769 

.9768 

.9767 

.9765 

.9764 

.9763 

.9762 

.9760 

.9759 

.9758 

.9757 

.9755 

.9754 

.9753 

.9751 

.9750 

.9749 

.9748 

.9746 

.9745 

.9744 

.9742 

.9741 

.9740 

.9738 

.9737 

.9736 

.9734 

.9733 

.9732 

.9730 

.9729 

.9728 

.9726 

.9725 

.9724 

.9722 

.9721 

.9720 

.9718 

.9717 

.9715 

.9714 

.9713 

.9711 

.9710 

.9709 

.9707 

.9706 

.9704 

.9703 

.2079 
.2084 
.2090 
.2096 
.2102 
.2108 
.2113 
.2119 N 
.2125 
.2130 
.2136 
.2142 
.2147 
.2153 
.2159 
.2164 
.2170 
.2176 
.2181 
.2187 
.2193 
.2198 
.2204 
.2210 
.2215 
.2221 
.2227 
.2232 
.2238 
.2244 
.2250 
.2255 
.2261 
.2267 
.2272 
.2278 
.2284 
.2289 
.2295 
.2300 
.2306 
.2312 
.2317 
.2323 
.2329 
.2334 
.2340 
.2346 
.2351 
.2357 
.2363 
.2368 
.2374 
.2380 
.2385 
.2391 
.2397 
.2402 
.2408 
.2414 
.2419 

78°0' 

58 

56 

54 

52 

50 

48 

46 

44 

42 

40 

38 

36 

34 

32 

30 

28 

26 

24 

22 

20 

18 

16 

14 

12 

10 

8 

6 

4 

2 

77°0' 

58 

56 

54 

52 

50 

48 

46 

44 

42 

40 

38 

36 

34 

32 

30 

28 

26 

24 

22 

20 

18 

16 

14 

12 

10 

8 

6 

4 

2 

76°0' 

14°0 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

32 

34 

36 

38 

40 

42 

44 

46 

48 

50 

52 

54 

56 

58 

1500' 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

32 

34 

36 

38 

40 

42 

44 

46 

48 

50 

52 

54 

56 

58 

I6°0' 

.9703 
.9702 
.9700 
.9699 
.9697 
.9696 
.9694 
.9693 
.9692 
.9690 
.9689 
.9687 
.9686 
.9684 
.9683 
.9681 
.9680 
.9679 
.9677 
.9676 
.9674 
.9673 
.9671 
.9670 
.9668 
.9667 
.9665 
.9664 
.9662 
.9661 
.9659 
.9658 
.9656 
.9655 
.9653 
.9652 
.9650 
.9649 
.9647 
.9646 
.9644 
.9642 
.9641 
.9639 
.9638 
. 9636 
.9635 
.9633 
.9632 
.9630 
.9628 
.9627 
.9625 
.9624 
.9622 
.9621 
.9619 
.9617 
.9616 
.9614 
.9613 

.2419 

.2425 

.24:41 

.2436 

.2442 

.2447 

.2453 

.2459 

.2464 

.2470 

.2476 

.2481 

.2487 

.2493 

.2498 

.2504 

.2509 

.2515 

.2521 

.2526 

.2532 

.2538 

.2543 

.2549 

.2554 

.2560 

.2566 

.2571 

.2577 

.2583 

.2588 

.2594 

.2599 

.2605 

.2611 

.2616 

.2622 

.2628 

.2633 

.2639 

.2644 

.2650 

.2656 

.2661 

.2667 

.2672 

.2678 

.2684 

.2689 

.2695 

.2700 

.2706 

.2712 

.2717 

.2723 

.2728 

.2734 

.2740 

.2745 

.2751 

.2756 

76°0' 

58 

56 

54 

52 

50 

48 

46 

44 

42 

40 

38 

36 

34 

32 

30 

28 

26 

24 

22 

20 

18 

16 

14 

12 

10 

8 

6 

4 

2 

75°0' 

58 

56 

54 

52 

50 

48 

46 

44 

42 

40 

38 

36 

34 

32 

30 

28 

26 

24 

22 

20 

18 

16 

14 

12 

10 

8 

6 

4 

2 

74°0' 

16°0' 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

32 

34 

36 

38 

40 

42 

44 

46 

48 

50 

52 

54 

56 

58 

17°0 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

32 

34 

36 

38 

40 

42 

44 

46 

48 

50 

52 

54 

56 

58 

18°0' 

.9613 

.9611 

.9609 

.9608 

.9606 

.9605 

.9603 

.9601 

.9600 

.9598 

.9596 

.9595 

.9593 

.9591 

.9590 

.9588 

.9587 

.9585 

.9583 

.9582 

.9580 

.9578 

.9577 

.9575 

.9573 

.9572 

.9570 

.9568 

.9566 

.9565 

.9563 

.9561 

.9560 

.9558 

.9556 

.9555 

.9553 

.9551 

.9549 

.9548 

.9546 

.9544 

.9542 

.9541 

.9539 

.9537 

.9535 

.9534 

.9532 

.9530 

.9528 

.9527 

.9525 

.9523 

.9521 

.9520 

.9518 

.9516 

.9514 

.9512 

.9511 

.2756 

.2762 

.2768 

.2773 

.2779 

.2784 

.2790 

.2795 

.2801 

.2807 

.2812 

.2818 

.2823 

.2829 

.2835 

.2840 

.2846 

.2851 

.2857 

.2862 

.2868 

.2874 

.2879 

.2885 

.2890 

.2896 

.2901 

.2907 

.2913 

.2918 

.2924 

.2929 

.2935 

.2940 

.2946 

.2952 

.2957 

.2963 

.2968 

.2974 

.2979 

.2985 

.2990 

.2996 

.3002 

.3007 

.3013 

.3018 

.3024 

,3029- 

.3035 

.3040 

.3046 

.3051 

.3057 

.3062 

.3068 

.3074 

.3079 

.3085 

.3090 

74°0' 

58 

56 

54 

52 

50 

48 

46 

44 

42 

40 

38 

36 

34 

32 

30 

28 

26 

24 

22 

20 

18 

16 

14 

12 

10 

8 

6 

4 

2 

73°0' 

58 

56 

54 

52 

50 

48 

46 

44 

42 

40 

38 

36 

34 

32 

30 

28 

26 

24 

22 

20 

18 

16 

14 

12 

10 

8 

6 

4 

2 

72°0' 


Dep. 

or 

E. W. 

Lat. 

or 

N. S. 



Dep. 

or 

E. W. 

Lat. 

or 

N. S. 



Dep. 

or 

e. w'. 

Lat. 

or 

N. 8. 


---- 

-- 

















































TRAVERSE TABLE 


183 


Traverse Table for a Distance = 1. (Continued.) 



Lat. 

or 

n. a. 

Dep. 

or 

E. W. 



Lat. 

or 

N. S. 

Dep. 

or 

E. W. 



Lat. 

or 

N. S. 

Dep. 

or 

E. W. 


&8°0' 

.9511 

.3090 

72°0' 

20 r 0' 

.9397 

.3420 

70°0 

22°0 

.9272 

.3746 

68°0 

2 

.9509 

.3096 

58 

2 

.9395 

.3426 

58 

2 

.9270 

.3751 

58 

4 

.9507 

.3101 

56 

4 

.9393 

.3431 

56 

4 

.9267 

.3757 

56 

6 

.9505 

.3107 

54 

6 

.9391 

.3437 

54 

6 

.9265 

.3762 

54 

8 

.9503 

.3112 

52 

8 

.9389 

.3442 

62 

8 

.9263 

.3768 

52 

10 

.9502 

.3118 

50 

10 

.9387 

.3448 

50 

10 

.9261 

.3773 

50 

12 

.9500 

.3123 

48 

12 

.9385 

.3453 

48 

12 

.9259 

.3778 

48 

14 

.9498 

.3129 

46 

14 

.9383 

.3458 

46 

14 

.9257 

.3784 

46 

16 

.9496 

.3134 

44 

16 

.9381 

.3464 

44 

16 

.9254 

.3789 

44 

18 

.9494 

.3140 

42 

18 

.9379 

.3469 

42 

18 

.9252 

.3795 

42 

20 

.9492 

.3145 

40 

20 

.9377 

.3475 

40 

20 

.9250 

.3800 

40 

22 

.9491 

.3151 

38 

22 

.9375 

.3480 

38 

22 

.9248 

.3805 

38 

24 

.9489 

.3156 

36 

24 

.9373 

.3486 

36 

24 

.9245 

.3811 

36 

26 

.9487 

.3162 

34 

26 

.9371 

.3491 

34 

26 

.9243 

.3816 

34 

28 

.9485 

.3168 

32 

28 

.9369 

.3497 

32 

28 

.9241 

.3821 

32 

30 

.9483 

.3173 

30 

30 

.9367 

.3502 

30 

30 

.9239 

.3827 

30 

32 

.9481 

.3179 

28 

32 

.9365 

.3508 

28 

32 

.9237 

.3832 

28 

34 

.9480 

.3184 

26 

34 

.9363 

.3513 

26 

34 

.9234 

.3838 

26 

36 

.9478 

.3190 

24 

36 

.9361 

.3518 

24 

36 

.9232 

.3843 

24 

38 

.9476 

.3195 

22 

38 

.9359 

.3524 

22 

38 

.9230 

.3848 

22 

40 

.9474 

.3201 

20 

40 

.9356 

.3529 

20 

40 

.9228 

.3854 

20 

42 

.9472 

.3206 

18 

42 

.9354 

.3535 

18 

42 

.9225 

.3859 

18 

44 

.9470 

.3212 

16 

44 

.9352 

.3540 

16 

44 

.9223 

.3864 

16 

46 

.9468 

.3217 

14 

46 

.9350 

.3546 

14 

46 

.9221 

.3870 

14 

48 

.9466 

.3223 

12 

48 

.9348 

.3551 

12 

48 

.9219 

.3875 

12 

50 

.9465 

.3228 

10 

50 

.9346 

.3557 

10 

50 

.9216 

.3881 

10 

52 

.9463 

.3234 

8 

52 

.9344 

.3562 

«j 

52 

.9214 

.3886 

8 

54 

.9461 

.3239 

6 

54 

.9342 

.3567 

6 

54 

. .9212 

.3891 

6 

56 

.9459 

.3245 

4 

56 

.9340 

.3573 

4 

56 

.9210 

.3897 

4 

58 

.9457 

.3250 

2 

58 

.9338 

.3578 

2 

58 

.9207 

.3902 

2 

19°0' 

.9455 

.3256 

71°0’ 

21°0' 

.9336 

.3584 

69°0 

23°0' 

.9205 

.3907 

67°0' 

2 

.9453 

.3261 

58 

2 

.9334 

.3589 

58 

2 

.9203 

.3913 

58 

4 

.9451 

.3267 

56 

4 

.9332 

.3595 

56 

4 

.9200 

.3918 

56 

6 

.9449 

.3272 

54 

6 

.9330 

.3600 

54 

6 

.9198 

.3923 

54 

8 

.9448 

.3278 

52 

8 

.9327 

.3605 

52 

8 

.9196 

.3929 

52 

10 

.9446 

.3283 

50 

10 

.9325 

.3611 

50 

10 

.9194 

.3934 

50 

12 

.9444 

.3289 

48 

12 

.9323 

.3616 

48 

12 

.9191 

.3939 

48 

14 

.9442 

.3294 

46 

14 

.9321 

.3622 

46 

14 

.9189 

.3945 

46 

16 

.9440 

.3300 

44 

16 

.9319 

.3627 

44 

16 

.9187 

.3950 

44 

18 

.9438 

.3305 

42 

18 

.9317 

.3633 

42 

18 

.9184 

.3955 

42 

20 

.9436 

.3311 

40 

20 

.9315 

.3638 

40 

20 

.9182 

.3961 

40 

22 

.9434 

.3316 

38 

22 

.9313 

.3643 

38 

22 

.9180 

.3966 

38 

24 

.9432 

.3322 

36 

24 

.9311 

.3649 

36 

24 

.9178 

.3971 

36 

26 

.9430 

.3327 

34 

26 

.9308 

.3654 

34 

26 

.9175 

.3977 

34 

28 

.9428 

.3333 

32 

28 

.9306 

.3660 

32 

28 

.9173 

.3982 

32 

30 

.9426 

.3338 

30 

30 

.9304 

.3665 

30 

30 

.9171 

.3987 

30 

32 

.9424 

.3344 

28 

32 

.9302 

.3670 

28 

32 

.9168 

.3993 

28 

34 

.9423 

.3349 

26 

34 

.9300 

.3676 

26 

34 

.9166 

.3998 

26 

36 

.9421 

.3355 

24 

36 

.9298 

.3681 

24 

36 

.9164 

.4003 

24 

38 

.9419 

.3360 

22 

38 

.9296 

.3687 

22 

38 

.9161 

.4009 

22 

40 

.9417 

.3365 

20 

40 

.9293 

.3692 

20 

40 

.9159 

.4014 

20 

42 

.9415 

.3371 

18 

42 

.9291 

.3697 

18 

42 

.9157 

.4019 

18 

44 

.9413 

.3376 

16 

44 

.9289 

.3703 

16 

44 

.9154 

.4025 

16 

46 

.9411 

.3382 

14 

46 

.9287 

.3708 

14 

46 

.9152 

.4030 

14 

48 

.9409 

.3387 

12 

48 

.9285 

.3714 

12 

48 

.9150 

.4035 

12 

50 

.9407 

.3393 

10 

50 

.9283 

.3719 

10 

50 

.9147 

.4041 

10 

52 

.9405 

.3398 

8 

52 

.9281 

.3724 

8 

52 

.9145 

.4046 

8 

54 

.9403 

.3404 

6 

54 

.9278 

.3730 

6 

54 

.9143 

.4051 

6 

56 

.9401 

.3409 

4 

56 

.9276 

.3735 

4 

56 

.9140 

.4057 

4 

58 

.9399 

.3415 

2 

58 

.9274 

.3741 

2 

58 

.9138 

.4062 

2 

20~0' 

.9397 

.3420 

70°0' 

22°0' 

.9272 

.3746 

68°0' 

24°0' 

.9135 

.4067 

66 c 0 


Dep. 

or 

E. W. 

Lat. 

or 

N. S. 



Dep. 

or 

E. W. 

Lat. 

or 

N. S. 



Dep. 

or 

E. W. 

Lat. 

or 

N. S. 























































184 


TRAVERSE TABLE 


Traverse Table for a Distance = 1. (Continued.) 



Lat. 

or 

N. S. 

Dep. 

or 

E. W. 



Lat. 

or 

N. S. 

24 0' 

.9135 

.4067 

66°0' 

2600' 

.8988 

2 

.9133 

.4073 

58 

2 

.8985 

4 

.9131 

.4078 

56 

4 

.8983 

6 

.9128 

.4083 

54 

6 

.8980 

8 

.9126 

.4089 

52 

8 

.8978 

10 

.9124 

.4094 

50 

10 

.8975 

12 

.9121 

.4099 

48 

12 

.8973 

14 

.9119 

.4105 

46 

14 

.1970 

IK 

.9116 

.4110 

44 

16 

.8967 

18 

.9114 

.4115 

42 

18 

.8965 

20 

.9112 

.4120 

40 

20 

.8962 

22 

.9109 

.4126 

38 

22 

.8960 

24 

.9107 

.4131 

36 

21 

.8957 

26 

.!,• 04 

.4136 

34 

26 

.8955 

28 

.9102 

.4142 

32 

28 

.8952 

30 

.9100 

.4147 

30 

30 

.8949 

32 

.9097 

.4152 

28 

32 

.8947 

34 

.9095 

.4158 

26 

34 

.8944 

36 

.9092 

.4163 

24 

36 

.8942 

38 

.9090 

.4168 

22 

38 

.8939 

40 

.9088 

.4173 

20 

40 

.8936 

42 

.9085 

.4179 

18 

42 

.8934 

44 

.9083 

.4184 

16 

44 

.8931 

46 

.9080 

.4189 

14 

46 

.8928 

48 

.9078 

.4195 

12 

48 

.8926 

50 

.9075 

.4200 

10 

50 

.8923 

52 

.9073 

.4205 

8 

52 

.8921 

54 

.9070 

.4210 . 

6 

54 

.8918 

56 

.9068 

.4216 

4 

56 

.8915 

58 

.9066 

.4221 

2 

58 

.8913 

25°0' 

.9063 

.4226 

65°0 

27°0 

.8910 

2 

.9061 

.4231 

58 

2 

.8907 

4 

.9058 

.4237 

56 

4 

.8905 

6 

.9056 

.4242 

54 

6 

.8902 

8 

.9053 

.4247 

52 

8 

.8899 

10 

.9051 

.4253 

50 

10 

.8897 

12 

.9048 

.4258 

48 

12 

.8894 

14 

.9046 

.4263 

46 

14 

.8892 

16 

.9043 

.4268 

44 

16 

.8889 

18 

.9041 

.4274 

42 

18 

.8886 

20 

.9038 

.4279 

40 

20 

.8884 

22 

.9036 

.4284 

38 

22 

.8881 

24 

.9033 

.4289 

36 

24 

.8878 

26 

.9031 

.4295 

34 

26 

.8875 

28 

.9028 

.4300 

32 

28 

.8873 

30 

.9026 

.4305 

30 

30 

.8870 

32 

.9023 

.4310 

28 

32 

.8867 

34 

.9021 

.4316 

26 

34 

.8865 

36 

.9018 

.4321 

24 

36 

.8862 

38 

.9016 

.4326 

22 

38 

.8859 

40 

.9013 

.4331 

20 

40 

.8857 

42 

•9011 

.4337 

18 

42 

.8854 

44 

.9008 

.4342 

16 

44 

.8851 

46 

.9006 

.4347 

14 

46 

.8849 

48 

.9003 

.4352 

12 

48 

.8816 

50 

.9001 

.4358 

10 

50 

.8843 

52 

.8998 

.4363 

8 

52 

.8840 

54 

.8996 

.4368 

6 

54 

.8838 

56 

.8993 

.4373 

4 

56 

.8835 

58 

.8990 

.4378 

2 

58 

.8832 

26 "O' 

.8988 

.4384 

64 °0' 

28°0' 

.8829 


Cep. 

Lat. 



Dep. 


or 

or 



or 


E. W. 

N. S. 



E. W. 


Dep. 

or 

E. W. 



Lat. 

or 

N. 8. 

Dep. 

or 

E. \r. 


.4384 

64°0' 

28°0' 

.8829 

.4695 

62°0' 

.4389 

58 

2 

.8827 

.4700 

56 

.4394 

56 

4 

.8824 

.4705 

56 

.4399 

54 

6 

.8821 

.4710 

54 

.4405 

52 

8 

.8819 

.4715 

52 

.4410 

50 

10 

.8816 

.4720 

50 

.4415 

48 

12 

.8813 

.4726 

48 

.4420 

46 

14 

.8810 

.4731 

46 

.4425 

44 

16 

.8808 

.4736 

44 

.4431 

42 

18 

.8805 

.4741 

42 

.4436 

40 

20 

.8802 

.4746 

40 

.4441 

38 

22 

.8799 

.4751 

38 

.4446 

36 

24 

.8796 

.4756 

36 

.4452 

34 

26 

.8794 

.4761 

34 

.4457 

32 

28 

.8791 

.47H(> 

32 

.4462 

30 

30 

.8788 

.4772 

30 

.4467 

28 

32 

.8785 

.4777 

28 

.4472 

26 

34 

.8783 

.4782 

26 

.4478 

24 

36 

.8780 

.4787 

24 

.4483 

22 

38 

.8777 

.4792 

22 

.4488 

20 

40 

.8774 

.4797 

20 

.4493 

18 

42 

.8771 

.4802 

18 

.4498 

16 

44 

.8769 

.4807 

16 

.4504 

14 

46 

.874441 

.4812 

14 

.4509 

12 

48 

.8763 

.4818 

12 

.4514 

10 

50 

.8760 

.4823 

10 

.4519 

8 

52 

.8757 

.4828 

8 

.4524 

6 

54 

.8755 

.4833 

6 

.4530 

4 

56 

.8752 

.4838 

4 

.4535 

2 

58 

.8749 

4843 

2 

.4540 

63°0' 

29°0' 

.8746 

.4848 

61<"0' 

.4545 

58 

2 

.8743 

.4853 

58 

.4550 

56 

4 

.8741 

.4858 

56 

.4555 

54 

6 

.8738 

.4863 

54 

.4561 

52 

8 

.8735 

.4868 

52 

.4566 

50 

10 

.8732 

.4874 

50 

.4571 

48 

12 

.8729 

.4879 

48 

.4576 

46 

14 

.8726 

.4884 

46 

.4581 

44 

16 

.8724 

.4889 

44 

.4586 

42 

18 

.8721 

.4894 

42 

.4592 

40 

20 

.8718 

.4899 

40 

.4597 

38 

22 

.8715 

.4904 

38 

.4602 

36 

24 

.8712 

.4909 

36 

.4607 

34 

26 

.8709 

.4914 

34 

.4612 

32 

28 

.8706 

.4919 

32 

.4617 

30 

30 

.8704 

.4924 

30 

.4623 

28 

32 

.8701 

.4929 

28 

.4628 

26 

34 

.8698 

.4934 

26 

.4633 

24 

36 

.8695 

.4939 

24 

.4638 

22 

38 

.8692 

.4944 

22 

.4643 

20 

40 

.8689 

.4950 

20 

.4648 

18 

42 

.8686 

.4955 

18 

.4654 

16 

44 

.8683 

.4960 

16 

.4659 

14 

46 

.8681 

.4965 

14 

.4664 

12 

48 

.8678 

.4970 

12 

.4669 

10 

50 

.8675 

.4975 

10 

.4674 

8 

52 

.8672 

.4980 

8 

.4679 

6 

54 

.864)9 

.4985 

6 

.4684 

4 

56 

.8666 

.4910 

4 

.4690 

2 

58 

.8663 

.4995 

2 

.4695 

62°0' 

S0°0' 

.8660 

.50(0 

60 "O' 

Lat. 

or 



Dep. 

or 

E. \V. 

Lat. 

or 

K. S. 

























































TRAVERSE TABLE 


185 


Traverse Table for a Distance = 1. (Continued.) 



Lat. 

or 

N. S. 

Dep. 

or 

E. W. 



Lat. 

or 

N. S. 

Dep. 

or 

E. W. 



Lat. 

or 

N. S. 

Dep. 

or 

E. W. 


30°0' 

.8660 

.5000 

60°0' 

32°0' 

■ 8480 

.5299 

5800' 

34°0' 

.8290 

.5592 

56'0' 

2 

.8657 

• .5005 

58 

2 

•8477 

•5304 

58 

2 

.8287 

.5597 

58 

4 

.8654 

.5010 

56 

4 

•8474 

.5309 

56 

4 

.8284 

.5602 

56 

6 

.8652 

.5015 

54 

6 

•8471 

•5314 

54 

6 

.8281 

.5606 

54 

8 

.8649 

.5020 

52 

8 

•8468 

•5319 

52 

8 

•8277 

.5611 

52 

10 

.8646 

.5025 

50 

10 

•8465 

•5324 

50 

10 

•8274 

.5616 

50 

12 

.8643 

.5030 

48 

12 

•8462 

•5329 

48 

12 

•8271 

.5621 

48 

14 

.8640 

.5035 

46 

14 

•8459 

■ 5334 

46 

14 

• 8268 

•5626 

46 

16 

.8637 

.5040 

44 

16 

•8456 

• 5339 

44 

16 

•8264 

.5630 

44 

18 

.8634 

.5045 

42 

18 

•8453 

•5344 

42 

18 

.8261 

• 5635 

42 

20 

.8631 

.5050 

40 

20 

•8450 

•5348 

40 

20 

.8258 

•5640 

40 

22 

.8628 

.5055 

38 

22 

•8446 

•5353 

38 

22 

.8254 

.5645 

38 

24 

.8625 

.5060 

36 

24 

•8443 

•5358 

36 

24 

.8251 

• 5650 

36 

26 

.8622 

.0065 

34 

26 

•8440 

•5363 

34 

26 

.8248 

.5654 

34 

28 

.8619 

.5070 

32 

28 

•8437 

• 5368 

32 

28 

.8245 

•5659 

32 

30 

.8616 

.5075 

30 

30 

•8434 

•5373 

30 

30 

.8241 

•5664- 

30 

32 

.8613 

.5080 

28 

32 

•8431 

•5378 

28 

32 

.8238 

.5669 

28 

34 

.8610 

.5085 

26 

34 

•8428 

•5383 

26 

34 

.8235 

•5674 

26 

36 

.8607 

.5090 

24 

36 

•8425 

•5388 

24 

36 

.8231 

.5678 

24 

38 

.8604 

.5095 

22 

38 

•8421 

•5393 

22 

38 

.8228 

•5683 

22 

40 

.8601 

.5100 

20 

40 

•8418 

•5398 

20 

40 

.8225 

.5688 

20 

42 

.8599 

.5105 

18 

42 

•8415 

•5402 

18 

42 

.8221 

.5693 

18 

44 

.8596 

.5110 

16 

44 

•8412 

•5407 

16 

44 

.8218 

.5698 

16 

46 

.8593 

.5115 

14 

46 

•8409 

•5412 

14 

46 

.8215 

.5702 

14 

48 

.8590 

.5120 

12 

48 

•8406 

•5417 

12 

48 

.8211 

.5707 

12 

50 

.8587 

.5125 

10 

50 

•8403 

• -5422 

10 

50 

.8208 

•5712 

10 

52 

.8584 

.5130 

8 

52 

•8399 

•5427 

8 

52 

.8205 

•5717 

8 

54 

.8581 

.5135 

6 

54 

•8396 

• 5432 

6 

54 

.8202 

•5721 

6 

56 

.8578 

.5140 

4 

56 

•8393 

•5437 

4 

56 

.8198 

.5726 

4 

58 

.8575 

.5145 

2 

58 

•8390 

•5442 

2 

58 

.8195 

.5731 

2 

31°0' 

.8572 

.5150 

59®0 

33°0' 

•8387 

•5446 

57°0' 

35°0' 

.8192 

.5736 

55°0' 

2 

.8569 

.5155 

58 

2 

•8384 

•5451 

58 

2 

.8188 

.5741 

58 

4 

.8566 

.5160 

56 

4 

•8380 

•5456 

56 

4 

.8185 

.5745 

56 

6 

.8563 

.5165 

54 

6 

•8377 

• 5461 

54 

6 

.8181 

.5750 

54 

8 

.8560 

.5170 

52 

8 

•8374 

• 5466 

52 

8 

.8178 

.5755 

52 

10 

.8557 

.5175 

50 

10 

•8371 

•5471 

50 

10 

•8175 

.5760 

50 

12 

.8554 

.5180 

48 

12 

•8368 

•5476 

48 

12 

.8171 

.5764 

48 

14 

.8551 

.5185 

46 

14 

•8364 

•5480 

46 

14 

.8168 

.5769 

46 

16 

.8548 

.5190 

44 

16 

•8361 

•5485 

44 

16 

.8165 

.5774 

44 

18 

.8545 

.5195 

42 

18 

•8358 

•5490 

42 

18 

.8161 

•5779 

42 

20 

.8542 

.5200 

40 

20 

•8355 

.5495 

40 

20 

.8158 

.5783 

40 

22 

.8539 

.5205 

38 

22 

•8352 

•5500 

38 

22 

.8155 

.5788 

38 

24 

.8536 

.5210 

36 

24 

•8348 

.5505 

36 

24 

.8151 

.5793 

36 

26 

.8532 

.5215 

34 

26 

•8345 

.5510 

34 

26 

.8148 

.5798 

34 

28 

.8529 

.5220 

32 

28 

•8342 

.5515 

32 

28 

.8145 

.5802 

32 

30 

.8526 

.5225 

30 

30 

•8339 

.5519 

30 

30 

.8141 

.5807 

30 

32 

.8523 

.5230 

28 

32 

•8336 

.5524 

28 

32 

.8138 

.5812 

28 

34 

.8520 

.5235 

26 

34 

•8332 

.5529 

26 

34 

.8134 

.5816 

26 

36 

.8517 

.5240 

24 

36 

• 8329 

.5534 

24 

36 

.8131 

.5821 

24 

38 

.8514 

.5245 

22 

38 

•8326 

.5539 

22 

38 

.8128 

.5826 

22 

40 

.8511 

.5250 

20 

40 

•8323 

.5544 

20 

40 

.8124 

.5831 

20 

42 

.8508 

.5255 

18 

42 

•a320 

.5548 

18 

42 

.8121 

.5835 

18 

44 


.5260 

16 

44 

•8316 

.5553 

16 

44 

.8117 

.5840 

16 

46 

.8502 

.5265 

14 

46 

• 8313 

.5558 

14 

46 

.8114 

.5845 

14 

48 

.8499 

.5270 

12 

48 

•8310 

.5563 

12 

48 

.8111 

.5850 

12 

50 

.8496 

.5275 

10 

50 

• 8307 

.5568 

10 

50 

.8107 

.5854 

10 

5‘2 

.8493 

.5279 

8 

52 

•8303 

.5573 

8 

52 

.8104 

.5859 

8 

54 

.8490 

.5284 

6 

54 

•8300 

.5577 

6 

54 

.8100 

.5864 

6 

56 

.8487 

.5289 

4 

56 

•8297 

.5582 

4 

56 

.8097 

.5868 

4 

58 

£484 

.5294 

2 

58 

•8294 

.5587 

2 

58 

.8094 

.5873 

2 

32°0' 

^8480 

.5299 

58°0- 

340Q' 

.8290 

.5592 

56°0’ 

36°0' 

.8090 

.5878 

54°0' 


Dep. 

or 

E.5T. 

Lat. 

or 

N.S. 



Dep. 

or 

E. IV. 

Lat. 

or 

N.S. 



Dep. 

or 

E. W. 

Lat. 

or 

N.S. 






































6 ° 0 ' 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

32 

34 

36 

38 

40 

42 

44 

46 

48 

50 

52 

54 

56 

58 

7°0' 

2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

24 

26 

28 

30 

32 

34 

36 

38 

40 

42 

44 

46 

48 

50 

52 

54 

56 

58 

OA' 


TRAVERSE TABLE 


Traverse Table for a Distance = 1. (Continued.) 


Dep. 

or 

E. W. 



Lat. 

or 

N. S. 

Dep. 

or 

E. W. 



Lht# 

or 

N. S. 

Dep. 

or 

E. W. 


.5878 

54°0' 

38°0' 

.7880 

.6157 

52°0 

40 °0' 

.7660 

.6428 

50 °0 

.5883 

58 

2 

.7877 

.6161 

58 

2 

.7657 

.6432 

58 

.5887 

56 

4 

.7873 

.6166 

56 

4 

.7653 

.6437 

56 

.5892 

54 

6 

.7869 

.6170 

54 

6 

.7649 

.6441 

54 

.5897 

52 

8 

.7866 

.6175 

52 

8 

.7645 

.6446 

52 

.5901 

50 

10 

.7862 

.6180 

50 

10 

.7642 

.6450 

50 

.5906 

48 

12 

.7859 

.6184 

48 

12 

.7638 

.6455 

48 

.5911 

46 

14 

.7855 

.6189 

46 

14 

.7634 

.6459 

46 

.5915 

44 

16 

.7851 

.6193 

44 

16 

.7630 

.6463 

44 

.5920 

42 

18 

.7848 

.6198 

42 

18 

.7627 

.6468 

42 

.5925 

40 

20 

.7844 

.6202 

40 

20 

.7623 

.6472 

40 

.5930 

38 

22 

.7841 

.6207 

38 

22 

.7619 

.6477 

38 

.5934 

36 

24 

.7837 

.6211 

36 

24 

.7615 

.6481 

36 

.5939 

34 

26 

.7833 

.6216 

34 

26 

.7612 

.6486 

34 

.5944 

32 

28 

.7830 

.6221 

32 

28 

.7608 

.6490 

32 

.5948 

30 

30 

.7826 

.6225 

30 

30 

.7604 

.6494 

30 

.5953 

28 

32 

.7822 

.6230 

28 

32 

.7600 

.6499 

28 

.5958 

26 

34 

.7819 

.6234 

26 

34 

.7596 

.6503 

26 

.5962 

24 

36 

.7815 

.6239 

24 

36 

.7593 

.6508 

24 

.5967 

22 

38 

.7812 

.6243 

22 

38 

.7589 

.6512 

22 

.5972 

20 

40 

.7808 

.6248 

20 

40 

.7585 

.6517 

20 

.5976 

18 

42 

.7804 

.6252 

18 

42 

.7581 

.6521 

18 

.5981 

16 

44 

.7801 

.6257 

16 

44 

.7578 

.6525 

16 

.5986 

14 

46 

.7797 

.6262 

14 

46 

.7574 

.6530 

14 

.5990 

12 

48 

.7793 

.6266 

12 

48 

.7570 

.6534 

12 

.5995 

10 

50 

.7790 

.6271 

10 

50 

.7566 

.6539 

10 

.6000 

8 

52 

.7786 

.6275 

8 

52 

.7562 

.6543 

8 

.6004 

6 

54 

.7782 

.6280 

6 

54 

.7559 

.6547 

6 

.604)9 

4 

56 

.7779 

.6284 

4 

56 

.7555 

.6552 

4 

.6014 

2 

58 

.7775 

.6289 

2 

58 

.7551 

.6556 

2 

.6018 

53 °0' 

39°0' 

.7771 

.6293 

51°0' 

41°0' 

.7547 

.6561 

49°0 

,6023 

58 

2 

.7768 

.6298 

58 

2 

.7543 

.6565 

58 

.6027 

56 

4 

.7764 

.6302 

56 

4 

.7539 

.6569 

56 

.6032 

54 

6 

.7760 

.6307 

54 

6 



54 

.6037 

52 

8 

.7757 

.6311 

52 

8 

.7532 

.6578 

52 

.6041 

50 

10 

.7753 

.6316 

50 

10 

.7528 

.6583 

50 

.6046 

48 

12 

.7749 

.6320 

48 

12 

.7524 

.6587 

48 

.6051 

46 

14 

.7746 

.6325 

46 

14 

.7520 

.6591 

46 

.6055 

44 

16 

.7742 

.6329 

44 

16 

.7516 

.6596 

44 

.6060 

42 

18 

.7738 

.6334 

42 

18 

.7513 

.6600 

42 

.6065 

40 

20 

.7735 

.6338 

40 

20 

.7509 

.6604 

40 

.6069 

38 

22 

.7731 

.6343 

38 

22 

.7505 

.6609 

38 

.6074 

36 

24 

.7727 

.6347 

36 

24 

.7501 

.6613 

36 

.6078 

34 

26 

.7724 

.6352 

34 

26 

.7497 

.6617 

34 

.6083 

32 

28 

.7720 

.6356 

32 

28 

.7493 

.6622 

32 

.6088 

30 

30 

.7716 

.6361 

30 

30 

.7490 

.6626 

30 

.6092 

28 

32 

.7713 

.6365 

28 

82 

.7486 

.6631 

28 

.6097 

26 

34 

.7709 

.6370 

26 

34 

.7482 


26 

.6101 

24 

36 

.7705 

.6374 

24 

36 

.7478 

.6639 

24 

.6106 

22 

38 

.7701 

.6379 

22 

38 

.7474 

.6644 

22 

.6111 

20 

40 

.7698 

.6383 

20 

40 

.7470 

.6648 

20 

.6115 

18 

42 

.7694 

.6388 

18 

42 

.7466 

.6652 

18 

.6120 

16 

44 

.7690 

.6392 

16 

44 

.7463 

.6657 

16 

.6124 

14 

46 

.7687 

.6397 

14 

46 

.7459 

.6661 

14 

.6129 

12 

48 

.7683 

.6401 

12 

48 

.7455 


12 

.6134 

10 

50 

.7679 

.6406 

10 

50 

.7451 

.6670 

10 

.6138 

8 

52 

.7675 

.6410 

8 

52 

.7447 

.6674 

8 

.6143 

6 

54 

.7672 

.6414 

6 

54 

.7443 

.6678 

6 

.6147 

4 

56 

.7668 

.6419 

4 

56 

.7439 

.6683 

4 

.6152 

2 

58 

.7664 

.6423 

2 

58 

. 7435 

.6687 

2 

.6157 

52°0 

10°0' 

.7660 

.6428 

50°0' 

42°0' 

.7431 

.6691 

48°0' 

Lat. 

or 

N. S. 



Dep. 

or 

E. W. 

Lat. 

or 

N. S. 



Dep. 

or 

E. W. 

i 

Lat. 

or 

N. S. 




























































TRAVERSE TABLE, 


187 


Traverse Table for a Distance = 1. (Concluded.) 



Lat. 

or 

N. S. 

Dep. 

or 

E. W. 



Lat. 

or 

N. S. 

Dep. 

or 

E. W. 



Lat. 

or 

N. S. 

Dep. 

or 

E. W. 


42°0 

.7431 

.6691 

48°0' 

43°0' 

.7314 

.6820 

47°0" 

14°0' 

.7193 

.6947 

46^0' 

2 

.7428 

.6696 

58 

2 

.7310 

.6824 

58 

2 

.7189 

.6951 

58 

4 

.7424 

.6700 

56 

4 

.7306 

.6828 

56 

4 

.7185 

.6955 

56 

6 

.7420 

.6704 

54 

6 

.7302 

.6833 

54 

6 

.7181 

.6959 

54 

8 

.7416 

.6709 

52 

8 

.7298 

.6837 

52 

8 

.7177 

.6963 

52 

10 

.7412 

.6713 

50 

10 

.7294 

.6841 

50 

10 

.7173 

.6967 

50 

12 

.7108 

.6717 

48 

12 

.7290 

.6845 

48 

12 

.7169 

.6972 

48 

14 

.7404 

.6722 

46 

14 

.7286 

.6850 

46 

14 

.7165 

.6976 

46 

16 

.7400 

.6726 

44 

16 

.7282 

.6854 

44 

16 

.7161 

.6980 

44 

18 

.7396 

.6730 

42 

18 

.7278 

.6858 

42 

18 

.7157 

.6984 

42 

20 

.7392 

.6734 

40 

20 

.7274 

.6862 

40 

20 

.7153 

.6988 

40 

22 

.7388 

.6739 

38 

22 

.7270 

.6867 

38 

22 

.7149 

.6992 

38 

24 

.7385 

.6743 

36 

24 

.7266 

.6871 

36 

24 

.7145 

.6997 

36 

26 

.7381 

.6747 

34 

26 

.7262 

.6875 

34 

26 

.7141 

.7001 

34 

28 

.7377 

.6752 

32 

28 

.7258 

.6879 

32 

28 

.7137 

.7005 

32 

30 

.7373 

.6756 

30 

30 

.7254 

.6884 

30 

30 

.7133 

.7009 

30 

32 

.7369 

.6760 

28 

32 

.7250 

.6888 

28 

32 

.7128 

.7013 

28 

34 

.7365 

.6764 

26 

34 

.7246 

.6892 

26 

34 

.7124 

.7017 

26 

36 

.7361 

.6769 

24 

36 

.7242 

.6896 

24 

36 

.7120 

.7021 

24 

38 

.7357 

.6773 

22 

38 

.7238 

.6900 

22 

38 

.7116 

.7026 

22 

40 

.7353 

.6777 

20 

40 

.7234 

.6905 

20 

40 

.7112 

.7030 

20 

42 

.7349 

.6782 

18 

42 

.7230 

.6909 

18 

42 

.7108 

.7034 

18 

44 

.7345 

.6786 

16 

44 

.7226 

.6913 

16 

44 

.7104 

.7038 

16 

46 

.7341 

.6790 

14 

46 

.7222 

.6917 

14 

46 

.7100 

, .7042 

14 

48 

.7337 

.6794 

12 

48 

.7218 

.6921 

12 

48 

,7096 

.7046 

12 

50 

.7333 

.6799 

10 

50 

.7214 

.6926 

10 

50 

.7092 

.7050 

10 

52 

.7329 

.6803 

8 

52 

.7210 

.6930 

8 

52 

.7088 

.7055 

8 

54 

.7325 

.6807 

6 

54 

.7206 

.6934 

6 

54 

.7083 

.7059 

6 

56 

.7321 

.6811 

4 

56 

.7201 

.6938 

4 

56 

.7079 

.7063 

4 

58 

.7318 

.6816 

2 

58 

.7197 

.6942 

2 

58 

.7075 

.7067 

2 

43 °0' 

.7314 

.6820 

47°0’ 

44°0 - 

.7193 

.6947 

46°0' 

45°0' 

.7071 

.7071 

45°0' 


Dep. 

Lat. 



Dep. 

Lat. 



Dep. 

Lat. 



or 

or 



or 

or 



or 

or 



E. W. 

N. S. 



E. W. 

N. S. 



E. W. 

N. S. 



When the angle exceeds 46°, the lats and deps are read upward from the bottom. 

Rem. —Since these lats and deps are for a dist 1, we may proceed as follows for greater dists. 
Thus, let the dist be 856.1. Add together 800 times, 50 times, 6 times, and -j-L time the correspond¬ 
ing lats and deps of the table. 


—What is the lat and dep for 856.1 feet; the angle being 43°? 

Here for 43° we have from the table, lat .7314; dep .6820. 

Hence, .7314 X 800 = 585.12; and .6820 X 800 - 545.60 

.7314 X 50= 36.57; and .6820 X 50= 34.10 

.7314 X 6= 4.39; and .6820 X 6 = 4.09 

.7314 X .1 = .07 ; and .6820 X .1 = .07 

Lat 626.15 Dep 583.86 

These multiplications may be made mentally. Or we may, with a little more trouble, mult the lat 
and dep of the table by the given dist. Thus, 

.7314 X 856.1 = 626.15 lat; and .6820 X 856.1 = 583.86 dep.* 


* Inasmuch as the engineer but rarelyneedsatraver.se table, we have thought it best to give a 
correct one, rather than the common one for * degrees. The first involves more trouble in using it; 
but the last is entirely unfit for o-her than the rude calculations for common surveying with compass 
courses taken to the nearest degree. 

To divide a scale of one mile into feet, first out ofTone-sixth of it; 

then divide the remainder into four equal parts. Each of these parts will be 1100 feet. 


















































188 


THE ENGINEER’S TRANSIT. 


THE ENGINEER’S TRANSIT. 



















































THE ENGINEER’S TRANSIT. 


189 


The details of the transit, like those of the level, are differently arranged by 
diff makers, and to suit particular purposes. We describe it in its modern form, 
as made by Heller and Brightly, of Philada. Without the long; bubble-tube 
F F, Fig i, under the telescope, and the graduated are g , it is their plain 
transit. With these appendages, or rather with a graduated circle in place of 
the arc, it becomes virtually a Complete Theodolite.* 

BDD, Fig 2, is the tripod-head. The screw-threads at® receive the screw 
of a wooden tripod-head-cover when the instrument is out of use. S S A is the 
lower parallel plate. After the transit has been set very nearly over the 
center of a stake, the shifting-plate, dd, cc, enables us, by slightly loosening 
the levelling-screws K, to shift the upper parts horizontally a trifle, and 
thus bring the plumb-bob exactly over the center with less trouble than by the 
older method of pushing one or two of the legs further into the ground, or spread¬ 
ing them more or less. The screws, K, are then tightened, thereby pushing up¬ 
ward the upper parallel plate m mm xx, and with it the half-ball f>, thus 
pressing c c tightly up against the under side of S. The plumb-line passe* 



through the vert hole in b. Screw-caps, /, g , protect the levelling-screws from 
dust &c The feet i, of the screws, work in loose sockets,.;, made flat at bottom, 
to preserve S from being indented. The parts thus far described are generally 
left attached to the legs at all times. Fig 1 shows the method of attachment. 

To set the upper parts upon the parallel plates. I lace the 
lower end of U U in x x, holding the instrument so that the three blocks on m m 
(of which the one shown at F is movable) may enter the three corresponding 

* The price of a first-class plain transit with shifting-plate and plumb-bob, by Heller & Brightly, 
in isse, is $185. One with vert arc g and long bubble-tube F t , 





















































































































































190 


THE ENGINEER’S TRANSIT. 


recesses in a, thus allowing a to bear fully on m, upon which the upper parts 
then rest. (The inner end of the spring-catch, l, in the meantime enters a groove 
around U, just below a, and prevents the upper parts from falling off, if the in- ! 
strument is now carried over the shoulder.) Revolve the upper parts horizontally 
a trifle, in either direction, until they are stopped by the striking of a small lug i 
on a against one of the blocks F. The recesses in a are now clear of the blocks. 
Tighten q, thereby pushing inward the movable block F, which clamps the 
bevelled flange a between it and the two fixed blocks on m m, and confines the j 
spindle U to the fixed parallel plates. It remains so clamped while the instrument 
is being used. 

To remove the upper parts from the parallel plates. Loosen 
q , bring the recesses in a opposite the blocks F. Hold back /, and lift the upper 
parts, which are then held together by the broad head of the screw inserted into 
the foot of the spindle w. 

T T is the outer revolving 1 spindle, cast in one with the support¬ 
ing-plate Z Z, to which is fastened the graduated limb O O. The limb 
extends beyond the coinpass-box, and thus admits of larger graduations than 
would otherwise be obtainable, w w is the inner revolving spindle. At 
its top it has a broad flange, to which is fastened the vernier plate P P. To 
the latter are fastened the compass-box C, one of the bubble-tabes M M 
(the one shown in Fig 2), the dust-box W W, the standards V V, supporting 
the telescope, &c. Each bubble-tube is supported and adjusted by two capstan- 
screws, one at each end. One is shown at r. The bent strip curving over the 
tube protects the glass. 

The clamp-screw, H, presses the split collar, 11, tightly against the fixed spindle, i 
U, but not against Z or T. The set-screws, G G, working in nuts that are cast in 
one with Z, hold between them the tongue, y, which projects from it, and the j 
graduated limb is thus held fast, except that by moving the screws, G, it may be 
made to revolve slightly. 

In Fig 1, the tangent-screw, 6, is seen passing through two towers, in which it 
works. One of the towers is fast to the lower one of the two small pieces at the 
foot of the clamp-screw e, Fig 2. When e is tightened, it draws the two small 
pieces together, confining between them an edge of the graduated limb, which is 
thus made fast to the above-mentioned tower. The other tower is fast to the 
vernier-plate; and the tangent-screw, b, holds the towers at a fixed dist apart. 
The clamping of e thus prevents the vernier-plate from revolving over the gradu¬ 
ated limb, except that it may be moved slightly by turning b, and thus changing 
the dist apart of the towers. In Heller and Brightly’s instruments, the screw, b, 
is provided with means for taking up its “ wear,” or “ lost-motion.” 

There are two verniers. One is shown at p, Fig 1. Both may be read, and 
their mean taken, when great accuracy is required. Ivory reflectors, c, facilitate 
their reading. Before the instrument is moved from one place to another, the 
coin pass-needle.*, Fig 2, should always be pressed up against the glass cover 
of the compass-box by means of the upright milled-head screw seen on the ver¬ 
nier-plate in Fig 1, just to the right of the nearest standard. The pivot-point is 
thus protected from injury. 

R, Fig 1, is a ring with a clamp (the latter not shown) for holding the telescope 
in any required position. It is best to let the eye-end, e, of the telescope revolve 
dotvnward , as otherwise the shade on O, if in use, may fall off. The tangent-screw, 
d, moves a vert arm attached to R, and is thus used for slightly changing the 
elevation of the telescope. In the arm is a slit like that seen in the vernier-arm 
l. When 0° of the vernier is placed at 30° on the arc, g , and the index of the 
opposite arm is placed over a small notch on the horizontal brace (not seen in our 
figs) of the standards, the two slits will be opposite each other, and may be used 
for laying off offsets, Ac, at right-angles to the line of sight. 

One end, R, of the telescope axis rests in a movable box, under which is a screw. 
By means of the screw, the box may be raised or lowered, and the axis thus ad¬ 
justed for very slight derangements of the standards. For E, B, O, and A, see 
Level, p 201. a is a dust-guard for the object-slide. 

Stadia Hairs. Immediately behind the capstan-screw, p, Fig 1, is seen a 
smaller one.. This and a similar one on the opposite side of the telescope, work 
in a ring inside the telescope, and hold the ring in position. Across the ring are 
stretched two additional horizontal hairs, called stadia hairs, placed at such a 
distance apart, vertically, that they will subtend say 10 divisions of a graduated rod 
placed 100 ft from the instrument, 15 divisions at 150 ft, &c. They are thus used for 
measuring hor and sloping distances. 

The long bubble-1 ube. F F, Fig 1, enables ns to use the transit as a level 
although it is not so well adapted as the latter to this purpose. 


THE ENGINEER’S TRANSIT. 


191 


To adjust a plain Transit. 

When either a level or a transit is purchased, it is a good precaution (but one 
which the writer has never seen alluded to) to first screw the object-glass firmly home 
to its place ; and then make a short continuous scratch upon the ring of the glass, and 
upon its slide; so as to be able to see at any time when at work, that the glass is 
always in the same position with regard to the slide. For if, after all the adjustments 
are completed, the position of the glass should become changed, (as it is apt to be if 
unscrewed, and afterward not screwed up to the same precise spot,) the adjustments 
may thereby become materially deranged; especially if the object-glass is eccentric, 
or not truly ground, which is often the case. Such scratches should be prepared bj 
the maker. In making adjustments, as well as when using a transit or level, be 
careful that the eye-glass and object-glass are so drawn out that there shall be ne 
parallax. The eye-glass must first be drawn out so as to obtain perfect distinctness 
of the cross-hairs; it must not be disturbed afterward; but the object-glass must 
be moved for different distances. 

First, to ascertain that the bnbble-tubes, M M, are placet! 
parallel to the vernier-plate, and that therefore when both bubbles are in 
the centers of their tubes the axis of the hist is vert. By means of the four levelling- 
screws, K, bring both bubbles to the centers of their tubes in one position of the 
inst; then turn the upper parts of the inst half-way round. If the bubbles do not 
remain in the center, correct half the error by means of the two capstan-screws 
rr; and the other half by the levelling-screws K. Repeat the trial until both 
bubbles remain in the center while the inst is being turned entirely around on 
its spindle. 

Second, to see that the standards have suffered no derange¬ 
ment ; that is, that they are of equal height and perpendicular to the vernier- 
plate, as they always are when they leave the maker’s hands. Level the inst 
perfectly; then direct the intersection of the hairs to some point of a high object 
(as the top of a steeple) near by; clamp the inst by means of screws H and e, 
and lower the telescope until the intersection strikes some point of a low object. 
(If there is none such drive a stake or chain-pin, Ac, in the line.) Then un¬ 
clamp either H or e, and turn the upper parts of the inst half-way round ; fix the 
intersection again upon the high point; clamp; lower the telescope to the low 
point. If the intersection still strikes the low point, the standards are in order. 
If not. correct one-quarter of the diff (same principle as in Fig 4) by means of the 
adjusting-block and screw at the end, R, of the telescope axis, Fig 1, and repeat 
the trial de novo, resetting the stake or chain-pin at each trial. If the inst has no 
adjusting-block for the axis, it. should be\eturned to the maker for correction of 
any derangement of the standards. 

A transit may be used for running straight lines, even if the standards become 
Slightly bent, by the process described at the end of the fourth adjustment. 

Third, to see that the cross-hairs are truly vert and lior 
when the inst is level. When the telescope inverts, the cross-hairs are 
nearer the eye-end than when it shows objects erect. The maker takes care to place 
the cross-hairs at right-angles to each other in their ring, or diaphragm ; and gene¬ 
rally he so places the ring in the telescope, that when levelled, they shall be vert 
and hor. Sometimes, however, this is neglected; or the ring may by accident be¬ 
come turned a little. To be certain that one hair is vert, (in which case the other 
m ust, by construction, be hor,) after having adjusted the bubble-tubes, level the in¬ 
strument carefully, and take sight with the telescope at a plumb-line, or other vert 

straight edge. If the vert hair coincides with this object, 
it is, so far, in adjustment; but if not, then loosen slightly 
only two adjacent screws of the four, p p i i, Fig 1; and 
with a knife, key, or other small instrument, tap very 
gently against the screw-heads, so as to turn the ring a 
little in the telescope; persevering until the hair be¬ 
comes truly vertical. When this is done, tighten the 
screw r s. In the absence of a plumb-line, or vert straight 
edge, sight the cross-hair at a very small distinct 
point; and see if the hair still cuts that point, when 
the telescope is raised or lowered by revolving it on 
its axis. 

The mode of performing the foregoing w ill be readily 



Fig. 3- 


understood from this Fig, which represents a section across the top part of the tele¬ 
scope, and at the cross-hairs. The hair-ring, or diaphragm, a; vert hair, v; tele¬ 
scope tube, g; ring outside of telescope tube, d; b is one of the four capstan-headed 
screws which hold the hair-ring, a, in its place, and also serve to adjust it. The 
lower ends of these screws work in the tnicltness of the liair-ring; so that when 
they are loosened somewhat, they do not lose their hold on the ring. Small loose 





















192 


THE ENGINEER’S TRANSIT. 


washers, c, are placed under the heads b cf the screws. A space yy is left around 
each screw vv here it passes through the telescope tube, to allow the screws and ring 
together to be moved a little sideways when the screws b are slightly loosened. 

Fourth, to see that the vertical hair is in the line of eolli- 
niation. Plant the tripod firmly upon the ground, as at a. Level the inst; 
clamp it; and direct the vert hair by means of tangent-screws G (figs. 1 and 2) f 
upon some convenient object b\ or il there is none such, drive a thin stake, or a 
chain-pin. Then revolving the telescope vert on its axis, 
observe some object, as c, where the vert hair now strikes; ^ a 

or if there is none, place a second pin. Unclamp the instru- •-* -*0 

ment by the clamp-screw H; and turn the w’hole upper • ^ %1) 

part of' it around until the vert hair again strikes b. Fig 1 . 4, 

Clamp again; and again revolve the telescope vert on its \ 

axis. If the vert hair now strikes c, as it did before, it shows that c is really 
at o ; and that b, a, c, are in the same straight line; and therefore this adjustment 
is in order. If not, observe where it does strike, say at m, (thedist a m being 
taken equal to a c,) and place a pin there also. Measure m c; and place a pm , 
at v, in the line rn c, making m v — one-fourth of m c. Also put a pin at o, half- . 
wav between rn and c, or in range with a and b. By means ol the two lior 
screws that move the ring carrying the cross-hairs, adjust the vert hair until it 
cuts v. Now repeat the entire operation; and persevere until the telescope, after 
being directed to b, shall strike the same object o, both times, when revolved on 
its axis. See whether the movement of the ring in this 4th adjustment has dis¬ 
turbed the verticality of the hair. If it has. repeat the 3d adjustment. Then re¬ 
peat the 4th, if necessary; and so on until both adjustments are found to be right 
at the same time. Thus a straight line may be run, even if the hairs are out of 
adjustment; but with somewhat more trouble. For at each station, as at a, two 
back-sights, and two fore-sights, a c and a rn, may be taken, as when making the 
adjustment; and the point o, half-way between cand in, will be in the straight line. 
The inst may then be moved to o, and the two back-sights be taken to a ; and so on. 

Angles measured by the transit, whether vert or hor, will evidently not be 
affected by the hairs being out of adjustment, provided either that the vert 
hair is truly vert, or that we use the intersection of the hairs when measuring. 

The foregoing are all the adjustments needed, unless the tran¬ 
sit is required for levelling, in which case the following one must be attended to: 



To adjust the Iona: bubble, tube, F F. Fig l, we first place the line 

of sight of the telescope hor, and then make the bubble-tube hor, so that the 
two are parallel. Drive two pegs, a. and b Fig. 5, with their tops at precisely 
the same level (see Rem. p. 193 ) and at least about 100 ft. apart; 300 or more 11 
will be better. Plant the inst firmly, in range w T ith them, as at c, making b c 1 
an aliquot part of a b. and as short as will permit focusing on a rod at b. The 
inst need not be leveled. .Suppose the line of sight to cut e and d Take the J ' 
readings b e and a d. Their diff is b e — a d = a n — a d = d n; and a b : a c : : !' 
d n: ds) s being the height of the target at a wdien the readings (a s, b o) on the 

two stakes are equal. as = ad + ds=ad + —• If the reading on a 

exceeds that on b (as when the line of sight is vfg) the diff of readings is=a^ — 

b f—a g — a i = g i\ and a s=ag — g s —ag — ^ c « Sight to s, bring the 

bubble to the cen of its tube by means of the two small nuts n n at one end of the 
tube, b ig. 1 , and assume that the telescope and tube are parallel.* The zeros of 


* rhis n(‘gleets a small error due to the curvature of the earth; for a hor line at v is v h tan¬ 
gential to the curved (or “ level ') surface of still water at v , whereas v s is tangential to water surf 
at, a point midway between a and h. Hence if the telescope at v points to s it will not be parallel to 

^o e 'raise l ^ e ;^::; 1° f ° F ^ for the air. which dimililkX 

error, raise the target on a to a point h above s. h s = .000000020.) x square of a c in ft • but wnen 

cross-UMr4n the tele^oS?* 11 ““ *** ° f “ iU ° h a “ d barely 00vers the “Pl-areut thickness of the 

























THE ENGINEER’S TRANSIT. 


193 



the \eit circle, and of its vernier, may now be adjusted, if they require it, 
by loosening the vernier screws and then moving the vernier until the two coin¬ 
cide. 

Item. If no level is at hand for levelling the two pegs a and b, it may be done 
b\ the transit itselt, thus: Carefully level the two short bubbles, by means of the 
h'Ve I ling-screws K. Drive a peg m, from 100 to .300 feet from the instrument o. 
llieu placing a target-rod on m, clamp the target tight at whatever height, as ev, 

the hor hair happens to cut it; it being of no im¬ 
portance whether the telescope is level or not; 
although it might as well be as nearly so as can 
conveniently be guessed at. Clamp the telescope 
0 ~ ^ in its position by the clamp-ring It, Fig 1. Re¬ 
pair g volve the inst a considerable way round: say 

. , * nearly or quite half wav. Place another peg ??, 

atprecmh/ the same (list from the instrument that m is; and continue to drive it un- 
1 til the hor hair cuts the target placed on it, and still kept clamped to the rod, at the 
same height as when it was on m. W hen this is done, the tops of the two pegs are 
on a level with each other, and are ready to be used as before directed. 

When a transit is intended to be used for surveying farms, &c, or for retracing 
lines of old surveys, it is very useful to set the compass so as to allow for the “va¬ 
riation” during the interval between the two surveys. For this purpose a 
“ variation-vernier ” is added to such transits; and also to the compass. 

When the graduations of a transit are figured, or numbered, so as to read both 

10 0 10 

ways from zero, thus, i I I I u i i I i i i i I i i i i 1 i i i i I n i the vernier also is made 


doublet that is. it also is graduated and numbered from its zero both ways. In this 
case, if the angle is measured from zero toward the right hand, the reading must be 
made from the right hand half of the vernier; and vice versa. If the figuring is 
single, or only in one direction, from zero to 360°, then only the single vernier is 
necessary, as the angles are then measured only in the direction that the figuring 
Counts. Engineers differ in their preferences for various manners of figuring the 
grt> illations. The writer prefers from zero each way to 180°, with two double ver¬ 
niers. 

To rejplace cross-Is airs in a level, or transit. Take out the tube 
from the eye end of the telescope. Looking in, notice which side of the cross¬ 
hair diaphragm is turned toward the eye end. Then loosen the four screws which 
hold the diaphragm, so as to let the latter fall out of the telescope. Fasten on new 
hairs with beeswax, varnish, glue, or gum-arabic water, &c. This requires care. 
Then, to return the diaphragm to its place, press firmly into one of the screw-holes 
on the circumf of the diaphragm itself, the end of a piece of stick, long enough to 
reach easily into the telescope as far as to where the diaphragm belongs. By this 
stick, as a handle, insert the diaphragm edgewise to its place in the telescope, and hold 
it there until two opposite screws are put in place and screwed. Then draw thestick 
out of the hole in the diaphragm ; and with it turn the diaphragm until the same 
side presents itself toward the eye end as before; then put in the other two screws. 

The so-called cross-hairs are actually spider-web, so fine as to be barely visible to 
the naked eye. Heller & Brightly use very fine platina wire, which is much better. 
Human hair is entirely too coarse. 

To rejslaee a spirit-level, or ImMsIe-gdass. Detach the level from 
the instrument; draw off its sliding ends; push out the broken glass vial, and the 
cement which held it; insert the new one, with the proper side up (the upper side 
is always marked with a file by the maker); wrapping some paper around its ends, 
.if it fits loosely. Finally, put a little putty, or melted beeswax over the ends of the 
vial, to secure it against moving in its tube. 

In purchasing instruments, especially when they are to be used far from a maker, 
it is advisable to provide extras of such parts as may be easily broken or lost; siich 
as glass compass-covers, and needles; adjusting pins; level vials; magnifiers, &c. 


Theodolite adjustments are performed like those of the level and transit. 
1st. That of the cross-hairs; the same as in the level. 

2d. The long bubble-tube of the telescope; also as in the level. 

2d. The two short bubble-tubes: as in the transit. 

4th The vernier of the vert limb; as in the transit with a vert circle. ‘ 

5th. To see that the vert hair travels vertically; as in the fourth adjustment 
of the transit, in some theodolites, no adjustment is provided for this; but in 
large ones it is provided for by screws under the feet of the standards. 
Sometimes a second telescope is added ; it is placed below the hor limb, and is 

13 









194 


THE BOX OR POCKET SEXTANT. 


called a watcher. It has its own clamp, and tangent-scrcvv. Its use is to ascertain 
whether the zero of that limb has moved during the measurement of hor angles. 
When previously to beginning the measurement, the zero and upper telescope are 
directed toward the first object, point the lower telescope to any small distant 
obiect and then clamp it. During the subsequent measurement, look through it, 
from time to Se, to be sure that it still strikes that object; thus proving that no 
slipping has occurred. 


1 


THE BOX OR POCKET SEXTANT. 




The portability of the pocket sextant, and the fact that it reads to single minutes, 
render it at times very useful to the engineer.* by it, angles can be measured " hue 
in a boat, or on horseback: and in many situations which preclude the use of a 
transit. It is useful for obtaining latitudes, by aid of an artificial horizon. ” hen 
closed, it resembles a cylindrical brass box, about 3 inches in diameter, and \ /2 
inches deep. This box is in two parts; 


by unscrewing which, then inverting 
one part, and then screwing them to¬ 
gether again, the lower part becomes a 
handle for holding the instrument. 

Looking down upon its top when thus 
arranged, we see, as in this figure, a 
movable arm 1 C, called the index, 
which turns on a center at C, and car¬ 
ries the vernier V at its other end. G 
G is the graduated arc or limb. It 
actually subtends about 73°, but is di¬ 
vided into about 146°. Its zero is at 
one end. Its graduations are not shown 
in the Fig. 

Attached to the index is a small mov¬ 
able lens, (not shown in the figure,) 
likewise revolving around C, for read¬ 
ing the fine divisions of the limb. When 
measuring an angle, the index is moved 
by turning the milled-liead P of a 
pinion, which works in a rack placed within the box. 
cular hole at the side of the box, near A. A 



The eye is applied to a cir- 
small telescope, about 3 inches long, 
accompanies the instrument; but may generally be dispensed with. When .so, the 
eye hole at A should be partially closed by a slide which has a very small eve-hole 
in it; and which is moved by the pin h, moving in the curved slot. Another slide, 
at the side of the box, carries a dark glass for covering the eye-hole when observing 
the sun. When the telescope is used, it is fastened on by the milled-liead screw T. 
The top part shown in our figure, can be separated from the cylindrical part, by 
removing 3 or 4 small screws around its edge; and the interior can then be exam¬ 
ined, and cleaned if necessary. Like nautical, and other sextants, this one has 
t wo principal glasses, both of them mirrors. One, the index-glass, is attached 
to the underside of the index, at C; its upper edge being indicated by the 
two dotted lines. The other, the liorizou-glttss, (because, when meas¬ 
uring the vert angles of celestial bodies, it is directed toward the horizon,) is also 
within the box; the position of its upper edge being shown by the dotted lines at 
R. The horizon-glass is silvered only half-way down; so that one of the observed 
objects may be seen directly through its lower half, while the image of the other 
object is seen in the upper half, reflected from the index-glass. That the instrument 
may be in adjustment, ready for use, these two glasses must be at right angles to tha 
plane of the instrument; that is, to the under side of the top of the box, to which they 
are attached ; and must also be parallel to each other, when the zeros of the vernier 
and of the limb coincide. The index-glass is already permanently fixed l>y the 
maker, and requires no other adjustment. Rut the horizon-glass has two adjust¬ 
ments, Which are made by a key like that of a watch, and having a milled-head K. 
It is screwed into the top of the box, so as to be always at hand for use. When 
needed, it is unscrewed. This key fits upon two small square-heads, (like that for 


* Price, with telescope, about $50. Made by Stackpole & Bro., 41 Fulton St., New York. 
























THE COMPASS. 


195 


winding a watch;) one of which is shown at S; while the other is near it, but on the 
side ot the box. These squares are the heads of two small screws. If the 
horizon glass II should, as in this sketch, (where it is shown endwise,) not beat 
right angles to the top U J of the box, it is brought right by turning the square- 
head S oi the screw S T; and if, after being so far rectified, it still is not parallel to 

the index-glass when the zeros coincide, it is moved 
a little backward or forward by the square head 
at the side. 

To « cl j list a box sextant, bring the two 
zeros to coincide precisely; then look through the 
eye-hole, and the lower or unsilvered part of the 
horizon-glass, at some distant object. If the instru¬ 
ment is in adjustment, the object thus seen directly, 
will coincide precisely with its reflected image, 
seen at the same time, at the same spot. But if it 
is not in adjustment, the tw o will appear separated 
either hor or vert, or both, thus, * *; in which case 
apply the key K to the square-head S ; and by turning it slightly in w hichever direc¬ 
tion may be necessary, still looking at the object and its image, bring the two into a hor 
position, or on a level with each other, thus, * *. Then apply the key to the square- 
head in the side of the box; and by turning it slightly, bring the two to coincide 
perfectly The instrument is then adjusted. 

Jn some instruments, the hor glass has a hinge at v, to allow it play while being 
; adjusted by the single screw ST; but others dispense with this hinge, and use two 
screws like S on top of the box, in addition to the one in the side. 

if a sextant is used for measuring vert angles by means of an tirtiliciai 
horizon, the actual altitude will be but one-half of that read otf on the 
limb; because we then read at once both the actual and the reflected angle. The 
great objection to the sextant for engineering purposes, is that it does not measure 
angles horizontally, as the transit does; unless wdien the observer, and the two ob¬ 
jects happen to be in the same hor plane. 
Thus an observer with a sextant at A, if 
measuring the angle subtended by the 
mountain-peaks B and C, must hold the 
graduated plane of the sextant in the 
plane of A B C; and must actually meas¬ 
ure the angle BAC; whereas what he 
wants is the hor angle n A m. This is 
greater than BAC, because the dists An 
and A m are shorter than A B and A C. 
The transit gives the hor angle n A in, be¬ 
cause its graduated plane is first fixed hor by the levelling-screws: and the subse¬ 
quent measurement of the angle is not affected by his directing merely the line of 
sight upward, to any extent, in order to fix it upon B and C. For more on this sub¬ 
ject; and for a method of partially obviating this objection to the sextant, see the 
note to Example 2, Case 4, of “ Trigonometry,” page 113. 

The nautical sextant, used on ships, is constructed on the same principle 
as the box sextant; and its adjustments are very similar. In it, also, the index- 
glass is permanently fixed by the maker; and the horizon-glass has the two adjust¬ 
ments of the box sextant. It also has its dark glasses for looking at the sun; and 
a small sight-hole, to be used when the telescope is dispensed with. 






THE COMPASS. 


To art just a Compass. 

The first adjustment is that, of the bubbles. Plant firmly ; and level the 
instrument, in any position ; that is, bring the bubbles to the centers of their tubes. 
Then turn the instrument half-way round. If the bubbles then remain at the ceil- 
ters they are in adjustment; but if not, correct one-half the diff in each bubble, 
by means of the adjusting-screws of the tubes. Level the instrument again; turn 
it half round; and if the bubbles still do not remain at the center, the adjusting- 
screws must be again moved a little, so as to rectify half the remaining diff. Gener- 










196 


THE COMPASS. 


ally several trials must be thus made, until the bubbles will remain at the center, 
while the compass is being turned entirely around. 

Second adjustment. Level the compass, and then see that the needle is 
hor; and if not, make it so by means of the small piece of wire which is wrapped 
around it; sliding the wire toward the high end. A needle thus horizontally ad¬ 
justed at one place, will not remain so if removed far north or south trom that place. 

If carried to the north, the north end will dip down; and if to the south, the south 
end will do so. The sliding wire is intended to counteract this. 

Tliird adjustment. This is always fixed right at first by the maker; that 
is, the sights, or slits for sighting through, are placed at right angles to the compass 
plate; so that when the latter is levelled by the bubbles, the sights 
are vert. To test whether they are so, hang up a plumb-line; and 
having levelled the compass, take sight at the line, and see if the 
slits coincide with it. If one or both slits should prove to be 
out of plumb, as shown to an exaggerated extent in this sketch, 
it should be unscrewed from the compass, and a portion of its foot 
on the high side be filed or ground otf, as per the dotted line; or 
as a temporary expedient, a small wedge may be placed under the ^ 
low side, so as to raise it. 

Fourill adjustment, to straighten the needle, if it should become bent. 
The compass beiiig levelled, and the needle hor, and loose on its pivot, see whether 
its two ends continue to point to exactly opposite graduations, (that is, graduations 
180° apart;) while the compass is turned completely around. If it does, the needle 
is straight; and its pin is in the center of the graduated circle ; but if it does not, 
then one or both of these require adjusting. First level the compass. Then turn it 
until some graduation (say 90°) comes precisely to the north end of the needle. If 
the south end does not then point precisely to the opposite 90° division, lilt off the 
needle, and bend the pivot-point until it does; remembering that every time said 
point is bent, the compass must be turned a hairsbreadth so as to keep the north end 
of the needle at its 90° mark. Then turn the compass half-way round, or until the 
opposite 90° mark comes precisely to the north end of the needle. Make a fine pen¬ 
cil mark where the south end of the needle now points. Then take off the needle, 
and bend it until its south end points half-way between its 90° mark and the pencil 
mark, while its north end is kept at 90° by moving the compass round a hairsbreadth. 
The needle will then be straight, and must not be altered in making the following i 
adjustment, although it will not yet cut opposite degrees. 

Fii'tli adjustment, of the pivot-pin. After being certain that the needle is 
straight, turn the compass around until a part is arrived at where the two ends of the 
needle happen to cut opposite degrees. Then turn the compass quarter way around, 
or through 90°. If the needle tlieu cuts opposite degrees, the pivot-point is already 
in adjustment; but if the needle does not so cut, bend the pivot-point, until it does. 
Repeat, if necessary, until the needle cuts opposite degrees whilebeing turned entirely 
around. 

Care and nicety of observation are necessary in making these adjustments properly ; 
because the entire error to be rectified is, in itself,a minute quantity; and the novice 
is very apt to increase Ins trouble by not knowing how to use bis iiiuk'hi 
when looking at the endof the needle and the corresponding graduations. The mag¬ 
nifier must always be held with its center directly over the point to be examined: and 
it must be held parallel to the graduated circle. Otherwise annoying errors of 
several minutes will be made in a single observation; and the accumulation of two 
or three such errors, arising from a cause unknown to him, may compel him to 
abandon the adjustments in despair. This suggestion applies also to the reading of 
angles taken by the transit, &c ; although the errors are not then likely to be so 1 
great as in the case of the compass. In purchasing a magnifier for a compass, see , 
that no part of it, as hinges, or rivets, are made of iron; for such would change the 1 
direction of the needle. 

If the sight-slits of a compass are not fixed by the maker in line with the two 
opposite zeros, the engineer cannot remedy the defect. This can be ascertained by 
passing a piece of fine thread through the slits, and observing whether it stands 
precisely over the zeros. 



'Variation ot the Coinpass.* 

The numerous disturbing influences to which the compass is subject, render its 

* For full information on this subject see that useful little book “ Magnetic Variation iu the U S. 
bv J B Stone, C E, 187S. It is iuvaluable in retracing old lines. 











CONTOUR LINES. 


197 


indications of bearings or courses very unreliable. The daily variation itself some¬ 
times amounts to \/^ of a degree; and always to at least several minutes. It is almost 
incessantly changing the direction of the needle, to one side or the other, at the rate 
of 1 or 2 minutes per hour, especially in summer. Local attraction, from iron in the 
soil, ferruginous gravel, trap rocks, Ac, is another source of inaccuracy; as are also 
the annual and the secular variations. Electricity, either atmospheric, or excited 
bv rubbing the glass cover, sometimes gives trouble. It may be removed by touch¬ 
ing the glass with the moist tongue, or finger. It is plain that none of these causes 
(except the last) will affect the measurement of angles by the compass. 

Ib» 1885 the line of no variation enters the U. S. near the N end of 
Lake Superior; passes through the Straits of Mackinaw ; 40 m W of Detroit. Mich ; 
50 m E of Columbus, 0; and reaches the Atlantic about half way between Charles¬ 
ton, S C, and Wilmington, N C. A compass placed anywhere in the vicinity of that 
line, will point nearly due north and south. To the eastward of this line, the varia¬ 
tion is westward; and vice versa; becoming greater, the farther the place is from 
I the line; until in some parts of Maine and along the Pacific coast it is as great as 
18° to 21 °. This line is moving westward, at an average rate of about 3 or 4 min¬ 
utes per year. 

The needle, if of soft metal, sometimes loses part of its magnetism, and consequently 
does not work well. It may be restored by simply drawing the north pole of a 
| common magnet (either straight or horseshoe; about a dozen times, from the center 
to the end of the south half of the needle: and the south pole, in the same way, along 
the north half; pressing the magnet gently upon the needle. After each stroke, 
remove the magnet several inches from the needle, while bringing it back to the 
center for making another stroke. Each half of the needle in turn, while being thus 
operated on, should be held fiat upon a smooth hard surface. Sluggish action of the 
needle is, however, more generally produced by the dulling or other injury of the 
point of the pivot. Demagnetizing will throw the needle out of balance; which must 
be counteracted by the sliding wire. 

In or«ler to prevent mistakes by reading- sometimes from one end, 
and sometimes from the other end of ilie needle, it is nest t<> always point the N of 
the compass-box toward the object whose bearing is to betaken; and to read off 
from the north end of the needle. This is also more accurate. 


CONTOUR, LINES. 


1 A contour line is a curved hor one, every point in which represents the same level; 
thus each of the contour lines 88c, 01c, 94c, &c, Fig 1, indicates that every point in 
the ground through which it is traced is at the same level; and that that level or 
height is everywhere 88, ill, or 94 ft above a certain other level or height called 
datum: to which all others are referred. 

Frequently the level of the starting point of a survey is taken as being 0, or zero, 
or datum; and if we are sure of meeting with no points lower than it, this answers 
every purpose. But if there is a probability ot many lower points, it is better to 
assume the starting point to be so far above a certain supposed datum, that none of 
these lower points shall become minus quantities, or below said supposed datum or 
zero. The only object in this is to avoid the liability to error w hich arises when 
some of the levels'are +, or plus ; and some —, or minus. Hence we may assume 
the level of the starting point to be 10, 100, 1000, &c, ft above datum, according to 
circumstances. 

The vert dists between each two contour lines are supposed to be equal; and in 
railroad surveys through well-known districts, where the engineer knows that his 
actual line of survey will not require to be much changed, the dist may bo 1 or 2 it 
only ; and the lines need not be laid down for widths greater than 100 or 200 ft on 
each fide of his center-stakes. But in regions of which the topography is compara¬ 
tively unknown; and where consequently unexpected obstacles may occur which 
require the line to be materially changed for a considerable dist back, the observa¬ 
tions should extend to greater widths; and for expedition the vertical dists apart 
may bs increased to 3, 5, or even 10 ft, depending on the character of the country, 
Ac. Also, when a survey is made for a topographical map ot a State, or ot a county, 
vert dists of 5 or 10 ft will generally suffice. 

Let the line A B, Fig 1, starting from 0, represent three stations (h 1, S 2. S 3,) ot 
the center line of a railroad survey; and let the numbers 100, 103, 101, 104, along 
that line denote the heig’hts at the stakes above datum, as determined by levelling. 
Then the use of the contour lines is to show in the ottic » what would bo the effect 
of changing the surveyed center line A B, by moving any part ot it jojhcjfight_or^ 











108 


CONTOUR LINES. 




left hand * Thus, if it should be moved 100 ft to the left, the starting point 0 would 
be on ground about 6 ft higher than at present; inasmuch as its level would then 
be about 106 ft above datum, instead of 100. Station 1 would be about 7 ft higher 
or 110 ft instead of 10:5 Station 2 would be about 7 ft higher, or 10s It instead of 
101. If the line be thrown to the right, it will plainly be on lower ground. 

The field observations for contour lines are som dimes made with the spirit-level: 
but more frequently by a slope-man, with a straight 12-ft graduated rod, anti a slope 
instrument, or clinometer. At each station he lays his rod upon the ground, as 



B 


Fig. 1. 


nearly at right angles to the center line A B as he can*judge by eye; and placing 
the slope instrument upon it, he takes the angle of the slope of the ground to the 
nearest ^ of a degree. lie also observes how tar beyond the rod the slope continues 
the same ; and with the rod he measures the dist. Then laying down the rod at that 
point also, he takes the next slope, and measures its length ; and so on as far as may 
be judged necessary. His notes are entered in his field book as shown in Fig 2; the 
angles of the slopes being written above the lines, and their lengths below; and 
should be accompanied by such remarks as the locality suggests; such as woods, 
rocks, marsh, sand, field, garden, across small run, &c, &c. 


* In thus using the words right and left we are supposed to have our backs turned to the starting 

point of the survey. In a river, the right bank or shore is that which 


is on the right hand as we riesreml it, that is, in speaking of its right or left 
bank, we are supposed to have our backs turned towards its head, or origin ; and so with a survey. 




























CONTOUR L^TES. 


199 



It is not absolutely necessary to represent the slopes roughly in the field-book, as 
in Fig 2; for by using the sign + to signify “up;” —“down;” and = “level,” 
the slopes may be writ¬ 
ten in a straight line, 
as in Fig 

The notes having been 
taken, the preparation 
of the contour lines by 
means of them, is of 
course office-work ; and 
is usually done at the 
same time as the draw¬ 
ing of the map, &c. The 
field observations at each 
station are then sepa¬ 
rately drawn by protrac¬ 
tor and scale, as shown 
in Fig 3 for the starting 
point 0. The scale should not be less than about yy inch to a ft, if anything like 
accuracy is aimed at. Suppose that at said station the slopes to the right, taken in 
their order, are, as in F’ig 2, 15°, 4°, and 26°; and those to the left, 20°, 10°, and 10°; 
and their lengths as in the same Fig. Draw a hor line ho, F'ig 3; and consider the 
center of it to be the station-stake. From this point as a center, lay off these angles 
with a protractor, as shown on the arcs in Fig 3. Then beginning say on the right 
hand, with a parallel ruler draw the first dist ac, at its proper slope of 15°; and of 
its proper length, 45 ft, by scale. Then the same with c.y and y t. Do the same with 
those on the left hand. We then have a cross-section of the ground at Sta O. Then 
on the map, as in F’ig 1, draw a line as m n, or h w, at right angles to the line of road, 
and passing through the station-stake. On this line lay down the /mr dists a d, d s,sv, 
a e, eg, g k, marking them with a small star, as is dune and lettered in Fig 1, at Sta 0. 

When extreme accuracy is pretended to, these hor dists must be found by measure 
on Fig 3; but as a general rule it will be near enough, when the slopes do not ex¬ 
ceed 10°, to assume them to be the same as the sloping dists measured in the field. 
Next ascertain how high each of the points cyt Ini is above datum. Thus, measure 
by scale the vert dist dc. Suppose it is found to be 5 ft; or in other words, that c 
is 5 ft below station-stake 0. Then since the level at stake 0 is 100 ft above datum, 
that at c must be 5 ft less, or 100 — 5 = 95 ft above datum ; which may be marked in 
light lead-pencil figures on the map, as at d , F’ig 1. Next for the point y, suppose 
we find sy to be 11 ft, or y to be 11 ft below stake 0; then its height above datum 
must be 100 — 11 = 89: which also write in pencil, as at s. Proceed in the same 
way with t. Next going to the left hand of the station-stake, we find el to be say 
2 ft; but l is above the level of the station-stake, therefore its height above datum is 



100 + 2 == 102 ft, as figured at e on the map. Let ng bo 5 ft; then is v, 100 + 5 — 
105 ft above datum, as marked at g’, and so on at each station. Y\ hen this has been 
done at several stations, we may draw in the contour lines of that portion by hand 
thus: Suppose they are to represent vert heights of 3 It. Beginning at Station O 
(of which the height above datum is 100 ft) to lay down a contour lino 103 ft above 
datum, we see at once that the height of 103 tt must be at t, or at % the dist from e 
to g. Make a light lead-pencil dot at t ; and then go to the next Station 1. Here 
we see that the height of 103 ft coincides with the station-stake itself; place a dot 
there, and go to Sta 2. The level at this stake is 101; therefore the contour for 103 











200 


CON'JOUR LINES. 


ft must evidently he 2 ft higher, or at i, % of the dist from Sta 2 to +104; therefore 
make a dot at i. Then go to Sta 3. Here the level being 104 above datum, the con¬ 
tour of 103 must be at y, or \ of the dist from Sta 3 to +99; put a dot at y. finally 
draw by hand a curving line through t, SI, i, and ?/; and the contour line ol It 
is done. All the others are prepared in the same way, one by one. The level ot each 
must be figured upon it at short intervals along the map, as at 103 c, 100c Ac. 

Or, instead of first placing the + points on the map, to denote the slope diets actu¬ 
al I v measured upon the ground, we may at once, and with less trouble, find and snow 
those only which represent the points t, S 1, t, y, &c, of the contours themselves. 
Thus, say that at any given station-stake, Fig 4, the level is 104; that the cross-sec¬ 
tion c s of the ground has been prepared as before ; and that we want the lior utsts 
from the stake, to contour lines for 94, 97,100 It, &c, 3 It apart-vert. 



Draw avert line v 1, through the station-stake, and on it by scale mark levels of 
94, 97,100, &c. tt. This is readily done, inasmuch its we have the level 104 of the 
stake already given. Through these levels draw the hor lines a, b, m, n, &c. to the 
ground-slopes. Then these lines, measured by the scale, plainly give the required 
dists. 

When the ground is very irregular transversely, the cross-sections must be taken 
in the field nearer together than 100 ft. The preparation of contour lines will be 
greatly facilitated by the use of paper ruled into small squares of uot less than about 

inch to a side, for drawing the cross-sections upon. 

’When the ground is very steep, it is usual to shade such portions of the map to 
represent hill-side. The closer together the contours come, the steeper of course is 
the ground between them; and the shading should he proportionally darker at such 
portions. But for working maps it is best to omit the shading. 

[n surveys of wide districts, the transit instrument with a graduated vertical cir¬ 
cle or arc, < 7 , p. 188, is used for measuring the angles of slope,instead of the common 
slope-instrument.* 


* The prepariug of contour lines is a slow and tedious office-work ; and the writer considers tin ni 
of hut little value in many cases; as when they are takeu lor ouly about 100 ft on each side of the 
line, with reference to slight changes of direction. He conceives that, ordinarily, every useful purpo: e 
is fulfilled if the leveller or the topographer enters into his field-book at each station, notes similar 


to the following: 

Sta 60.— 3. 1 R. + 2. 1 L. 

61.+2. 2R. — 1.3L. 

62............ — 1. R. -+ 4- 2 L. 

63.“ 


AVhich means that at station 60, the slope of the ground on the right, as nearly as he can judge by 
eye, or by liis hand-level, is about 3 ft downward, for 1 chain, or 100 ft; and on the lett, about 2 It 
upward in 1 chain. At 61, 2 ft. up, in 2 chains to the right; and 1 tt down iu 3 chains to the left. 
At 62, level for 1 chain to the right; and ascending 4 ft in 2 chains to the left. At 03, the same as at 
62. At some spots it will be well to add a sketch of a cross-section, like Fig 2 ; only, instead of the 
angles, use ft of rise or fall, to indicate the slopes, as judged by eye, or by a hand-level. Bv this 
method, the result at every station will be somewhat in error; but these small errors will balance 
each other so nearly that "the total may be regarded as sufficiently correct for all the purposes of a 
preliminary estimate of the cost of a road. ^1 hen the final stakes for guidingthe workmen are placed, 
the slopes should be carefully taken, in order to calculate the quantity of excavation accurately for 
payment. 

















THE LEVEL. 


201 


THE LEVEL. 


Although the levels of different makers vary somewhat in their details, still their 
principal parts will be undei’stood from the following figure.* The telescope T T 
rests upon two supports YY, called Ys; out of which it can be lifted, first removing 
the pins s s which confine the semicircular clips c c, and then opening the clips. 
The pins should be tied to the Ys, by pieces of string, to prevent their being lost. 
The slide of the object-glass O, is moved backward or forward by a rack and pinion, 
by means of the milled head A. The slide of the eye-glass E, is moved in the same 
way by the milled head e. A cylindrical tube of brass, called a shade , is usually 
furnished with each level. It is intended to be slid on to the object-end O of the 
telescope, to prevent the glare of the sun upon the object-glass, when the sun is 
low. At B is an outer ring encircling the telescope, and carrying 4 small capstan- 
beaded screws; two of which, pp, are at top and bottom; while the other two, 
of which i is one, are at the sides, and at right angles top p. Inside of this outer 
ring is another, inside of the telescope, and which has stretched across it two 
spider-webs, usually called the cross-hairs. These are much finer than they ap¬ 
pear to be, being considerably magnified. They are at right angles to each other; 
and, in levelling, one is kept vert, and the other hor. They are liable at times to be 



thrown out of this position by a partial revolution of the telescope, when carrying 
the level, or when setting the tripod down suddenly upon the ground; but since, in 
levelling, the intersection of the hairs is directed to the target-rod, this derangement 
does not affect the accuracy of the work. Still it is well to keep them nearly vert 
and hor, by keeping the bubble-tube D D as nearly directly over the bar V ¥ as can 
be judged by eye. This enables the leveller to see that the rod-man holds his rod 
nearly vert, which is absolutely essential for correct levelling. If perfect verticality 
is desired, as is sometimes the case, when staking out work, it may be obtained (if 
the instrument is in perfect adjustment , and levelled) by sighting at a plumb-line, or 
other vert object, and then turning the telescope a little in its Ys, so as to bring the 
hair to correspond. When this is done, a short continuous scratch may be made on 
the telescope and Y, to save that trou; le in future. Heller & Brightly, however, 
provide their levels with a small projection inside of the Ys, and a corresponding 
stop on the telescope, the contact of which insures the verticality of the hair. 
Should the hairs be broken by accident, they may be replaced as directed here¬ 
after. 

The small holes around the heads of the 4 small capstan-screws p, tjust reterred to, 
are for admitting the end of a small steel pin, or lever, for turning them. If first 
the upper screw p be loosened, and then the lower one tightened, the interior ring 
will be lowered, and the horizontal hair with it. But on looking throusrh the tele- 


* The price of a first-class level, by Heller & Brightly, is $145. It is 

bad economy to buy inferior instruments. 


































202 


THE LEVEL. 


scope they will appear to be raised. If first the lower one be loosened, and tbe upper 
one tightened, the hor hair will he actually raised, but apparently lowered. This is 
because the glasses in the eye-piece E reverse the apparent position of objects inside 
of the telescope; which effect is obviated, as regards exterior objects, by means of 
the object-glass 0. This must be remembered when adjusting the cross-hairs ; for if a 
hair appears to strike too high, it must be raised still higher; if it appears to tie 
already too far to the right or left, it must be actually moved still more in the same 
direction. 

This remark, however, does not apply to telescopes which make objects appear 
inverted. 

There is no danger of injuring the hairs by these motions, inasmuch as the four 
screws act against the ring only, and do not come in contact with the hairs them¬ 
selves. 

Under the telescope is the bubble-tubf. D D One end of this tube can be raised or 
lowered slightly by means of the two capstan-headed nuts n n, one of which must 
be loosened before the other is tightened. On top of the bubble-tube are scratches 
for showing when the bubble is central in the tube. Frequently these scratches, or 
marks, are made on a strip of brass placed above the tube, as in our fig. There are 
several of them, to allow lor the lengthening or shortening of the bubble by changes 
of temperatuie. At the other end of the bubble-tube are two small capstan-screws, 
placed on opposite sides horizontally. The circular head of one of them is shown 
near t. lly means of these two screws, that end of the tube can be slightly moved 
hor, or to right or left. Under the bubble-tube is the bar V F; at one end of which, 
as at V,are two large capstan-nuts tv iv, which operate upon a stout interior screw 
which forms a prolongation of the Y. The holes in these nuts are larger than the 
others, as they require a larger lever for turning them. If the lower nut is loosened 
and the upper one tightened, the Y above is raised; and that end of the telescope 
becomes farther removed from the bar; and vice versa. Some makers placea similar 
screw and nuts under both Ys; while others dispense with the nuts entirely, and 
substitute beneath one end of the bar a large circular milled head, to be turned by 
the fingers. This, however, is exposed to accidental alteration, which should be 
avoided. 

When the portions above m are put upon m, and fastened by the screw Y, all 
the upper part may be swung round hor, in either direction, by loosening the 
clamp-screw H; or such motion may be prevented by tightening that screw. 
It frequently happens, after the telescope has been sighted very nearly upon an 
object, and then clamped by H, that w T e wish to bring the cross-hairs to coincide 
more precisely with the object than we can readily do by turning the telescope by 
hand; and in this case we use the tangent-serew b, by means of which a 
slight but steady motion may be given after the instrument is clamped. For 
fuller remarks on the clamp and tangent-screws, see “Transit.” 

The parallel plates m and S are operated by four levelling-screws ; 
three of which are seen in the figure, at K K. The screws work in sockets R; 
which, as well as the screws, extend above the upper plate. When the instrument 
is placed on the ground for levelling, it is well to set it so that the lower parallel 
plate S shall be as nearly horizontal as can be roughly judged by eye; in order 
to avoid much turning of the levelling screws K K iri making the* upper plate 
m hor. The lower plate S, and the brass parts below it, are together called the 
tripod-head; and, in connection with three wooden legs Q Q Q, constitute 
the tripod. In the figure are seen the heads of wing-nuts J which confine the 
legs to tiie tripod-head. Under the center of the tripod-head should always be 
placed a small ring, from which a plumb-bob may be suspended. This is not 
needed in ordinary levelling, but becomes useful when ranging center-stakes, Ac. 

To adjust a Level. 

This is a quite simple operation, but requires a little patience. Be careful to avoid 
straining any of the screws. The large Y nuts w w sometimes require some force to 
start them; but it should be applied by pressure, and not by blows. Before begin¬ 
ning to adjust, attend to the object-glass, as directed in the first sentence under “To 
adjust a plain transit,” p. 191. 

Three adjustments are necessary ; and must be made in the following order: 

First, that of the cross-hairs : to secure that their intersection shall 
continue to strike tbe same point of a distant object, while the telescope is being 
turned round a complete revolution in its Ys. This is called adjusting the line 
of col I i (nation, or sometimes, the line of sight; but it is not strictly the line 
of sight until all the adjustments are finished; for until then, the line of collimation 
will not serve for taking levelling sights. If cross-hairs break, see p 193. 

Second, tliaf of the bubble-tube D I), to place it parallel to the lino 





TIIE LEVEL. 


203 


of oollimation. previously adjusted ; so that when the bubble stands at the centre of 
its tube, indicating that it is level, we know that our sight through the telescope is 
hor. To replace broken bubble tube, see p 193. 

Third, that of the Ys, by which the telescope and bubble-tube are supported; 
mo that the bubble-tube, and line of sight, shall he perp to the vert axis of the instru¬ 
ment; so as to remain hor while the telescope is pointed to objects in diff directions, 
as when taking back and fore sights. 

To make t he first, adjustment, or that of the cross-hairs, plant the 
tripod firmly upon the ground. In this adjustment it is not necessary to level the 
instrument. Open the clips of theYs; unclamp; draw out the eye-glass E. until 
the cross-hairs are seen perfectly clear; sight the telescope toward some clear dis¬ 
tant point of an object; or still better, toward some straight line, whether vert or 
not. Move the object-glass 0, by means of the milled head A, so that the object shall 
be clearly seen, witiiout parallax, that is, without any apparent dancing 
about of the cross-hairs, if the eye is moved a little up or down or sideways. To 
secure this, the object-glass alone is moved to suit different distances: the eye-glass 
is not to be changed after it is once properly fixed upon the cross-hairs. The neglect 
of parallax is a source of frequent errors in levelling. Clamp ; and, by means of the 
tangent-screw b, bring either one of the cross-Jiairs to coincide precisely with the 
object. Then gently, and without jarring, revolve the telescope half-way round in 
its Ys. When this is done, if the hair still coincides precisely with the object, it is 
in adjustment; and we proceed to try the other hair. But if it does not coincide, 
then by means of the 4 screws p, i, move the ring which carries the hairs, so as to 
rectify, as nearly as can be judged by eye, only one-half of the error; remembering 
that the ring must be moved in the direction opposite to what appears to be the 
right one: unless the telescope is air inverting one. Then turn the telescope back 
again to its former position: and again by the tangent-screw bring the cross-hair to 
coincide with the object. Then again turn the telescope half-way round as before. 
The hair will now be found to be more nearly in its right place, but, in all probabil¬ 
ity, not precisely so ; inasmuch as it is difficult to estimate one-half the error accu¬ 
rately by eye. Therefore a little more alteration of the ring must be made; and it 
may be necessary to repeat the operation several times, before the adjustment is 
perfect. Afterward treat the other hair in precisely the same manner. When both 
are adjusted, their intersection will strike the same precise spot while the telescope 
is being turned entirely round in its Ys. This must be tried before the adjustment 
can be pronounced perfect; because at times the adjustment of the second hair, 
slightly deranges that of the first one; especially if both w-ere much out in the be¬ 
ginning. 



To make the second adjustment, or to place the bubble-tube parallel 
to the line of collimation. This consists of two dis¬ 
tinct adjustments, one vert, and one hor. The first 
of these is effected by means of the two nuts n n on 
the vert screw at one end of the tube ; and the second 
by the two hor screws at the other end, t, of the tube. 

Looking at the bubble-tube endwise, from t in the 
foregoing Fig, its two hor adjusting-screws t l are 
seen as in this sketch. The larger capstan-headed 
nut below, has nothing to do with the adjustments; 
it merely holds the end of the tube in its place. 

To make the vert adjustment of the bubble-tr.be, b}' means of the two nuts nn. Place 
the telescope over a diagonal pair of the levelling-screws K K; and clamp it there. 
Open the clips of the Ys; and by means of the levelling-screws bring the bubble to 
the center of its tube. Lift the telescope gently out of the Ys, turn it end for end, and 
put it back again in its reversed position. This being done, if the bubble still remains 
at the center of its tube, this adjustment is in order ; but if it moves toward one end, 
that end is too high, and must be lowered; or else the other end must be raised. 
First, correct half the error by means of the levelling-screws K K, and then the re¬ 
maining half by means of the two small capstan-headed nuts nn. To raise the end 
n, first loosen the upper nut and then tighten the lower one; to do which, turn each 
nut so that the near side moves toward your right. To lower it, first loosen the lower 
nut then tighten the upper one, moving the near side of each nut toward your left. 
Having thus brought the bubble to the middle again, again lift the telescope out of 
its Ys - turn it end for end, and replace it. The bubble will now settle nearer the 
center’than it did before, but will probably require still further adjustment. If so, 
correct half the remaining error by the levelling-screws, and half by the nuts, as be¬ 
fore- and so continue to repeat the operation until the bubble remains at the center 
in both positions. For another method, see “ To adjust the long bubble-tube,” p 192. 

Horizontal adjustment of bubble tube; to see that its axis is in the same plane 
with that of the telescope, as it usually is in new instruments. It is not easily de- 





















204 


THE LEVEL. 


ranged, except by blows. Have tbe bubble-tube, as nearly as may be, directly under 
the telescope, or over the center of the bar V F. Bring the telescope over two of the 
levelling-screws K K; damp it there; center the bubble with said screws; turn the 
telescope in its Ys, say about % inch, bringing the bubble-tube out from over the 
center of the bar, first on one side, then on the other. If the bubble stays centered 
while so swung out, this adjustment is correct. It it runs toward opposite ends of its 
tube when swung out on opposite sides of the center, move the end t of the tube by 
the two horizontal screws tt. until the bubble stays centered when the tube is swung 
out on either side. If the bubble runs toward the same end of its tube on both sides, 
the tube is not truly cylindrical, but slightly conical,* so that if the telescope is 
turned in its Ys the bubble will leave the center, even when the horizontal adjust¬ 
ment is correct. It is known to be correct, in such tubes, if the bubble runs the same 
dish nice from the center when swung out (lie same distance on each side. 

Having made the horizontal adjustment, turn the telescope hack in its Ys until the 
bubble-tube is over the bar. Repeat the vertical adjustment (p 203), which may have 
become deranged in making this horizontal one. Rersevere until both adjustments 
are found to be correct at the same time. 

To make tl»e third adjustment, or to adjust the heights of the Ys, so 

as to make the line of-collimation parallel to the bar Y F, or perp to the vert axis 
of the instrument. The other adjustments being made, fasten down the clips of the 
Ys. Make the instrument nearly level by means of all four of the levelling-screws 
K. Place the telescope over two of the levelling-screws which stand diagonally; 
and leave it there unclamped. Then bring the bubble to the center of its tube, by 
the two levelling-screws. Swing the upper part of the instrument half-way around, 
so that the telescope shall again stand over the same two screws; but end for end. 
This done, if the bubble leaves the center, bring it half-way back by the large cap¬ 
stan nuts w, w ; and the other half by the two levelling-screws. Remember that to 
raise the Y, and the end of the bubble over w, w, the lower w must be loosened : and 
the upper one tightened; and vice versa. Now place the telescope over the other 
diagonal pair of levelling-screws: and repeat the whole operation with them. Hav¬ 
ing completed it, again try with the first pair; and so keep on until the bubble re¬ 
mains at the center of its tube, in every position of the telescope. 

Correct levelling may be performed even if all the foregoing adjustments are 
out of order: provided each fore-sight be taken at precisely the same distance, from 
the instrument as the back-sight is. But a good leveller will keep his instrument al ways 
in adjustment; and will test the adjustments at least once a day when at work. As 
much, however, depends upon the rodman, or target-man, as upon tile leveller. A rod- 
man who is careless about holding the rod vert, or about reading the sights correctly, 
should be discharged without mercy. 

The levelling-screws in many instruments become very hard to turn if dirty. Clean 
with water and a tooth-brush. Use no oil on field instruments. 

Forms for level note-hook's. When the distance is short, so as not to 
require two sets of books, the following is perhaps as good as any. 


No. of 
Station. 


Back 

sights 


Fore 

sights. 


Diff. 


Level. Grade. 


Cut. 


Fill. 


But on public works generally the original field-books have only the first five co’s 
After the grades have been determined by means of the profile drawn from these 
the results are placed in another book, which has only the first col and the last four! 
In both cases, the right-hand page is reserved for memoranda. The writer considers 
it best, both with the level and with the transit, to consider the term “ Station” to 
apply to the whole dist between two consecutive stakes; and that its number shall 
be that written on the last stake. Thus, with the transit, Station 6 means the dist 
from stake 5 to stake 6; that it has a bearing or course of so and so; and its length 
is so and so. And with the level, Station 6 also means the dist from stake 5 to stake 
(5; the back-sight for that dist 'being taken at stake 5, and the fore-sight on stake 
6; and that the level, grade, cut, or fill is that at stake 6. The starting-point of the 
survey, whether a stake, or any thing else, we call and mark simply 0. 


* This defect can be remedied only by removing the tube and inserting a correctl v- 
shaped one, and this is best done by an instrument-maker; but correct work can 
be done in spite of it, thus: Make ail the adjustments as nearly correct as possible. 
Level the instrument. By turning the telescope in its Ys, make the vertical hair 
coincide with a plumb-line or other vertical line, and make a short continuous knife- 
scratch on the collar nearest the object-glass, and on the adjoining Y. Lift the tele¬ 
scope out of its Ys, turn it end for end, replace it in its Ys; again bring the upright 
hair vertical, and make on the other Y a scratch coinciding with that on the collar. 
Then, in levelling or in adjusting, always see that the scratch on the collar coincides 
with that on the adjoining Y when the bubble-tube is under the telescope. 











THE HAND-LEVEL. 


205 




THE HAND-LEVEE. 



This very useful little instrument, as arranged by Professor Locke, of Cincinnati, is 
but about, five orsix inches long. Simply holdingit in one hand, and looking through 
It in any direction, we can ascertain at once, approximately, what objects are at the 
same level with the eve. E is the eye end: and 0 the object end. L is a small 
level, enclosed in a kind of brass boxing t g, the bottom of which is open, with a cor¬ 
responding opening under it, through the top of the main tube E 0. Immediately 
at the bottom of the small level L, is a cross-wire, stretched across said opening, and 
carried by a small plate, which, for adjusting the wire, can be pushed backward a 
trifle by tightening the screw f, or pushed forward by a small spring within the box¬ 
ing, near g. when the screw t is loosened. At m is a small semicircular mirror a a, 
silvered on the back to. This is placed at an angle of 45°. and occupies one-half the 
width of the tube E 0. Through the forementioned openings, the images of the 
cross-wire and of the level-bubble are reflected down on the unsilvered face a a of 
the mirror, and thence to the eye, as shown by the single dotted lines c and w: and 
when the instrument is adjusted, and held level, the wire will appear to be at the 
center of the bubble. At lc is one-half of a plano-convex lens, at the inner end of a 
short tube k. p, which may be moved backward or forward by a pin v, projecting 
through a short slit in the main tube By this means the image of the cross-wire is 
rendered distinct; and the half lens must be moved until, when viewing an object, 
the wire shall show no parallax; but appear steady against the object when the eyo 
is slightly moved up or down. At each end of the tube E 0 is a circular piece of 
plain glass for excluding dust. 



To adjust tlie hand-level, first fix two precisely level marks, say from 
50 feet to 100 yards apart. This being done, rest the instrument against one of the 
level marks, and take sight at the other. If, then, the wire does not appear to be 
precisely at the center of the bubble, move it slightly backward or forward, as the 
case may be, by the screw t, until it does so appear. 

The two level marks may be fixed by means of the 
hand-level itself, even if it is entirely out of adjust¬ 
ment, thus: First, by the pin n arrange the half lens 
k, so as to show the wire distinctly and without paral¬ 
lax. Then holding the level steadily, at any selected 
object, as a, so that the wire appears to cut the center 

of the bubble, see where it cuts any other convenient object, as b. Then go to b, 
and from it, in like manner, sight back toward a. If the instrument is in adjust¬ 
ment, the wire will cut a; but if not, it will strike either above it or below it, as at c. 
In either case, make a mark to, half-way between c and a. Then b and to will be the 
two level marks required. With care, these adjustments, when once made, will 
remain in order for years. The instrument generally has a small ring r, for hanging 
it around the neck: it is not adapted to very accurate work, but admirably so for 
exploring a route. The height of a bare hill can be found by beginning at the foot, 
and sighting ahead at any little chance object which the cross-wire may strike, as a 
pebble, twig, &c; then going forward, stand at that object, and fix the wire on 
another one still farther on, and so to the top. At each observation we plainly rise 
a height equal to that of the eye, say 5^ feet, or whatever it may be. Whether 
going up or down it, if the hill is covered with grass, bushes, &c, a target rod must 
be used for the fore-sights; and the constant height of the eye may be regarded as 
the back-sight at each station. An attachment may be made for screwing the level 
to a small ball and socket on top of a cane, or of a longer stick, for occasional use, 
when rather more accuracy is desired.* 


* Price, about $10 or $12; with about $3 more for attachable ball and socket. 



































206 


LEVELS 


To adjust a builder’s plumb- 
level, t b d% stand it upon any two sup¬ 
ports to and n, and mark where the plumb- 
line cuts at o. Then reverse it. placing the 
foot t upon n, and d upon /», and mark where 
the line now cuts at c. Half-way between o 
and c make the permanent mark. Whenever 
the line cuts this, the feet t and d are on a 
level. 


b 



To ad just a slope-instrument, or clinometer. As usually made, 

the bubble-tube is attached to the movable bar by a screw near each end, and the 
head of one of the screws conceals a small slot in tiie bar, which allows a slight vert 
motion to the screw when loose, and with it to that end of the tube. Therefore, in 
order to adjust the bubble, this screw is first loosened a little, and then moved up 
or down a trifle, as may be reqd. It is then tightened again. 















LEVELLING BY THE BAROMETER. 


207 


LEVELLING BY THE BAROMETER. 

1. Many circumstances combine to render the results of this kind of levelling un¬ 
reliable where great accuracy is required. This fact was most conclusively proved 
by the observations made by Captain T. J. Cram, of the U. S. Coast Survey. See 
Report of U. S. C. S., vol. for 1854. It is difficult to read off from an aneroid (the 
kind of barom generally employed for engineering purposes) to within from two to 
five or six ft, depending on its size. The moisture or dryness of the air affects the 
results; also winds, the vicinity of mountains, and the daily atmospheric tides, 
which cause incessant and irregular fluctuations in the barom. A barom hanging 
quietly in a room will often vary of an inch within a few hours, corresponding 
to a cliff of elevation of nearly 100 ft. No formula can possibly be devised that shall 
embrace these sources of error. The variations dependent upon temperature, lati¬ 
tude, &c, are in some measure provided for; so that with very delicate instruments, a 
skilful observer may measure the diff of altitude of two points close together, such 
as the bottom and top of a steeple, with a tolerable confidence that he is within two 
or three feet of the truth. But if as short an interval as even a few hours elapses 
between his two observations, such changes may occur in the condition of the atmo¬ 
sphere that he may make the top of the steeple to be lower than its bottom ; or at 
least, cannot feel by any means certain that he is not ten or twenty ft in error; and 
this may occur without any perceptible change in the atmosphere. Whenever prac¬ 
ticable, therefore, there should be a person at each station, to observe at both points 
at the same time. Single observations at points many miles apart, and made on dif¬ 
ferent days, and in different states of the atmosphere, are of little value. In such 
cases the mean of many observations, extending over several days, weeks, or months, 
and made when the air is apparently undisturbed, will give tolerable approximations 
to the truth. In the tropics the range of the atmospheric pres is much less than 
in other regions, seldom exceeding y 2 inch at any one spot; also more regular in 
time, and, therefore, less productive of error. Still, the barometer, especially either 
the aneroid, or Bourdon’s metallic, may be rendered highly useful to the civil engi¬ 
neer, in cases where great accuracy is not demanded. By hurrying from point to 
point, and especially by repeating, he can form a judgment as to which of two sum¬ 
mits is the lowest. Or a careful observer, keeping some miles ahead of a surveying 
party, may materially lessen their labors, especially in a rough country, by select¬ 
ing the general route for them in advance. The accounts of the agreement within 
a few inches, in the measurements of high mountains, by diff observers, at diff 
periods ; and those of ascertaining accurately the grades of a railroad, by means of 
an aneroid, w'hile riding in a car, will be believed by those only who are ignorant 
of the subject. Such results can happen only by chance. 

When possible, the observations at different places should be taken at the same 
time of day, as some check upon the effects of the daily atmospheric tides; and in 
very important cases, a memorandum should be made of the year, month, day, and 
hour, as well as of the state of the weather, direction of the wind, latitude of the 
place, &c, to be referred to an expert, if necessary. 

The effects of latitude are not included in any of our formulas. When 
reqd they may be found in the table page 209. Several other corrections must be 
made when great accuracy is aimed at; but they require extensive tables. 

In rapid railroad exploring, however, such refinements may be neglected, inas¬ 
much as no approach to such accuracy is to be expected; but on the contrary, errors 
of from 1 to 10 or more feet in 100 of height, will frequently occur. 

As a very rough average we may assume that the barometer falls ^ 
inch for every 90 feet that we ascend above the level of the sea, up to 1000 ft. But 
in fact its rate of fall decreases continually as we rise; so that at one mile high it 
falls inch for about 100 ft rise. Table 2 shows the true rate. 





208 


LEVELLING BY THE BAROMETER, 




To ascertain the <litl' of height between two points. 

Rule 1. Take readings of the barom and therm (Fah) in tlie shade at both 
stations. Add together the two readings of the barom, and div their sum by 2, for 
their mean ; which callfc. Do the same with the two readings of the therraom, and 
call the mean t. Subtract the least reading of the barom from the greatest; and call 
the diff d. Then mult together this diff d; the number from the next Table No. 1, 
opposite t; and the constant number 30. Div the prod by b. Or 

Height = Diff (d) of x Tabular number opposite x Constant 30 . 

1U leet l»mm ^ mmn It. 1 of thermom 


barom 


mean ( t ) of thermom 


mean (bj of barom. 

Example. Reading of the barom at lower station, 26.64 ins; and at the upper 
sta 20.82 ins. Thermom at lowest sta, 70°; at upper sta, 40°. What is the dill' in 
height of the two stations? Here, 

Barom, 26.64 Therm, 70° 

“ 20.82 “ 40° 


2)47.46 


Also, 


2)110 


23.73 mean of bar, or b. 

The tabular number opposite 55°, is 917.2. 

Bar. Bar. 


65° mean of 
therm, or t. 


Again, 26.64 — 20.82 = 5.82, diff of bar; or d. Hence, 


d, Tab No. Con. 

Height _ 5.82 X 917.2 X 30 160143.12 = 6748 . 5 ft answer . 

in feet 2 3.73 (or b) 23.73 

Then correct for latitude, if more accuracy is reqd, by rule on next page. 

The screw' at the back of an aneroid is for adjusting the index by a stand¬ 
ard barom. After this has been done it must by no means he meddled with. In 
6oine instruments specially made to order with that intention, this screw 1 may bo 
used also for turning the index back, after having risen to an elevation so great that 
the index has reached the extreme limit of the graduated arc. After thus turning 
it back, the indications of the index at greater heights must bo added to that at¬ 
tained when it was turned back. 


TABLE 1. For Rule 1. 


Mean 

of 

Ther. 

No. 

Mean 

of 

Ther. 

No. 

Mean 

of 

Ther. 

No. 

Mean 

of 

Ther. 

No. 

0° 

801.1 

300 

864.4 

60° 

927.7 

90° 

991.0 

1 

803.2 

31 

866.5 

61 

929.8 

91 

993.1 

2 

805.3 

32 

868.6 

62 

931.9 

92 

995.2 

3 

807.4 

33 

870.7 

63 

934.0 

93 

997.3 

4 

809.5 

34 

872.8 

64 

936.1 

94 

999.4 

5 

811.7 

35 

874.9 

65 

938.2 

95 

1001.6 

6 

813.8 

36 

877.0 

66 

940.3 

96 

1003.7 

7 

815.9 

37 

879.2 

67 

942.4 

97 

1005.8 

8 

818.0 

38 

881.3 

68 

944.5 

98 

1007.9 

9 

820.1 

39 

883.4 

69 

946.7 

99 

1010.0 

10 

822.2 

40 

885.4 

70 

948.8 

100 

1012.1 

11 

824.3 

41 

887.5 

71 

950.9 

101 

1014.2 

12 

826.4 

42 

889.6 

72 

953.0 

102 

1016.3 

13 

828.5 

43 

891.7 

73 

955.1 

103 

1018.4 

14 

830.6 

44 

893.8 

74 

957.2 

104 

1020.5 

15 

832.8 

45 

896.0 

75 

959.3 

105 

1022.7 

16 

834.9 

46 

898.1 

76 

961.4 

106 

1024.8 

17 

837.0 

47 

900.2 

77 

963.5 

107 

1026.9 

18 

839.1 

48 

902.3 

78 

965.6 

108 

1029.0 

19 

841.2 

49 

904.5 

79 

967.7 

109 

1031.1 

20 

843.3 

50 

906.6 

80 

969.9 

110 

1033.2 

21 

845.4 

51 

908.7 

81 

972.0 

111 

1035.3 

22 

847.5 

52 

910.8 

82 

974.1 

112 

1037.4 

23 

849.6 

53 

913.0 

83 

976.2 

113 

1039.5 

24 

851.8 

54 

915.1 

84 

978.3 

114 

1041.6 

25 

853.9 

00 

917.2 

85 

980.4 

115 

1043.8 

26 

856.0 

56 

919.3 

86 

982.6 

116 

1045.9 

27 

858.1 

57 

921.4 

87 

984.7 

117 

1048.0 

28 

860.2 

58 

923.5 

88 

986.8 

118 

1050.1 

29 

862.3 

59 

925.6 

89 

988.9 

119 

1052.2 

























LEVELLING BY THE BAROMETER. 


209 


Ri le Belville’s short approx rule is the one best adapted to rapid 
neld use, namely, add together the two readings of the barom only. Also And the 
dirt between said two readings; then, as tile sum of the two readings 
is to their tliff, so is 55000 feet to the reqtl altitude. 

Ex. Same as before. Here the sum = 26.64 4- 20.82 = 47.46 ins: and the diff = 
26.64 — 20.S2 = 5.82 ins. 

Sum. Diff. Feet. Feet. 

Hence, as 47.46 : 5.82 :: 55000 : 6744.6 reqd alt; instead of the 6748.5 of Rule 1. 

< orrection for latitude. The lat of the place affects the result to some 

extent ; but is usually omitted wheu great accuracy is not reqd. To apply the correction, first find 
the altitude by the rule, as before. Then divide it by the number in the following table opp the lat 
of the place. Add the quotient to the alt if the lat is less than 45°; or subtract it if the lat is more 
than 45°. If the two places are in diff lats, use their mean. 


Table of corrections for latitude. 


Lat. 

0 ° 

352 

Lat. 

140 

399 

Lat. 

283 

630 

Lat. 

42° 

3367 

Lat. 

54° 

1140 

Lat. 

68 ° 

490 

2 

354 

16 

416 

30 

705 

44 

10101 

56 

941 

70 

460 

4 

356 

18 

436 

32 

804 

45 

0 

58 

804 

72 

436 

6 

360 

20 

460 

34 

941 

46 

10101 

60 

705 

74 

416 

8 

367 

22 

490 

36 

1140 

48 

3367 

62 

630 

76 

399 

10 

375 

24 

527 

38 

1458 

50 

2028 

64 

572 

78 

886 

12 

386 

26 

572 

40 

2028 

52 

1458 

66 

527 

80 

375 


Levelling- by Barometer; or by the boiling point. 

Rule 3. The following table. No. 2, enables us to measure heights either by means 
of boiling water, or by the barom. The third column shows the approximate alti¬ 
tude above sea-level corresponding to diff heights, or readings of the barom; and to 
the dill degrees of Fahrenheit’s thermom,at which water boils in the open air. Thus 
when the barom, under undisturbed conditions of the atmosphere, stands at 24.08 
inches, or when pure rain or distilled water boils at the temp of 201° Fah ; the place 
is about 5764 ft above the level of the sea, as shown by the table. It is therefore 
very easy to find the diff of altitude of two places. Thus : take out from table No 2, 
the altitudes opposite to the two boiling temperatures ; or to the two barom readings. 
Subtract the one opposite the lower reading, from that opposite the upper reading. 
The rem will be the reqd height, as a rough approximation. To correct this, add 
together the two therm readings; and div the sum by 2, for their mean. From table 
for temperature, p 211, take out the number opposite this mean. Mult the ap¬ 
proximate height just found, by this tabular number. Then correct for lat if reqd. 

Ex. The same as preceding; namely, barom at lower sta, 26.64; and at upper sta, 
20.82. Thermom at lower sta, 70° F.,h; and at the upper one, 40°. What is the diff 
of height of the two stations ? 

Alt. 


Here the tabular altitudes are, for 20.82. 9579 

and for 26.64. 3115 


6464 ft, approx height. 

70° 4- 40° 110° 

To correct this, we have ——-= —— == 55° mean; and in table p 211, opp to 

A A 

55°, we find 1.048. Therefore 6464 X 1-048 = 6774 ft, the reqd height. 

This is about 26 ft more than by Rule 1 ; or nearly .4 of a ft in each 100 ft. 

At 70° Fah, pure water will boil at 1° less of temp, for an average of about 550 ft 
of elevation above sea-level, up to a height of % a mile. At the height of 1 mile, 1° 
of boiling temp will correspond to about 560 ft of elevation. In table p 210 the 
mean of the temps at the t wo stations is assumed to be 32° Fah : at which no correc¬ 
tion for temp is necessary in using the table; hence the tabular number opposite 
32°, in table p 211, is 1. 

This diff produced in the temp of the boiling point , by change of elevation, must 
not be confounded with that of the atmosphere, due to the same cause. The air be¬ 
comes cooler as we ascend above sea-level, at the rate (very roughly) of about 1° Fah 
for every 200 ft near sea-level, to 350 ft at the height of 1 mile. See “ Air,” p 215. 

Tlie following- table, No. 2, (so far as it relates to the barom.) was de¬ 
duced by the writer from the standard work on the barom by Lieut.-Col. R. S. Wil¬ 
liamson, U. S. army.* 

* Published bv permission of Government in 1868 by Van Nostraud, N. Y. 

14 





























210 


LEVELLING BY TIIE BAItOMETEIt, ETC 


TABLE 2. 

Levelling by Barometer; or by the boiling 1 point. 

Assumed temp in the shade 32° Fah. If not 32°, mult barom alt as per Table, j 211 


Boil 
point 
iu ileg 
Fah. 

Barom. 

Ins. 

Altitude 
above 
sea level 
Feet. 

Boil 
point 
in deg 
Fab. 

Barom. 

Ins. 

Altitude 
above 
sea level 
Feet. 

Boil 
point 
in deg 
Fah. 

Barom. 

Ins. 

Altitude 
above 
sea level 
Feet. 

Boil 
point 
in deg 
Fall. 

Barom. 

Ins. 

Altitude 
above 
sea level 
Feet. 

181° 

16.79 

15221 

.3 

19.66 

11083 

.6 

22.93 

7048 

.9 

26 59 

3164 

.1 

16.83 

15159 

.4 

19.70 

11029 

.7 

22.98 

6991 

206 

26-64 

3115 

.2 

16.86 

15112 

.5 

19.74' 

10976 

.8 

23.02 

6945 

.1 

26 69 

3066 

.3 

16.90 

15050 

.6 

19 78 

10923 

.9 

23.07 

6888 

.2 

26-75 

3007 A 

.4 

16 93 

15003 

.7 

19.82 

10870 

199 

23.11 

6843 

.3 

26-80 

2958 

.5 

16 97 

14941 

.8 

19.87 

10804 

.1 

23.16 

6786 

.4 

26-86 

2899 f 

.6 

17.00 

14895 

.9 

19.92 

10738 

.2 

23.21 

6729 

.5 

26-91 

2850 

.7 

17 04 

14833 

192 

19.96 

10685 

.3 

23.26 

6673 

.6 

26-97 

2792 

.8 

17.08 

14772 

.1 

20.00 

10633 

.4 

23.31 

6617 

.7 

27-02 

2743 

.9 

17.12 

14710 

.2 

20.05 

10567 

.5 

23.36 

6560 

.8 

27-08 

2685 

185 

17.16 

14049 

.3 

20.10 

10502 

.6 

23.40 

6516 

.9 

27.13 

2637 

.1 

17.20 

14588 

.4 

20.14 

10450 

.7 

23 45 

6460 

207 

27.18 

2589 

.2 

17.23 

14543 

.5 

20.18 

10398 

.8 

23.49 

6415 

.1 

27.23 

2540 

.3 

17.27 

14482 

.6 

20.22 

10346 

.9 

23 54 

6359 

.2 

27.29 

2483 

.4 

17.31 

14421 

.7 

20.27 

10281 

200 

23 59 

$104 

.3 

27.34 

2435 

•5 

17.35 

14381 

.8 

20.31 

10230 

.1 

23 64 

6248 

.4 

27.40 

2377 

.6 

17.38 

14315 

.9 

20.35 

10178 

.2 

23.69 

6193 

.5 

27.45 

2329 

.7 

17.42 

14255 

193 

20.39 

10127 

.3 

23.74 

6137 

.6 

27.51 

2272 

.8 

17.46 

14195 

.1 

20.43 

10075 

.4 

23.79 

6082 

.7 

27.56 

2224 

.9 

17.50 

14135 

.2 

20.48 

10011 

.5 

23 84 

6027 

.8 

27.62 

2167 t 

186 

17.54 

14075 

Ji 

20.53 

9947 

.6 

23.89 

5972 

.9 

27.67 

2120 

.1 

17.58 

14015 

A 

20.57 

9896 

.7 

23.94 

5917 

208 

27.73 

2063 

.2 

17 62 

13956 

.5 

20.61 

9845 

.8 

23.98 

5874 

.1 

27.78 

2016 


17.66 

13896 

.6 

20.65 

9794 

.9 

24.03 

5819 

.2 

27.84 

1959 

.4 

17.70 

13837 

.7 

20.69 

9743 

201 

24.08 

5764 

.3 

27.89 

1912 

.5 

17.74 

13778 

.8 

20.73 

9693 

.1 

24.13 

5710 

.4 

27.95 

1856 


17.78 

13718 

.9 

20.77 

9642 

.2 

24.18 

5656 

.5 

28.00 

1809 

.7 

17.82 

13660 

194 

20.82 

9579 

.3 

24.23 

5602 

.6 

28.06 

1753 

.8 

17.86 

13601 

.1 

20.87 

9516 

.4 

24.28 

5547 

.7 

28.11 

1706 

.9 

17.90 

13542 

.2 

20.91 

9466 

.5 

24.33 

5494 

.8 

28.17 

1650 

187 

17.93 

13498 

.3 

20.96 

9403 

.6 

24.38 

5440 

.9 

28.23 

1595 

• 1 

17.97 

13440 

.4 

21.00 

9353 

.7 

24.43 

5386 

209 

28.29 

1539 

.2 

18.00 

13396 

.5 

21.05 

9291 

.8 

24.48 

5332 

.1 

28.35 

1483 

•3 

18.04 

13338 

.6 

21.09 

9241 

.9 

24.53 

5279 

.2 

28.40 

1437 

.4 

18.08 

13280 

.7 

21.14 

9179 

202 

24.58 

5225 

.3 

28.45 

1391 

•5 

18.12 

13222 

.8 

21.18 

9130 

.1 

24.63 

5172 

.4 

28.51 

1336 

.6 

18 16 

13164 

.9 

21.22 

9080 

.2 

24.68 

5119 

.5 

28.56 

1290 

.7 

18.20 

13106 

195 

21.26 

9031 

.3 

24.73 

5066 

.6 

28.62 

1235 

.8 

18.24 

13049 

a 

21.31 

8969 

.4 

24.78 

5013 

.7 

28.67 

1189 

.9 

1828 

12991 

.2 

21.35 

8920 

.5 

24.83 

4960 

.8 

28.73 

1134 

188 

18.32 

12934 

.3 

21.40 

8859 

.6 

24.88 

4907 

.9 

28.79 

1079 

.1 

18.36 

12877 

.4 

21.44 

8810 

.7 

24.93 

4855 

210 

28.85 

1025 

.2 

18.40 

12820 

.5 

21.49 

8749 

.8 

24.98 

4802 

.1 

28.91 

970 

.3 

18.44 

12763 

.6 

21.53 

8700 

.9 

25.03 

4750 

.2 

28.97 

916 

.4 

18.48 

12706 

.7 

21.58 

8639 

203 

25.08 

4697 

.3 

29.03 

862 

J) 

18.52 

12649 

.8 

21.62 

8590 

.1 

25.13 

4645 

.4 

29.09 

808 

.6 

18.56 

12593 

.9 

21.67 

8530 

.2 

25.18 

4593 

.5 

29.15 

754 

.7 

18.60 

12536 

196 

21.71 

8481 

.3 

25.23 

4541 

.6 

29.20 

709 

.8 

18.64 

12480 

.1 

21.76 

8421 

.4 

25.28 

4489 

.7 

29.25 

664 

.9 

18.68 

12424 

.2 

21.81 

8361 

.5 

25.33 

4437 

.8 

29.31 

610 

189 

18.72 

12367 

.3 

21.86 

8301 

.6 

25.38 

4386 

.9 

29.36 

565 

.1 

18.76 

12311 

.4 

21.90 

8253 

.7 

25.43 

4334 

211 

29.42 

512 

.2 

18.80 

12256 

.5 

21.95 

8193 

.8 

25.49 

4272 

.1 

29.48 

458 

.3 

18.84 

12200 

.6 

21 99 

8145 

.9 

25.54 

4221 

.2 

29.54 

405 

.4 

18.88 

12144 

.7 

22.04 

8086 

204 

25.59 

4169 

.3 

29.60 

852 * 

.5 

18.92 

12089 

.8 

22.08 

8038 

.1 

25.64 

4118 

.4 

29.65 

308 

.6 

18.96 

12033 

.9 

22.13 

7979 

.2 

25.70 

4057 

.5 

29.71 

255 

.7 

19 00 

11978 

197 

22.17 

7932 

.3 

25.76 

3996 

.6 

29.77 

202 

.8 

19.04 

11923 

.1 

22.22 

7873 

.4 

25.81 

3945 

.7 

29 83 

149 

.9 

19.08 

11868 

.2 

22.27 

7814 

.5 

25.86 

3894 

.8 

29.88 

105 

199 

19.13 

11799 

.3 

22.32 

7755 

.6 

25.91 

3844 

.9 

29.94 

52 

.1 

19.17 

11745 

.4 

22.36 

7708 

.7 

25.96 

3793 

212 

30.00 


.2 

19.21 

11690 

.5 

22.41 

7649 

.8 

26.01 

3742 

Below sea level. 

.3 

19.25 

11635 

.6 

22.45 

7602 

.9 

26.06 

3392 

.1 

30.06 

_ 52 

.4 

19.29 

11581 

.7 

22.50 

7544 

205 

26.11 

3642 

.2 

30.12 

—104 

.5 

19.33 

11527 

.8 

22.54 

7498 

.1 

26.17 

3582 

.3 

30.18 

—158 

.6 

19.37 

11472 

.9 

82.59 

7439 

.2 

26.22 

3532 

.4 

30.24 

—209 

.7 

19.41 

11418 

198 

22.64 

7381 

.3 

26.28 

3472 

.5 

30.30 

— *261 

.8 

19.45 

11364 

.1 

22.69 

7324 

.4 

26.33 

3422 

.6 

30.35 

—^04 

.9 

19.49 

11310 

.2 

23.74 

7266 

.5 

26.38 

8372 

.7 

30.41 


191 

19.54 

11243 

.3 

22.79 

7208 

.6 

26.43 

3322 

.8 

30.47 

4 OH 

.1 

19.58 

11190 

.4 

22.84 

7151 

.7 

26.48 

3273 

.9 

30.53 

459 

.2 

19.62 

11136 

-5 

22.89 

7093 

.8 

26.54 

3213 

213 

30.59 

-511 




































SOUND 


211 


< R r nfoV°»?/ 0r t< ‘ n, P CPat,,p e; to be used in connection with 

m otion « !llV a I f CHr » c 5 r * s "eceswry. Also in con- 

neuion with table 2 when the temp is not 32°. 


Mean 
temp 
in the 
shade. 

Mult 

by 

Mean 
temp 
in the 
shade. 

Mult 

by 

Mean 
temp 
in the 

shade. 

Mult 

by 

Mean 
temp 
in the 
shade. 

Mult 

by 

Zero. 

.943 

28° 

.992 

56° 

1.050 

84° 

1.108 

2° 

.937 

jj 30 

.996 

58 

1.054 

86 

1.112 

4 

.912 

32 

1.000 

60 

1 058 

88 

1.117 

6 

.946 

34 

1004 

62 

1.062 

90 

1.121 

8 

.950 

36 

1.008 

64 

1.066 

92 

1.125 

10 

.954 

38 

1.012 

66 

1 071 

94 

1.129 

12 

.958 

40 

1 016 

68 

1.075 

96 

1.133 

14 

.9-12 

42 

1.020 

70 

1.079 

98 

1.138 

16 

.967 

44 

1.024 

72 

1.083 

100 

1.142 

18 

.971 

46 

1.028 

74 

1.087 

102 

1.146 

20 

.975 

48 

1.032 

76 

1.091 

104 

1.150 

22 

.979 

50 

1.036 

78 

1.096 

106 

1.154 

24 

.9i3 

52 

1.041 

80 

1.100 

108 

1.158 

26 

.987 

54 

1.046 

82 

1.104 

110 

1.163 


SOUND. 


Its velocity, in quiet open air, has been experimentally determined to 
be very approximately 1090 ft per sec, when the temp is at freezing point, or .12° 
Fah; and that it increases about Vy£ ft per sec for every degree above 02°, or, by 
some authorities, 1 ft for every 2°. The first would make it 


1090 ft at 32° 

1150 ft at 80° 

1100 *• 40° 

1162*6 - 90° 

1112*6 ** 50° 

1175 “ 100 s * 

1125 “ 600 

1187*6 “ 110° 

1137*6 “ 70° 

1200 “ 120° 

At 32° sound would travel a mile in 4.84 sec; 

at 80°, in 4.6: and at 120°. in 4.4 sec. 


k „ If the air is calm, fogs, or rain do not appreciably affect the result; but winds do. There is some 
reason to believe that very loud souuds travel somewhat faster than low ones. The watchword of 
sentinels has been heard across still water, on a calm night, 10*6 miles; and a cannon 20 miles. 
Separate sounds, at intervals of -j-L of a sec, cannot be distinguished, but appear to be connected. 
The dists at which a speaker can be understood, in front, on one side, and behind him, are about as 4, 
3, and 1. 

Dr. Charles M. Cresson informs the writer that, by repeated trials, he found that in a Philadelphia 
gas main 20 ins diam. and 10000 ft long, laid and covered in the earth, but empty of gas and having 
one horizontal bend of SHP, and of 40 ft radius, the sound of a pistol-shot travelled 16000 ft in pre¬ 
cisely 16 sec, or 1000 ft per sec. The arrival of the sound was barely audible ; but was rendered very 
apparent to the eye by its blowing oif a diaphragm of tissue-paper placed over the end of the main. 

Two boats anchored some dist apart may serve as a base line for triang¬ 
ulating objects along the coast; the dist between them being first found by firing guns on board one 
of them. . . . 

Ill water the vel is about 4708 ft per sec, or about 4 times that in air. In 
woods, it is from 10 to 10 times; and lu metals, from 4 to 10 times greater 
than in air, according to some authorities. 























212 


HEAT 


Approximate expansion of solids by beat : and tbelr melt¬ 
ing- points by Fahrenheit’s thermometer. $ 



For 1 degree. 

For 180 degrees.* 

Melting 
point 
iu Deg4 

fire-brick. 

1 part in 

365220 

% inch in 
3804 ft. 

1 part in 
2029 

inch in 

21.14 ft. 

., .. ( from 

187560 

1954 

1042 

10.85 


<* ranlte . j to 

228060 

2375 

1267 

13.20 


Glass rod . 

221400 

2306 

1230 

12.81 


4 lass tube. 

214200 

2231 

1190 

12.40 


'• crown . 

211500 

2203 

1175 

12.24 


plate. 

209700 

2184 

1165 

12.13 

4593 

Platina. 

208800 

2175 

1160 

12.08 

Marble, granular, white, dry.. 

173000 

1802 

961 

10.00 


“ “ “ moist. 

128000 

1333 

711 

7.41 


“ ‘ black, compact 

405000 

4219 

2250 

23.44 

955 

Antimony . 

166500 

1722 

925 

9.63 

Cast iron. 

16200011 

1688 

900 

9.38 

1920 to 

Slate . 

173000 

1802 

961 

10.00 

2800 

Steel. 

151200 

1575 

840 

8.75 

2370 to 

“ blistered. 

159840 

1665 

888 

9.25 

2550 

“ un tempered. 

167400 

1744 

930 

9.69 


“ tempered yellow. 

131400 

1369 

730 

7.60 


“ hardened. 

1468S0 

1530 

816 

8.50 


“ annealed. 

147600 

1537 

820 

8.54 


Iron, rolled. 

14994011 

1562 

833 

8.68 

3000 to 

“ soft, forged. 

147420 

1536 

819 

8.53 

3500 

“ wire. 

146340 

1524 

813 

8 55 


Bismuth. 

129600 

1350 

720 

7.50 

506 

Gold, annealed. 

123120 

1282 

684 

712 

2016 

Copper.average 

104400 

1088 

580 

6.04 

2000 

Sandstonej-. 

103320 

1076 

574 

5 98 


Brass. average 

97740 

1018 

543 

5.66 

1873 

wire. 

94140 

981 

523 

5.45 


Silver. 

95040 

990 

528 

5.50 

1861 

Tin. average 

87840 

915 

488 

5.08 

444 

Lead. average 

63180 , 

658 

351 

3.66 

612 

Few ter. 

78840 

821 

438 

4.56 


Zinc (most of all metals). 

61920 

645 

344 

3.58 

680 to 772 

White pine. 

440530 

4588 

2447 

25.49 



Heat of a common wood-fire, variously estimated from 800 to 1140 deg. That of a charcoal 

cue about 2200 ’; coal about 2400°. 

Each 12° to 15° of heat produce in wrot iron an expansion equal to 

that produced by a tension of about 1 ton per sq inch of section ; varying with the quality of the iron. 
For temperature, expansion, conducting power, Ac, of air, see p 215. 


* Ry adding yL part to the lengths in the two cols under 180°, we get the lengths corresponding to 
a number of degrees yf less than 180°; or to 163°.63 deg, which may be taken as about the extremes 
of temp in the colder portions of the United States. In the Middle States the extremes rarely reach 
135°, or *4 Part less thau 180°. 

No dependence whatever is to be placed on results obtained by Wedgewood’s pyrometer. 

t The table shows that the contraction and expansion of stone will cause 

open joints in winter; and crushing of the mortar in summer, at the ends of long coping-stones. 

J The melting points are quite uncertain. We give the mean of 

the best authorities. Assuming that with a change of temp of about I63 3 , wrought iron will alter its 
length 1 part in 916; this in a mile amounts to 5.764 ft, or about 5 ft 9 hi ins, and in lot) fi to .109 of a 
foot; or 1 % ins; so that a ditf of 5 ft, or more, can readily result from measuring a mile in winter 
mol to summer with the same chain; and a 25 ft rail will change its length full % of an inch. 

II Hence wrought expands by heat about oue eleventh part more than cast ; whereas uuder 
Oension within elastic limits cast stretches twice as much as wrt. 




















































THERMOMETERS, 


213 


THERMOMETERS. 


RpIow about 35° below zero of Fah. the mercurial thermometer and barometer become too irregular t« 
be depended on. Mercury begins to freeze at about-40° Fuh. Below—40° pure alcohol is need. 


To change degrees of Fahrenheit to the correspond hag de¬ 
grees of Centigrade; take a Fall reading 32° lower than the given oneT n.tilt 
this lower reading by 5 ; divide the prod by 9. Thus : +14° Fah = (H —32) Xori»= —18 X 5 7 - 9 — 
—10° Oeut. Again, —13° Fah = (—13 —32) X 5-f9 = —45 X 5-f 9 = —25° Cent. 

To change fah toReau; take a Fall reading 32° lower than the given 
one ; mult this lower reading by 4; div the prod bv 9. Thus : +14° Fah — (14 —32) X 4 -j-9 ~ —18 X 
4-f-9 = — 8 ° Reau. Again, —13° Fah = (—13 —32) X 4 7 -9 = — 45 X 4-r9 — —20° Reau. 

To change Cent to Fah; mult the Cent reading by 9; divide l>y 5. Take a 
Fah reading 32“ higher than the quot. Thus: -f-10° Cent = (10 X 9 7 -5) 4-32 = 18-4-32 =-4-50° Fah. 
Again, —20° Cent =(-20 X 9 7 - 5) 4-32 = — 4° Fah. 

To change Cent to Kean; mult by 4; div by 5. Thus: -t-10° Cent =H>X 

4-r5 = -J-8°ReaU. Again,—10°CeUt =—10 X 4 t- 5 =—8° RtSau. 

To change Reau to Fall; mult by 9; div by 4. Take a Fah reading 32° 

'higher. Thus: 4-16° Reau = (16 X9 t- 4) •+■ 82= 86 -4- 32 = +68° Fah. Again, —8° Reau = (—8 XSr 
4) -4-32 — — 18-4-32 = -4*14° 1 ah. 

Tochange Reau to Cent; mult by 5; div by 4. Thus : 4 - 8 ° Reau =• 4 - 8 ° 

X5-r4=-|-10 o Ceut. Again, —8°Reau =—8X5 t- 4= —10°Cent. • 


TABLE 1. Fahrenheit compared with Centigrade and Reau¬ 
mur. In this table the Cent and Reau readings are given to the nearest decimal. 


F. 

c. 

R. 

F. 

c. 

R. 1 

F. | 

c. 

O 

o 

O 

O 

o 

O | 

O 

0 

212 

100 

80 0 

158 

70.0 

56.0 

104 

40 0 

211 

99.4 

796 

157 

69.4 

56.6 

103 

39.4 

210 

98.9 

79.1 

156 

68.9 

55.1 

102 

38.9 

209 

98 3 

78.7 

153 

68 3 

54.7 

101 

88 3 

208 

97.8 

78.2 

154 

67.8 

54.2 

100 

37 8 

207 

97.2 

77.8 

153 

67.2 

53.8 

99 

37.2 

206 

96.7 

77.3 

152 

66.7 

53.3 

98 

36.7 

305 

96.1 

76.9 

151 

66.1 

52.9 

97 

36.1 

204 

95.6 

76.4 

150 

65.6 

52.4 

96 

35 6 

201 

95.0 

76 0 

149 

65.0 

52.0 

95 

35.0 

202 

94.4 

75.6 

148 

64.4 

51.6 

94 

34.4 

201 

93.9 

75.1 

147 

63.9 

51.1 

93 

33.9 

200 

93.3 

74.7 

146 

63.3 

50.7 

92 

33.3 

199 

92.8 

74.2 

145 

62.8 

50.2 

91 

32.8 

198 

92.2 

73.8 

144 

62 2 

49.8 

90 

32.2 

197 

91.7 

73.3 

143 

61.7 

49.3 

89 

31.7 

196 

91.1 

72.9 

142 

61 1 

48.9 

88 

31.1 

195 

90.6 

72.4 

141 

60.6 

48.4 

87 

30.6 

194 

90.0 

72 0 

140 

60.0 

48.0 

86 

30.0 

193 

89.4 

71.6 

139 

59.4 

47.6 

85 

29.4 

192 

88.9 

71.1 

138 

58.9 

47.1 

84 

28.9 

191 

88.3 

70.7 

137 

58.3 

46.7 

83 

28.3 

190 

87 8 

70.2 

136 

57.8 

46.2 

82 

27.8 

1 s9 

87.2 

69.8 

135 

57 2 

45.8 

81 

27.2 

188 

86.7 

69.3 

134 

56.7 

45.3 

80 

26.7 

187 

86 1 

68.9 

133 

56.1 

44.9 

79 

26.1 

186 

85.6 

68.4 

132 

55.6 

44.4 

78 

25.6 

1H5 

85.0 

68.0 

131 

55.0 

44.0 

77 

25.0 

184 

81.4 

67.6 

130 

54.4 

43.6 

76 

24.4 

18! 

83.9 

67.1 

129 

53.9 

43.1 

75 

23.9 

182 

83.3 

66.7 

128 

53.3 

42.7 

74 

23.3 

181 

82.8 

66.2 

127 

52.8 

42.2 

73 

22.8 

180 

82.2 

65.8 

126 

52.2 

41.8 

72 

22 2 

179 

81.7 

65.3 

125 

51.7 

41.3 

71 

21.7 

178 

81.1 

64.9 

124 

51.1 

40.9 

70 

21.1 

177 

80.6 

64.4 

123 

50.6 

40.4 

69 

20.6 

176 

80.0 

64.0 

122 

50.0 

40.0 

68 

20.0 

175 

79.4 

63.6 

121 

49.4 

39.6 

67 

19.4 

174 

78.9 

63.1 

120 

48.9 

39.1 

66 

18.9 

173 

78.3 

62.7 

119 

48.3 

38.7 

65 

18.3 

172 

77.8 

62.2 

118 

47.8 

38.2 

64 

17.8 

171 

77.2 

61.8 

117 

47.2 

37.8 

63 

17.2 

170 

76.7 

61.3 

116 

46.7 

37.3 

62 

16.7 

169 

76.1 

60.9 

115 

46 1 

86.9 

61 

16.1 

168 

75.6 

60 4 

114 

45 6 

36.4 

60 

15.6 

167 

75.0 

60.0 

113 

45.0 

36.0 

59 

15.0 

160 

74.4 

59.6 

112 

44.4 

35 6 

58 

14.4 

165 

73 9 

59.1 

111 

43.9 

35.1 

57 

13.9 

164 

73.3 

58.7 

110 

43.3 

34.7 

56 

13.3 

163 

72.8 

58.2 

109 

42 8 

34.2 

55 

12.8 

162 

72.2 

57.8 

108 

42.2 

33 8 

54 

12.2 

161 

71.7 

57.3 

107 

41.7 

33.3 

53 

11.7 

160 

71.1 

56 9 

106 

41 1 

32.9 

52 

11.1 

159 

70.6 

56.4 

105 

40.6 

32.4 

51 

10.6 


R. 

F. 

c. 

R. 

F. 

c. 

R. 

O 

O 

o 

O 

0 

o 

O 

32.0 

50 

10.0 

8.0 

—3 

-19.4 

—15 6 

31.6 

49 

9.4 

7.6 

—4 

—20.0 

— 16.0 

31.1 

48 

8.9. 

7.1 

—5 

-20.6 

—16.4 

30.7 

47 

8.3 

6.7 

—6 

—21.1 

—16.9 

30 2 

46 

7.8 

6.2 

—7 

—21.7 

— 17.3 

29.8 

45 

7.2 

5.8 

—8 

—22.2 

—17.8 

29.3 

44 

6.7 

5.3 

—9 

—22.8 

— 18.2 

28.9 

43 

6.1 

4.9 

—10 

—23.3 

—18.7 

28.4 

42 

6.6 

4.4 

—11 

- 23.9 

— 19.1 

28.0 

41 

5.0 

4.0 

—12 

—24.4 

— 19.6 

27.6 

40 

4.4 

3.6 

—13 

-25 0 

—20.0 

27.1 

39 

3.9 

3.1 

—14 

-25.6 

-20,4 

26.7 

38 

3.3 

2.7 

— 15 

-26.1 

—20.9 

26.2 

37 

2.8 

2.2 

—16 

—26.7 

-21.3 

25.8 

36 

2.2 

1.8 

—17 

- 27.2 

—21.8 

25.3 

35 

1.7 

1.3 

—18 

—278 

— 22 2 

24.9 

34 

1.1 

0.9 

—19 

—28.3 

—22.7 

24.4 

33 

0.6 

0.4 

—20 

—28.9 

—23.1 

24.0 

32 

0.0 

0.0 

—21 

—29.4 

—23.6 

23.6 

31 

—0.6 

—0.4 

—22 

—30.0 

— 24.0 

23.1 

30 

— 1.1 

-0.9 

—23 

—30.6 

—24.4 

22.7 

29 

—1.7 

—1.3 

—24 

—31.1 

—24 9 

22.2 

28 

—2.2 

— 1.8 

—25 

—31.7 

—25.3 

21.8 

27 

—2.8 

—2.2 

—26 

—32.2 

— 25 8 

21.3 

26 

—3.3 

—2.7 

—27 

—32.8 

—26.2 

20.9 

25 

—3.9 

—3.1 

—28 

—83.3 

- 26.7 

20.4 

24 

—4.4 

—3.6 

—29 

—33.9 

—27.1 

20.0 

23 

-5.0 

—4.0 

—30 

-34.4 

-27.6 

19 6 

22 

-5.6 

—4.4 

—31 

—35.0 

—28.0 

19.1 

21 

-6.1 

—4.9 

—32 

—35.6 

—28.4 

18.7 

20 

—6.7 

-5.3 

—33 

—36.1 

—28.9 

18.2 

19 

-7.2 

—5.8 

—34 

—36.7 

—29 3 

17.8 

18 

—7.8 

— 6.2 

—35 

—37.2 

—29.8 

17.3 

17 

—8.3 

-6.7 

—36 

—37.8 

—30.2 

16.9 

16 

—8.9 

—7.1 

-37 

—38.3 

—30.7 

16.4 

15 

—8.4 

—7.6 

—38 

—38.9 

—31.1 

16.0 

14 

—100 

—8.0 

—39 

—39.4 

-31.6 

15 6 

13 

—10.6 

—8.4 

—40 

—40.0 

—32.0 

15.1 

12 

—11.1 

—8.9 

—41 

— 40.6 

-32.4 

14.7 

11 

—11.7 

—9.3 

—42 

-41.1 

—32.9 

14.2 

10 

—12.2 

—9.8 

—43 

—41.7 

—33.8 

13.8 

9 

—12.8 

—10.2 

—44 

—42.2 

I 33.8 

13.3 

8 

—13.3 

—10.7 

—45 

—42.8 

—34 2 

12.9 

7 

—13.9 

—11.1 

—46 

-43.3 

—34.7 

12 4 

6 

— 14.4 

—11.6 

-47 

—43.9 

1-35.1 

12.0 

5 

—15 0 

—12.0 

—48 

—44.4 

—35.6 

11.6 

4 

—15.6 

—12.4 

—49 

—45.0 

—36.0 

11.1 

3 

—16.1 

—12.9 

—50 

—45 6 

—36.4 

10.7 

2 

—16 7 

—13.3 

—51 

—46.1 

—36.9 

10.2 

1 

—17.2 

—13.8 

—52 

— 46.7 

—37.3 

9.8 

0 

— 17.8 

—14.2 

—53 

—47.2 

—37.8 

9.3 

—1 

— 18.3 

—14.7 

—54 

—47.8 

—38.2 

8.9 

8.4 

—2 

—18.9 

—15.1 

—55 

—48.3 

-38.7 













































214 


THERMOMETERS 


TABLE 2. Centigrade compared with Fahrenheit and 

Reaumur. _______ 


c . 

F. 

R. 

c . 

F . 

R. 

c . 

F. i 

R. 

‘-I 

F. 

R. 

o 

O 

O 

0 

0 

O 

0 

O 1 

o 

0 

° 

O 


Exact. 

Exact. 


Exact. 

Exact. 


Exact. 

Exact. 


Exact. 

Exact. 

100 

212.0 

80.0 

62 

113.6 

49.6 

24 

75.2 

19.2 

—11 

6.8 

— 11.2 

99 

210.2 

79.2 

61 

141.8 

48.8 

23 

73.1 

18.4 

-15 

5.0 

— 12.0 

08 

208.1 

78.1 

60 

140.0 

18.0 

22 

71.6 

17.6 

— 16 

3.2 

- 12.8 

37 

206.6 

77.6 

69 

138.2 

47.2 

21 

69 8 

16.8 

-17 

1.1 

— 13.6 

% 

201 8 

76 8 

58 

136.1 

46.1 

20 

68.0 

16.0 

-18 

— 0.4 

— 14.1 

95 

203.0 

76.0 

57 

131.6 

45.6 

19 

66.2 

15.2 

—19 

—2.2 

— 15.2 

91 

201.2 

75.2 

56 

132.8 

44.8 

18 

64.1 

11.1 

—20 

— 4.0 

— 16.0 

98 

199.1 

71 1 

55 

131.0 

44.0 

17 

62.6 

13.6 

—21 

— 5.8 

—16.8 

92 

197.6 

73.6 

51 

129.2 

43 2 

16 

60.8 

12.8 

—22 

— 7.6 

— 17.6 

91 

195.8 

72.8 

53 

127.1 

42.1 

15 

59 0 

12.0 

—23 

-9.4 

- 18.1 

90 

191.0 

72.0 

52 

125.6 

41.6 

14 

57.2 

11.2 

—24 

— 11.2 

— 19.2 

89 

192.2 

71.2 

51 

123.8 

10.8 

13 

55.1 

10.1 

—25 

— 13.0 

— 20.0 

88 

190 1 

70.1 

50 

122.0 

40.0 

12 

53.6 

9.6 

—26 

— 14.8 

—20 8 

87 

188.6 

69.6 

49 

120.2 

39.2 

11 

51.8 

8.8 

—27 

— 16.6 

— 21.6 

86 

1 .86.8 

68 8 

18 

118.4 

38.1 

10 

50.0 

8.0 

-28 

— 18.4 

—22.4 

85 

185.0 

68.0 

47 

116.6 

37.6 

9 

48.2 

7.2 

—29 

— 20.2 

—23.2 

81 

183.2 

67.2 

46 

111.8 

36.8 

8 

46.1 

6.1 

—30 

— 22.0 

— 21.0 

88 

1 *1.1 

66.1 

45 

113.0 

36.0 

7 

44 6 

5.6 

-31 

—23.8 

—24.8 

82 

179.6 

65.6 

u 

111.2 

35.2 

6 

42.8 

4.8 

—32 

—25.6 

—25 6 

81 

177 8 

64 8 

43 

109.4 

34.1 

5 

41.0 

4.0 

—33 

- 27.1 

—26 1 

80 

176.0 

64.0 

12 

107 6 

33.6 

1 

39.2 

3.2 

—34 

— 29.2 

— 27.2 

79 

171.2 

63.2 

41 

105.8 

32.8 

3 

37.1 

2.4 

—35 

— 31.0 

— 28.0 

78 

172.1 

62.4 

10 

104.0 

32.0 

2 

35.6 

1.6 

—86 

—82.8 

—28.8 

77 

170 6 

61.6 

39 

102.2 

31.2 

1 

33.8 

0.8 

—37 

—31,6 

—29.6 

70 

168.8 

60.8 

38 

100.4 

30.1 

0 

32.0 

0.0 

-38 

—36.1 

—30.1 

75 

167.0 

60.0 

37 

98.6 

29.6 

— 1 

30.2 

—0.8 

—39 

—38.2 

—31.2 

71 

165.2 

59.2 

36 

96.8 

28.8 

—2 

28.1 

-1.6 

—40 

—40.0 

—32,0 

73 

163.1 

58.4 

.35 

95.0 

28.0 

—3 

26.6 

—2.4 

—41 

—41.8 

-32.8 

72 

161 6 

57.6 

34 

93.2 

27.2 

—4 

24.8 

—3.2 

—42 

-43.6 

—33.6 

71 

159 8 

56.8 

33 

91.1 

26.1 

—5 

23.0 

— 1.0 

—43 

— 45.1 

—34.1 

70 

158.0 

56.0 

32 

89.6 

25.6 

-6 

21.2 

—4.8 

— 44 

— 17.2 

—35.2 

69 

156.2 

55.2 

31 

87.8 

21 8 

-7 

19.1 

—5.6 

— 45 

—49.0 

—36.0 

68 

151.1 

51.4 

30 

86.0 

24.0 

—8 

17.6 

— 6.4 

—46 

—50.8 

—36.8 

67 

152.6 

53.6 

29 

84.2 

23.2 

—9 

15.8 

—7.2 

—47 

—52.6 

— 37.6 

66 

150.8 

52 8 

28 

82.4 

22.1 

— 10 

14.0 

-8.0 

— 48 

—54.1 

—38.1 

65 

119.0 

52.0 

27 

80.6 

21.6 

— 11 

12.2 

—8.8 

— 49 

-56.2 

—39.2 

64 

117 2 

51.2 

26 

78.8 

20.8 

— 12 

10.1 

—9.6 

—50 

—58.0 

—10 0 

63 

145.1 

50.4 

25 

77.0 

20.0 

—13 

8.6 

— 10.4 





i 


1 


1 


TABLE 3. Reaumur compared with Fahrenheit and 

Lent! (trade. 


R. 

F. 

C. 

R. 

F. 

C. 

R. 1 

F. 

c. 1 R- | 

F. 

c. 

C 

O 

Exact. 

O 

Exact. 

o 

o 

Exact. 

o 

Exact. 

O 

O 

Exact. 

O 

Exact. 

0 

O 

Exact. 

o 

Exact. 

80 

212.00 

100.00 

49 

112.25 

61.25 

19 

71.75 

23.75 

—11 

7.25 

— 13.75 

79 

209.75 

98.75 

48 

140 00 

60 00 

18 

72.50 

22.50 

-12 

5.00 

— 15.00 

78 

207.50 

97.50 

17 

137.75 

58.75 

17 

70.25 

21.25 

—13 

2.75 

—16.25 

1 1 

205.25 

96.25 

16 

135 50 

57.50 

16 

68.00 

20.00 

—11 

0.50 

—17.50 

76 

203.00 

95.00 

15 

13.3.25 

56.25 

15 

65.75 

18.75 

—15 

—1.75 

-18.75 

75 

200.75 

93.75 

11 

1.31.00 

55 00 

11 

6H.50 

17.50 

—16 

—4 00 

—20.00 

71 

198.50 

92.50 

13 

128.75 

53.75 

13 

61.25 

16.25 

— 17 

—6 25 

—21.25 

73 

196 25 

91.25 

42 

126.50 

52 50 

12 

59 00 

15.00 

—18 

—8.50 

— 22.50 

72 

194 00 

90.00 

41 

121.25 

51.25 

11 

56.75 

13.75 

—19 

—10.75 

— 23.75 

71 

191.75 

88.75 

40 

122.00 

50.00 

10 

54 50 

12.50 

—20 

—13.00 

- 25.00 

70 

189.50 

87.50 

39 

119 75 

48.75 

9 

52.25 

11.25 

—21 

—15.25 

—26.25 

69 

187.25 

86.25 

38 

117.50 

17.50 

8 

50.00 

10.00 

—22 

—17 50 

—27.50 

68 

185.00 

85 00 

37 

115.25 

46.25 

7 

47.75 

8.75 

—23 

— 19.75 

—28.75 

67 

182.75 

83.75 

36 

113.00 

45.00 

6 

45.50 

7.50 

—21 

—22.00 

—30.00 

66 

180.50 

82.50 

35 

110.75 

13.75 

5 

43.25 

6.25 

—25 

—24.25 

— 31.25 

65 

178 25 

81.25 

31 

108 50 

42.50 

4 

41.00 

5.00 

—26 

—26.50 

—32.50 

64 

176.00 

80 00 

33 

106.25 

41.25 

3 

38.75 

3.75 

—27 

—28.75 

—33.75 

63 

173.75 

78.75 

32 

101.00 

10 00 

2 

36.50 

2.50 

- 28 

—31.00 

—35 00 

62 

171.50 

77.50 

31 

101.75 

38.75 

1 

34.25 

1.25 

—29 

—33.25 

—36.25 

61 

169.25 

76.25 

30 

99.50 

37 50 

0 

32.00 

0 00 

—30 

—35.50 

—37.50 

60 

167.00 

75.00 

29 

97.25 

36.25 

—1 

29.75 

-1.25 

—31 

—37.75 

—38.75 

59 

164 75 

73.75 

28 

95.00 

35.00 

—2 

27.50 

-2.50 

—32 

—40.4K) 

—40 00 

58 

162.50 

72.50 

27 

92.75 

33 75 

—3 

25.25 

—3.75 

—33 

—42.25 

—41.25 

57 

160 25 

71.25 

26 

90.50 

32 50 

—1 

23.00 

- 5.00 

—31 

—44.50 

—42.50 

56 

158.00 

70.00 

25 

88.25 

31.25 

—5 

20.75 

—6.25 

-35 

—46.75 

—43.75 

55 

155.75 

68.75 

21 

86.00 

30.00 

—6 

18 50 

—7.50 

—36 

- 49 00 

—45 00 

54 

153.50 

67.50 

23 

83.75 

28.75 

—7 

16.25 

—8.75 

-37 

—51.25 

—43.25 

53 

151.25 

66.25 

22 

81.50 

27.50 

- 8 

11.00 

—10.00 

—38 

—53.50 

—47.50 

52 

149.00 

65.00 

21 

79.25 

26.15 

—9 

11.75 

—11.25 

—39 

—55.75 

— 48 75 

51 

50 

146.75 

111.50 

63.75 

62.50 

20 

77.00 

25.00 

—10 

9.50 

-12 50 

—40 

— 58.00 

—50.00 

























































AIR. 


215 


AIR—ATMOSPHERE. 

Tlie atmosphere is known to extend to at least 45 miles 

above the earth. Its composition is about .79 measures of nitrogen gas 
and .21 of oxygen gas; or about .77 nitrogen, .23 oxygen, by weight. It gen¬ 
erally contains, however, a trace of water; carbonic acid, and carburetted 
hydrogen gases; and still less ammonia. 

'When the barometer is at 30 inches, and the temperature 60° Fah, air 
weighs about one-815th part as much as water; or 535 grains = 1.224 commercial 
ounces=.0765 commercial lb. per cubic foot. Or 13.072 cubic feet weigh 1 lb; 
and a cubic yard, 2.066 lbs. Or a cube of 30.82 feet on each edge, 1 ton. When 
colder it weighs more per cubic foot, and vice versa, at the rate of about a grain 
per degree of Fah. The average weight of the entire atmospheric column, (at 
least 45 miles high,) at sea level, is 14% lbs avoirdupois per square inch ; or 2124 
lbs per square foot* = weight of a column of water 34 feet, or of mercury 30 
inches, high. This is what is usually called the “pressure of the air.” At % 
mile above sea level it is but 14.02 lbs per square inch: at % mile, 13.33; at % 
mile, 12.66; at 1 mile, 12.02; at 1% mile, 11.42; at 1% mile, 10.88; and at 2 miles, 
9.80 lbs. Therefore, a pump in a high region will not lift water to as great a 
height as in a low one. The pressure of air, like that of water, is, at any given 
point, equal in all directions. 

Iris often stated that the temperature of tlie atmosphere lowers 
or becomes colder, at the rate of 1° Fah for eacli 300 feet of ascent above the 
earth's surface; but this is liable to many exceptions, and varies much 
with local causes. Actual observation in balloons seems to show that up to the 
first 1000 feet, about 200 feet to 1°, is nearer the truth ; at 2000 feet, 250; at 4000 
feet.,' 300 ; and at a mile, 350. 

In breathing, a grown person at rest requires from .25 to .35 of a cubic 
foot of air per minute; which, when breathed, vitiates from 3% to 5 cubic feet. 
When walking or hard at work, he breathes and vitiates two or three times as 
much. About 5 cubic feet of fresh air per person per minute is required for the 
perfect ventilation of rooms in winter; 8 in summer. Hospitals 40 to 80. 

Beneath the general level of’the surface of the earth in temperate 
regions, a tolerably uniform temperature of about 50° to 60° Fah exists at 
the depth of about 50 to 60 feet; and increases about 1° for each additional 50 
to 60 feet; all subject, however, to considerable deviations from many local 
causes. In the Rose Bridge colliery, England, at the depth of 2424 feet, t he tem¬ 
perature of the coal is 93%° Fah ; and at the bottom of a boring 4169 feet deep, 
near Berlin, the temperature is 119°. 

The air is a very slow conductor of heat; hence hollow walls 
serve to retain the heat in dwellings: besides keeping them dry. It rushes 
into a vacuum near sea level with a velocity of about 1157 feet per second ; 
or 13% miles per minute; or about as fast as sound ordinarily travels through 
quiet air. See Sound, p 211. 

hike all other elastic fluids, it expands equally with equal 
increases of temperature. Every increase of 5° Fah, expands the hulk 
of anv of them slightly more than 1 per cent of that which it has at 0° Fah; 
or 500° about doubles its bulk at zero. The bulk of any of them diminishes 
inversely in proportion to the total pressure to which it is subjected. Thus, if 
we have a cylinder open at top, and 1 foot deep, full of air at its natural pres¬ 
sure of about 15 lbs per square inch; if by means of a piston we apply an addi¬ 
tional pressure of 15 lbs per square inch, making 30 lbs in all, or twice as much 
as the natural pressure, then the air will be compressed into 6 inches of depth 
of the cylinder, or one-half of what it occupied before. Or if we apply 45 lbs 
additional, making 60 lbs in all, or 4 times the natural pressure, then the air will 
be compressed into % of the depth of the cylinder. Experiment shows that 
this holds good with air at least up to pressures of about 750 lbs per square inch, 
or 50 times its natural pressure; the air in this case occupying one-fiftieth of 
its natural bulk. In like manner the bulk will increase as the total pressure is 
diminished; so that if we remove our additional 45 lbs per square inch, the air 
in the cylinder will regain its original bulk, and will precisely fill the cylinder. 
Substances which follow these laws, are said to be perfectly elastic. Under a 
pressure of about 5% tons per square inch, air would become as dense, or would 
weigh as much per cubic foot, as water. Since the air at the surface of the earth 
is pressed 14% lbs per square inch by the atmosphere above it, and since this 
is equal to the weight of a column of water 1 inch square and 34 feet high, it 
follows that at the depths of 34, 68,102 feet, Ac, below water, air will be com- 

* = 1.033 kilogrammes per square centimetre. 










216 


WIND. 


pressed into y 2 , y, %, Ac, of its bulk at the surface; because at those depths it I 
is exposed to pressures equal to 2, It, 4, Ac, times 14% lbs per square inch, in as- f 
much as the pressure of the atmosphere on the surface, is in each case to be 
added to that of the water. The pressure of the water alone at those depths 
would be but 1, 2, 3, Ac, times 14% lbs per square inch. 

In si diving-bell, men, after some experience, can readily work for sev¬ 
eral hours at a depth of 51 feet; or under a pressure of 2% atmospheres; or 
3iy lbs per square inch. But at 90 feet deep; or under 3.64 atmospheres; or 
nearly 55 lbs per square inch, they can work for but about an hour, without 
serious suffering from paralysis; or even danger of death. Still at the St. Louis 
bridge some work was done at a depth of 11034 feet; pressure 63.7 lbs per square 
inch. 

The dew point is that temperature (varying) at which the air deposits its 
vapor. 

The greatest heat of the air in the sun probably never exceeds 
145° Fab; nor the greatest cold—74° at night. About 130° above, and 40° below 
zero, are the extremes in the U. S. east of the Mississippi; and 65° below in the 
N. W.; all at common ground level. It is stated, however, that —81° has been 
observed in N. E. Siberia; and -)-101° Fab in the shade in Paris; and -(-153° in 
the sun at Greenwich Observatory, both in July, 1881. It has frequently ex¬ 
ceeded 4-100° Fah in the shade in Philadelphia during recent years. 


WIND. 

The relation between the velocity of wind, and its press- « 

ure against an obstacle placed either at right angles to its course, or inclined 
to it, has not been well determined ; and still less so its pressure against curved 
surfaces. The pressure against, a large surface is probably proportionally greater 
than against a small one. It is generally supposed to vary nearly as the squares 
of the velocities; and when the obstacle is at right angles to its direction, the 
pressure in lbs per square foot of exposed surface is considered to be equal to 
the square of the velocity in miles per hour, divided by 200. On this basis, 
which is probably quite defective, the following table, as given by Sraeaton, is 
prepared. 


Vel. in Miles 
per Hour. 


1 

2 

3 

4 

5 

to 

124 $ 

15 

•20 

25 

30 

40 

50 

tiO 

so 

100 


Vel. in Ft. 
per Sec. 


1.467 

2.933 

4.400 

5.867 

7.33 

14.67 

18.33 

22 . 

29.33 

39.67 
44 . 

58.67 

73.33 

88 . 

117.3 

146.7 


Pres, in Lbs. 
per Sq. Ft. 


.005 
.020 
.0 45 
.080 
.125 
.5 

.781 

1.125 

2 . 

3.125 
4.5 
8 . 

12.5 

18 . 

32 . 

50 . 


Remarks. 


Hardly perceptible. 
Pleasant. 


Fresh breeze. 



Brisk wind. 

Strong wind. 

High wind. 

Storm. 

Violent storm. 

Hurricane. 

Violent hurricane, uprooting large trees 


The pres against 
a semicyliudrical 
surface a c b nom 
is about half that 
against the hat 
surf abnm. 


Tredgold recommends to allow 40 lbs per sq ft of roof for the 

pres of wind against it.; but as roofs are constructed with a slope, and consequently do not receive 
the full force of the wind, this is plainly too much.* Moreover, only one-half of a roof is usually ex¬ 
posed, even thus partially, to the wind. Probably the force in such cases varies approximately as the 
sines of the angles of slopes. According to observations in Liverpool, in 1860, a wind of 38 miles per 
hour, produced a pres of 14 lbs per sq ft against an object perp to it; and one of 70 miles per hour, 
(the severest gale on record at that city,) 42 lbs per sq foot. These would make the pres per sq ft, 
more nearly equal to the square of the vel in miles per hour, div by 100; or nearly twice as great as 
given in Smeaton s table. We should ourselves give the preference to the Liverpool observations. A 
very violent gale in Scotland, registered by an excellent anemometer, or wind-gauge. 45 lbs per sq 
“ 18 that as high as 55 lbs has been observed at Glasgow. High winds often lift roofs. 

The gauge at Girard College, Philada, broke under a strain of 42 lbs per sq ft; a tornado passing 
at the moment, within % mile. 

By inversion of Smeaton's rule, if the force in lbs per sq ft, be mult by 200, the sq rt of the prod 
will give the vel in miles per hour. Smeaton’s rule is used by the U. S. Signal Service. 


* The writer thinks 8 lbs per sq foot of ordinary double-sloping roofs ) or 16 lbs for shed-roofs suiti 
cieut allowance for pres of wind. J ' 

























WATER. 


217 


WATER. 


Pitre water, as boiled and distilled, is composed of the two gases, hydro¬ 
gen and oxygen; in the proportions of 2 measures hydrogen to 1 of oxygen ; 
or 1 weight of hydrogen to 8 of oxygen. Ordinarily, however, it contains sev¬ 
eral foreign ingredients, as carbonic and other acids; and soluble mineral, or 
organic substances. When it contains much lime, it is said to be hard; and will 
not make a good lather with soap. The air in its ordinary state contains 
about 4 grains of water per cubic foot. 

The average pressure of tlie air at sea level, will balance a 
column of water 84 feet high ; or about 30 inches of mercury. At its boil¬ 
ing point of 212° Fab, its bulk is about one twenty-third greater than at 70°. 

Its weig'ht per cubic foot is taken at 62}^ lbs,or 1000 ounces avoir; but 62^ 
lbs would be nearer the truth, as per table below. It is about 815 times heavier 
than air, when both are at the temperature of 62°; and the barometer at 30 
inches. With barometer at 30 inches the weight of perfectly pure water is as 
follows. At about 39° it has its maximum density of 62.425 lbs per cubic foot. 


Temp, Fall. 

32°. 

40°. 

50°. 

60°. 


Lbs per Cub Ft. 

.62.417 

.62.423 

.62.409 

.62.367 


Temp, Fah. 

Lbs per Cub Ft. 

70°. 

.62.302 

80°. 

.62.218 

90°. 

.62.119 

212°.. 

.59.7 


Weig'ht of sea water 64.02 to 64.27 lbs per cubic foot, or say 1.6 to 1.9 
lbs more than fresh. 

Water has its maximum density when its temperature is a little above 
39° Fah ; or about 7° above the freezing point. By best authorities 39.2°. From 
about 39° it expands either by cold, or by heat. When the temperature of 32° 
reduces it to ice. its weight is but about 57.2 lbs. per cubic foot; and its specific 
gravity about .9175, according to the investigations of L. Du four. Hence, as 
ice, it has expanded one-twelfth of its original bulk as water; and the sudden 
expansive force exerted at the moment of freezing, is sufficiently great to 
split iron water-pipes; being probably not less than 30U00 lbs per square inch. 
Instances have occurred of its splitting cast tubular posts of iron bridges, and 
of ordinary buildings, when full of rain water from exposure. It also loosens 
and throws down masses of rock, through the joints of which rain or spring 
water has found its way. Retaining-walls also are sometimes overthrown, or 
at least bulged, by the freezing of water which has settled between their backs 
and the earth filling which they sustain; and walls which are not founded at a 
sufficient depth, are often lifted upward by the same process. 

It is said that in a glass tube % inch in diameter, water will not 
freeze until the temperature is reduced to 23°; and in tubes of less than ^ 
inch, to 3° or 4°. Neither will it freeze until considerably colder than 32° in 
rapid running streams. Anchor ice, sometimes found at depths as great as 
25 feet, consists of an aggregation of small crystals or needles of ice frozen at 
the surface of rapid open water; and probably carried below bv the force of the 
stream. It does not form under frozen water. For crushing strength of 
ice. see page 437. 

Since ice floats in water; and a floating body displaces a weight of the 
liquid equal to its own weight, it follows that a cubic foot of floating ice weighing 
57.2 lbs, must displace 57.2 tbs of water. But 57.2 lbs of water, one foot square, is 11 
inches deep: therefore, floating ice of a cubical or parallelopipedal shape, will 
have of its volume under water; and only fa above; and a square foot of ice 
of any thickness, will require a weight equal to ^ of its own weight to sink it 
to the*surface of the water. In practice, however, this must be regarded merely 
as a close approximation, since the weight of ice is somewhat affected by en¬ 
closed air-bubbles. 

Pure water is usually assumed to boil at 212° Fah in the open air, at the 
level of the sea; the barometer being at 30 inches; and at about l c less for every 
520 feet above sea level, for heights within 1 mile. In fact, its boiling point 
varies like its freezing point, with its purity, the density of the air, the material 
of the vessel, &c. In a metallic vessel, it may boil at 210°; and in a glass one, 
at. from 212° to 220° ; and it is stated that if all air be previously extracted, it 
requires 275°. For leveling by the boiling point, see page 209. 

It evaporates at all temperatures; dissolves more substances than any 
other agent: and has a greater capacity for heat than any other known substance. 

It is compressed at the rate of about one-21740th. (or about. of an 
inch in 18 T V feet,) by each atmosphere or pressure of 15 lbs per square inch. 
When the pressure is removed, its elasticity restores its original bulk. 














218 


WATER. 


Effect on metals. The lime contained in many waters, forms deposits in 
metallic water-pipes, and in channels of earthenware, or of masonry ; especially 
if the current be slow. Some other substances do the same; obstructing the 
flow of the water to such an extent, that it is always expedient to use pipes of 
diameters larger than would otherwise be necessary. The lime also forms very 
hard incrustations at tlie bottoms of boilers; very much impair¬ 
ing their efficiency; and rendering them more liable to hurst. Such water is 
unfit for locomotives. We have seen it stated that the Southwestern R R Co, 
England, prevent this lime deposit, along their limestone sections, by dissolving 
1 ounce of sal-ammoniac to 90 gallons of water. The salt of sea water forms 
similar deposits in boilers; as also does mud, and other impurities. 

Water, either when very pure, as rain water; or when it contains carbonic 
acid, (as most water does,) produces carbonate of lead in lead 
pipes; and as this is an active poison, such pipes should not be used for such 
waters. Tinned lead pipes may he substituted for them. If, however, sulphate 
of lime also be present, as is very frequently the case, this effect is not always 
produced; and several other substances usually found in spring and river 
water, also diminish it to a greater or less degree. Fresh water corrodes 
wrought iron more rapidly than cast; hut the reverse appears to 
he the case with sea water; although it also affects wrought iron very 
quickly; so that thick flakes may be detached from it with ease. The corrosion 
of iron or steel hv sea water increases with the carbon. Cast-iron cannons 
from a vessel which had been sunk in the fresh water of the Delaware River 
for more than 40 years, were perfectly free from rust. Gen. Pasley, who had 
examined the metals found in the ships Royal George, and Edgar, the first of 
which bad remained sunk in the sea for 62 years, and the last for 133 years, 
“stated that the cast iron had generally become quite soft; ana in some cases 
resembled plumbago. Some of the shot when exposed to the air became hot ; 
and burst into many pieces. The wrought iron was not so much injured, 
except when in contact with copper , or brass gun-metal. Neither of these last was 
much affected, except when in contact with iron. Some of the wrought iron 
was reworked by a blacksmith,and pronounced superior to modern iron.” “Mr. 
Cottam stated that some of the guns had been carefully removed in their soft 
state, to the Tower of London; and in time (within 4 years) resumed their orig¬ 
inal hardness. Brass cannons from the Mary Rose, which had been sunk in the 
sea for 292 years, were considerably honeycombed in spots only; (perhaps where 
iron had been in contact with them.) The old cannons, of wrought-iron liars 
hooped together, were corroded about % inch deep; but had probablv been pro¬ 
tected by mud. The cast-iron shot became redhot on exposure to the air: and 
fell to pieces like dry clay!” 

“Unprotected parts of cast-iron sluice-valves, on the sea gates of the Cale¬ 
donian canal, were converted into a soft plumbaginous substance, to a depth 
of <>f an inch, within 4 years; but where they had been coated with common 
Swedish tar, they were entirely uninjured. This softening effect on cast iron 
appears to be as rapid even when the water is but slightly brackish; and that 
only at intervals. It also takes place on cast iron imbedded in salt earth. Some 
water pipes thus laid near the Liverpool docks, at the expiration of 20 vears 
were soft enough to be cut. by a knife; while the same kind, on higher ground 
beyond the influence of the sea water, were as good as new at. the end of 50 vears ” 

Observation has, however, shown that the rapidity of this action 
depends much on the quality of the Iron ; that which is dark- 
colored. and contains much carbon mechanically combined with it, corrodes 
most rapidly: while hard white, or light-gray castings remain secure for a long 
tune. Some cast-iron sea-piles of this character, showed no deterioration in 40 
years. See note, page 615. 

< ontaet with brass or copper is said to induce a galvanic action 

which greatly hastens decay in either Iresli or salt water. Some muskets were 
tecovered from a wreck which had been submerged in sea water for 70 vears 
near New l ork. The brass parts were in perfect condition ; but the iron parts 
had entirely disappeared. <*alvanizinu; (coating with zinc) acts as a pre- 
servative to the iron, but at the expense of the zinc, which soon disappears. 
J lie iron then corrodes. If iron be well heated, and then coated with hot 
coal-tar, it will resist the action of either salt or fresh water for many vears. 
It is very important that the tar be perfectly purified. See p 291. Such a' coat¬ 
ing, or one ol paint, will not prevent barnaeles and other shells from 
attaching themselves to the iron. Asphaltum, if pure, answers as well as 

topper and bronze are very little affected by sea water. 

No galvanic action has been detected where brass ferules are inserted into 
the water-pipes in Philadelphia. 
















TIDES. 


219 


The most prejudicial exposure for iron, as well as for wood, is 
that to alternate wet and dry. At some dangerous spots in Long Island Sound, 
it lias been the practice to drive round bars of rolled iron about 4 inches diam¬ 
eter, for supporting signals. These wear away most rapidly bet ween high and 
low water; at the rate of about an inch in depth in 20 years; in which time the 
4-inch bar becomes reduced to a 2-inch one, along that portion of it. Under 
fresh water especially, or under ground, a thin coating of such tar, so applied, 
(see preceding paragraph,) will protect iron, such as water-pipes, &c, for a long 
time. See page 291. The sulphuric acid contained in the water from coal mines 
corrodes iron pipes rapidly. In the fresh water of canals, iron boats 
have continued in service from 20 to 40 years. Wood remains sound foi 
centuries under either fresh or salt water, if not exposed to be worn away by 
the action of currents; or to be destroyed by marine insects. 

Sea water differs a little in wei^lit, at different, places; but at the 
same place it is appreciably the same at all depths; and may be generally as 
sumed at about 64 lbs; or 1% lbs per cubic loot, more than fresh. The additiona 
1% ft>s, or one 36.6th of its entire weight, is chiefly common salt. Sea water 
freezes at 27° Fah: the ice is fresh. 

A teaspoonful of powdered alum, well stirred into a bucket of dirty W'ater, 
will generally purify it sufficiently within a few hours to be drinkable. If a 
hole 3 or 4 feet deep be dug in the sand of the sea-shore, the infiltrating water 
will usually be sufficiently fresh for washing with soap; or even for drinking. 
It is also stated that water may be preserved sweet for many years by placing 
in the containing vessel 1 ounce of black oxide of manganese for each gallon 
of water. 

It is said that water kept in zinc tanks; or flowing through iron 
tubes galvanized inside, rapidly becomes poisoned by soluble salts of zinc 
formed thereby; and it is recommended to coat zinc surfaces with asphalt 
varnish to prevent this. Yet, in the city of Hart ford, Conn, service pipes of 
iron, galvanized inside and out, were adopted in 1855, at the recommendation 
of the water commissioners; and have been in use ever since. They are like¬ 
wise used in Philadelphia and other cities to a considerable extent. In many 
hotels and other buildings in Boston, the “Seamless Drawn Brass Tube” of the 
American Tube Works at Boston, has for many years been in use for service 

E ipe; and has given great satisfaction. It is stated that the softest water may 
e kept in brass vessels for years without any deleterious result. 

The action of lead upon some waters (even pure ones) is highly poison¬ 
ous. The subject, however, is a complicated one. An injurious ingredient may 
be attended by another which neutralizes its action. Organic matter, whether 
vegetable or animal, is injurious. Carbonic acid, when not in excess, is harm¬ 
less. See near bottom of page 419. 

Ice may be so Impure that, its water is dangerous to drink. 

The popular notion that hot water freezes more quickly 
than cold, with air at the same temperature, is erroneous. 


TIDES. 

The tides are those well-known rises and falls of the surface of the sea 
and of some rivers, caused by the attraction of the sun and moon. There are 
two rises, floods, or high tides; and two falls, ebbs, or low tides, everv 24 hours 
and 50 minutes (a lunar day); making the average of 6 hours 12 'M minutes 
between high and low water. These intervals are, however, subject to 
areat variations; as are also the heights of the tides; and this not only 
at different places, but at the same place. These irregularities are owing to the 
shape of the coast line, the depth of water, winds, and other causes. Usually at 
new and full moon, or rather a day or two after, (or twice in each lunar month, 
at intervals of two weeks,) the tides rise higher, and fall lower than at other 
times; and these are called spring tides. Also one or two days after the 
moon is in her quarters, twice in a lunar month, they both rise and fall less t han 
at other times; and are then called neap tides. From neap to spring they 
rise and fall more daily ; and vice versa. The time of high water at any 
place, is generally two or three hours after the moon has passed over either 
the upper or lower meridian; and is called the establishment of that 
place; because, when this time is established, the time of high water on any 
other dav may be found from it in most cases. The total height of spring tides 
is qenerally from to 2 times as great as that of neaps. The great tidal 
wave is merely an undulation , unattended by any current, or progressive mo ton 
of the particles of water. Each successive high tide occurs about -4 minutes 
later than the preceding one; and so with the low tides. 



220 


RAIN. 


RAIN. 

Tlie quantity that falls annually in any one place, varies 

greatly from year to year; the extremes being frequently greater titan 2 to 1. 
In making calculations for collecting water in reservoirs, whether for feeding 
canals, or lor supplying cities, we cannot safely assume more than the minimum 
fall observed for many years; or rather, somewhat less. And from even this 
must be deducted the amount, (a quite considerable one) lost by evaporation and 
leakage after it has been collected. The following table shows in some cases, 
the average annual falls; and in others, the least and the greatest ones observed 
at several places; including snow water. It is highly probable that most of the 
results are merely approximate. See Evaporation, p 222. 


Inches 
per an. 


Augusta, Georgia. 

Albany, N. York. 

Arkausas. 

Bath, Maine. 

Baltimore, M<1. 

Boston, Mass. 

Charleston, S. C. 

Canada... 

Carlisle, Penna. 

Detroit, Michigan. 

Frankford, Penna. 

Fort Gaston, California, iu 9 months. 

Fort Yuma, Cal. 

Port (not Fort) Orford, Oregon.... .. 

Fort Pike, Louisiana. 

Fort Pierce, E. Florida. 

Fort Conrad, New Mexico. 

Fort Kent, Maine. 

Fort Preble, “ . 

Fort Constitution, N. Hamp. 

Fort Adams, Rhode Island. 

Fort Hamilton, N. York Harbor. 

Fort Niagara, N.Y. 

Fort Monroe, Virg. 

Fort Kearney, Nebraska. 


23 

31 to 51 
41 

30 to 50 
40 

25 to 46 

40 to 7b 1 
36 
34 
30 

33 to 54 
129 
3% 
69 
72 
63 


C % 
36 li 
45 % 
35 

sax 

43 % 

31% 

51 

28 


Inches 
per an. 


Fort Laramie, Nebraska. 

Fort Worth, Texas. 

Fort McIntosh, “ . 

Fort Dallas, Oregon. 

Key 'West, Florida. 

Lebanon, Penna. 

Michigan. 

Monterey, Cal. 

Marietta, Ohio. 

New Orleans, Louisiana. 

New Fane, Vermont. 

New England. average.. 

Natchez, Miss. 

Now York State. average.. 

Ohio. “ 

Philadelphia, Penna. 

“ av for 54 years, to 1&84... 

Pennsylvania. average.. 

Savannah, Georgia. 

Stow, Mass. 

St. Louis, Mo... 

Washington, D. C. 

WestChester, Pcuna. 

Williamstowu, Mass. 


20 

41 

18 % 

14 % 

30 to 39 

34 to 45 

35 
12 % 

35 to 54 
51 

36 to 741 
47 

37 to 58 
36% 

36 

34 to 61 
45.2 * 

41 

30 to 60 
33 to 49 

42 
41 

39 to 54 
26 to 39 


Tlie greatest fall recorded in one day in Philadelphia, was 

7.3 inches, Aug. 13, 1873. The greatest in any month, was 15.8 inches, Aug., 1867. 
In July, 1842, 6 inches fell in 2 hours. It has not readied 9 inches per month, 
more than 7 or 8 times, in 25 years. During a tremendous rain at Norristown, 
Pennsylvania, in 1865, tlie writer saw evidence that at least 9 indies fell within 

5 hours. At Genoa, Italy,on one occasion, 32 inches fell in 24 hours ; at Geneva, 
Switzerland, 6 inches in 3 hours; at Marseilles, France, 13 inches in 14 hours; 
iu Chicago, Sept., 1878, .97 inch in 7 minutes. 

Near London. England, the mean total fall for many years is 23 inches. 
On one occasion, 6 inches fell in 1% hours! In the mountain districts of the 
English lakes, the fall is enormous ; reaching in some years to 180 or 240 inches; 
or from 15 to 20 feet! while, in the adjacent neighborhood, it is hut 40 to 60 
indies. At Liverpool, the average is 34 indies; at Edinburgh, 30; Glasgow, 22 ; 
Ireland, 36; Madras. 47; Calcutta, 60; maximum for 16 years, 82; Delhi, 21; 
Gibraltar, 30; Adelaide, Australia, 23; West Indies, 36 to 96; Rome, 39. On the 
Khassya lulls north of Calcutta, 500 inches,or41 feet 8 inches,have fallen in the 

6 rainy months! In other mountainous districts of India, annual falls of 10 to 
20 feet are common. 

It requires a quite heavy rain, for 24 hours, to yield the depth of an inch; 
still, inasmuch as at rare intervals falls of as much as from 1 to 3, or even 6 
indies per hour occur, these latter depths must lie considered in planning 
sewers, culverts, etc. See Art. 23, p 279c. 

As a general rule, more rain falls in warm than in cold 
countries; and more in elevated regions than in low ones. Local peculiar- 


*In 1869, during which occurred the greatest drought known iu Philadelphia 
for at least 50 years, it was 48.84 inches. 
























































SNOW. 


221 


ities, however, sometimes reverse this; and also cause great differences in the 
amounts in places quite near each other; as in the English lake districts just 
alluded to. It is sometimes difficult, to account for these variations. In some 
lagoons in New Granada, South America, the writer has known three or four 
heavy rains to occur weekly for some months, during which not a drop fell on 
hills about 1000 feet high, within 10 miles’ distance, and within full sight. At 
another locality, almost a dead-level plain, fully three-quarters of the rains that 
fell for 2 years, at a spot 2 miles from his residence, occurred in the morning; 
while those which fell about 3 miles from it, in an opposite direction, were in 
the afternoon. 

“The returns of several rain gauges in the Longdendale district, England, for 
1847, gave the rainfalls at different altitudes above the sea, as follows:” 

At 500 feet altitude. 46.6 ins. At 1750 feet altitude. 56.5 ins. 

800 “ . 50.5 “ 1800 “ .. 62.1 “ 

1700 “ . 52.1 “ 

“ The annual average fall at Edinburgh, 200 feet above the sea, in three succes¬ 
sive years, was 30 inches. In the Pentland hills, a few miles south, and 700 feet 
above sea, 37.4 inches: and at Carlops, similarly situated near the last, but 900 
feet, above sea, 49.2 inches.” 

There are probably but few places In the United States, 

where an annual fall of 2 feet may not be safely relied on; and since, as an 
average, about half of it may be collected into reservoirs, a square mile of 
drainage (= 27878400 square feet) should yield annually 27878400 cubic 
feet; equal to 76379 cubic feet per day. Allowing 4 cubic feet, or 30 gallons, per 
day for each person; and making no deduction for evaporation and filtration, 
this would supply a population of 19095 persons ; or a square of 38% feet on a 
side, would in like manner suffice for one person. From two-tenths to eight- 
tenths of all the water annually resulting from rain and snow, passes off into 
the neighboring rivulets: and thence into the larger streams and rivers; or 
may be collected into reservoirs. Under ordinary circumstances of locality, 
about one-half may usually be thus secured. The difference is owing chiefly 
to the distance which the water may have to run; the rates of absorption of 
various soils; the rate of descent of the sides of the valleys leading to the 
streams; the season of the year, &e, &c. See p 222. 

An inch of rain amounts to 3630 cubic feet; or 27155 U. S. gal¬ 
lons; or 101.3 tons per acre; or to 2323200 cubic feet; or 17378743 U. S. gallons; 
or 64821 tons per square mile at 62% lbs per cubic foot. 

The most destructive rains are usually those which fall upon snow, under 
which the ground is frozen, so as not to absorb water. 


SNOW. 

Trials at different times by the writer, showed the weight of freshly 
fallen snow to vary from about 5 to 12 lbs per cubic foot; apparently depend¬ 
ing chieflv upon the degree of humidity of the air through which it had passed. 
On one occasion when mingled snow and hail had fallen to the depth of 6 inches, 
he found its weight to be 31 lbs per cubic foot. It was very dry and incoherent. 
A cubic foot of heavy snow may, by a gentle sprinkling of water, be converted 
into about half a cubic foot of slush, weighing 20 lbs; which will not slide or 
run ofT from a shingled roof sloping 30°, if the weather is cold. A cubic 
block of snow saturated with water until it weighed 45 lbs per cubic foot, just 
slid on a rough board inclined at 45°; on a smoothly planed one at 30°; and on 
slate at 18°; all approximate. A prism of snow, saturated to 52 lbs per cubic 
foot; one inch square, and 4 inches high, bore a weight of 7 lbs; which at 
first compressed it about one-quarter part of its length. European engineers 
consider 6 lbs per square foot of roof, to be sufficient allowance for the 
weight of snow; and 8 lbs for the pressure of wind; total, 14 lbs. The 
writer thinks that in the U. S. the allowance for snow should not be taken at 
less than 12 lbs; or the total for snow and wind, at 20 lbs. There is no danger 
that, snow on a roof will become saturated to the extent just alluded to; because 
a rain that would supply the necessary quantity of water, would also by its 
violence wash away the snow; but we entertain no doubt whatever that the 
united pressures from snow and wind, in our Northern States, do actually at 
times reach, and even surpass, 20 lbs per square foot of roof. See Table 4, p581, 
of Trusses. The limit of perpetual snow at the equator is at the height 
of about 16000 feet, or say 3 miles above sea-level; in lat. 45° north or south, it 
is about half that height; while near the poles it is about at sea-level. 







222 


HYDROSTATICS, 


EVAPORATION, FILTRATION, AND LEAKAGE. 


Tlie amount of evaporation from surfaces of water exposed to 

the natural effects of the open air, is of course greater iu summer thau iu winter; although it is quite 
perceptible iu eveu the coldest weather. It is greater in shallow water than iu deep, inasmuch as the 
bottoui also becomes heated by the sun. It is greater in ruuuiug, than in standing water, on much 
the same principle that it is greater during winds than calms. It is probable that Hie average daily 
loss from a reservoir of moderate depth, from evaporation alone, throughout the 3 warmer months 
ot the year, (June, July, August.) rarely exceeds about inch, iu any part of the United States. j* 
- iuoh duriug the 9 colder mouths; except iu the Southern States. These two averages would 
a daily one of .15 inch ; or a total auuual loss of 55 ius, or 4 ft 7 ins. It probably is 3.5 to 4 ft. , pg 

By some I rials by tlie writer, iu the tropics, ponds of pure V <! 

8 ft deep, in a stiff retentive clay, aud fully exposed to a very hot sun all day, lost during the dry . 
sou, precisely 2 ius in 16 days ; or % inch per day ; while the evaporation from a glass tumbler was 
% inch per day. The air iu that region is highly charged with moisture; and the dews are heavy. 
Every day duriug the trial the thermometer reached from 115° to 125° iu the sun. 

The total annual evaporation in several parts of Knglaud and Scotland is stated to average from 22 
to 38 ius; at Paris, 34; Boston, Mass, 32 ; many places iu the U. S., 30 to 36 ins. This last would give 

a daily average of -Jq inch for the whole year. Such statements, however, are of very little value, 
unless accompanied by memoranda of the circumstances of the case; such as the depth, exposure, 
size and nature of the vessel, pond. &c. which contains the water. <fcc. Sometimes the total annual 
evaporation from a district of country exceeds the rain fall; aud vice versa. 

On canals, reservoirs. &c, it is usual to combine the loss by evaporation, 
with that by filtration. The last is that which soaks into the earth; and of which some portion 
passes entirely through the banks, (when in embaukt;) and if in very small quantity, may be dried 
up by the sun and air as fast as it reaches the outside; so as not to exhibit itself as water; but if in 
greater quantity, it becomes apparent, as leakage. 


E. II. Gill. C E, states the average evaporation an«l filtra¬ 
tion on the Sandy and Beaver canal. Ohio. (38 ft wide at water sur¬ 
face; 26 ft at bottom: and 4 ft deep.) to he but 13 cub ft per mile per minute, in a dry season. Here 
the exposed water surf in one mile is 200640 sq ft; and in order, with this surf, to lose 13 cub ft per 
min, or 18720 cub ft per day of 24 hours, the quantity lost must be = - 0933 ft.= 1 H inch ir 

depth per day. Moreover, one mile of the canal contains 675840 cub ft ; therefore, the number of ri» 
reqd for the combined evaporation aud filtration to amount to as much as all the water in the canal 

6 / 5 8 4 0 _ gg ,j av9t Observations in warm weather on a 22 mile reach of the Chenango cant 
18720' B ' ’ 

York, (40; 28; and 4 ft.) gave 6514 cub ft per mile per min ; or 5 times as much as iu the preceding 
case. This rate would empty the canal iu about 8 days. Besides this there was an excessive leakage 
at the gates of a lock, (of only 5 ^ ft lift.) of 479 cub fi per min. 22 cub ft per mile per min ; and at 
aqueducts, and waste-weirs, others amounting to 19 cub ft per mile per min. The leakage at other 
locks with lifts of 8 ft, or less, did not exceed about 350 cub ft per min. at each. On otper canals, it 
has been found to be from 50, to 500 ft per min. On the Chesapeake and Ohio canal, (where 50. 32, 
and 6 ft,) Mr. Fisk, C E, estimated the loss by evap and filiration iu 2 weeks of warm weather, to be 
equal to all the water in the canal. Professor IS <1II K i lit* assumes 2 ills per 
<1 ay. for leakage of canal bed, and evaporation, on English 

canals. J. B. Jervis. C E. estimated the loss from evap. filtration, and leakage through lock- 
gates, on the original Erie canal. (40, 28. and 4 ft.) at 100 cub ft per mile per min : or 144000 cub ft 
per day. The water surf in a mile is 211200 sq ft; therefore, the daily loss would be equal to a depth of 

o i t ! o o = ft,= %% ins. See end of Rain, p 221. 

Z 1 1 Z 0 u 

On the Delaware division of the Pennsylvania canals, when 

the supply is temporarily shut otf from any long reach, the water falls from 4 to 8 ins per dav. The 
filtration will of course be much greater on embankts, than in cuts. In some of our cmials tlie depth 
at h gh embankts becomes quite considerable ; the earth, from motives of eoonomv. not being filled in 
Jevel under the bottom of the canal; hut merely left to form its own natural slopes. 4 t one spot at 
least, on the Ches and Ohio canal, where one side is a natural face of vertical rock, this depth is 40 
ft. Such depths increase the leakage very greatly ; especially when, as is frequently the case theein- 
bankts are not puddled; and the practice is not to he commended, for other reasons also 


The total average loss from reservoirs of moderate depths 

In case the earthen dams he constructed with proper care, and well settled by time, will not exceed 
about from % to 1 inch per day ; but in new ones, it will usually be considerably greater. 

The loss from ditches, or channels of small area, is much 

not-Hitn *??!! . frnn ; " av,eahlfi canals; “ that long canal feeders usually deliver but a small pro- 

pottion of the water which enters them at their heads. H 


HYDROSTATICS. 


Ar 1 t - Hydrostatics treats of the pressure of quiet water • 
and other liquids. The p-es of hqnids against anv point of any " * 

«Inch they act,whether said sui t be curved or plane, is always perp to the surf a •> 





HYDROSTATICS 


223 


point. At any given depth, the pres of water is equal in every direction • and is in 

depth belowthe8nrf - Hi all cases whatever, the total pres 
nfwRtfr Vi and perp to an y 8urf > is e T ial to the wt of a uniform column 

■uvV .f ti ,! ' ni f 0t Wh °, 8e cro j s ' s l ect i° n P arall el to its base, is everywhere equal to the 
^ , !• !, j pre8Sod \ a ! ld whose height is equal to the vert depth of the cen of 

/ t l‘ e S " rf prt ; 88ed ’ beluw the hor surf of the water. This fact is one of those 
important ones ot frequent application, which the young student should impress 
[; m K S7' ,n hls mem u °. r y- . the wt of a cub ft of fresh water is usually assumed to 
he..02/5 lbs avoir; which is sufficiently correct for ordinary engineering purposes: 
>ugli b-^4 is nearer the truth for ordinary temperatures of about 7U° Tali, lienee’ 


iU 


1 


fiml the total pres of quiet water against, and perp to 
7 surl whatever, as a dam, emblct, lock-gate, die; or the bottom, side , or ton 
,fU - v cnnt ' mi ng vessel, water-pipe, dc, whether said surf be vert, hor, or inclined at 
my angle whatever ; or whether it be flat, or curved ; or whether it reach to the surf of 
he water, or be entirely below it: J J 

Of !.iW er tPe f area - ln sq ft - of the surf pressed ; the vert depth in ft 

reqd pres m p 8 ouuds?*" ^ SU ° f the Water! a “ d the constant “umber 62.5. The prod will be the 

Kx. 1. The wall A, Fig 1, Is 50 ft long ; and the depth, no, of water pressing against its vert back is 
uniformly 10 ft. What pres does the wall sustain? 

1 be area of surf pressed is 50 X 10 — 500 sq ft. And the vert depth of its cen of gray below the surf 
of the water is 5 ft; hence, 


500 X 5 X 62.5 = 156250 pounds, or about 70 tons, the pres reqd. 

The pres in this case being perp to a 
vert surf, is horizontal; teudiug either 
to overturn the wall; or to make it 
slide on its base. The center of press¬ 
ure is at c; or one-third of the vert 
depth from the bottom. 

Ex. 2. As in the foregoing case, 
the wall B, Fig 114, is 50 ft long; aud 
the vert depth of water is 10 ft; but it 
presses against the sloping side of the 
wall; n o being 15 ft. What is the 
total pres, or the pres perp to n o; or 
in the direction of the arrow? 

'e the area of surf pressed is 50 X 15 = 750 sq ft. And the vert depth of its cen of grav below 
the surf of the water is 5 ft, as before; hence, 

750 X 5 X 62.5 = 234375 pounds, or about 105 tons, the total pres reqd. 

The cen of pres as before, is at c, one-third of the depth from the bottom. 

In such cases, the total pres perp to n o, may be considered as resolved into two pressures; one of 
them acting hor, either to overthrow the wall, or to make it slide; and the other actiug vert to hold it 
in its place. Aud if the sloped line n o be taken at any scale to represent the total pres, then will the 
vert line m o, measured by the same scale, represent the hor pres ; and the hor line m n, the vert 
one. See Art 34, Force in Rigid Bodies. Therefore, so long as the vert depth of water remains the 
ame, the hor pres remains the same, no matter what may be the slope of no; but the vert, as well 
•s the total pres, will increase with n o. See Art 4. In Fig 2, the pres teuds to lift the wall. 

Rem. 1. This total pres of the water is of course distributed over the entire depth of the wetted 
iart of the back of the wall ; being least at top, and gradually increasing toward the bottom; but so 
far as regards the united action of every portion of it, in tending to overthrow the wall, considered as 
a single mass of masonry, incapable of being bent or broken, it may all be assumed to be applied at 
c: dist from the bottom of the water one-third of its vert depth ; or, which is the same thing, at 
one-third of the sloping dist o n, Figs 1J4 and 2. See Art 57, of Force in Rigid Bodies. 



Rem. 2. It follows, from the foregoing rule, that the amount 
of pres of water against any surf is entirely independent of 
the quantity of the water, so long as the area pressed, and the vert 
depth of its cen of grav below the level surf remain unchanged. The wall A or B would sustain as 
great a pres from a layer of water only an inch thick behind it. as if the water had extended back 
for miles. From this cause, retaining-walls of mortar masonry carelessly backed, have been bulged, 
and cracked, by the infiltration of rain behind them ; while walls of dry masonry would have per¬ 
mitted the water to escape through the open joints ; and would therefore have stood safely. 

Also in vessels a, h, Fig 2J4, of any size or shape whatever, if they contain 
1 the same vert depth of water ; and have equal bases o o, pressed hy said depths 

o " of water, the pressures on the bases will all he equal, without any regard to 

the quantity of water. Or, if we have two water-pipes of the same diara, both 
full of water, one standing vertically, 10 feet long ; and the other 20 miles long, 
and laid at an inclination of J4 ft per mile, so as to make its vert depth of water 
also 10 ft, the pressures at the lower ends of the two pipes will be equal. 
This fact, that the pres on a giveu surf at a given depth is independent of 
the quantity of water, is called the hydrostatic paradox. In the vessel a, the 
pres on the base is much greater than the wt of the water ; but in ft, it is less. 


CZ .1 Tl 


.GO 




Rem. 3. Since the pres of water against any point is at right augles to the surf at that point, it fol- 


* This js strictlv true as regards the pres of the water alone ; and this is usually all that is required. 
r • -,<■ --,jst he borne in mind that the surf of the water is itself pressed by the. air; to the average 
!l - ‘ ■■ the level of the sea) of about 14.7 lbs per sq inch ; or 2H7 lb«. or nearly 1 ton per sq foot. 

‘ '•Uivtfe to find the true total pres, we should limit the area in sq ft. of the surf pressed by the water, 
2117 tbs ; and add the prod to the water-pres given by the rule. But in ordinary engineering cases, 















224 


HYDROSTATICS 


lows that props p p, for strengthening such structures as the sloping dam D, Fig 3, should be placed 
at right angles to them iu ol der to oppose the greatest possible resistauce to the pres. Other consid¬ 
erations mav at times prevent our doing so; thus the outer prop, p. if so placed, would be iu danger 
of being broken by ice, or logs tumbling over the dam ; and therefore, had better be more nearly 

vertical. 

Rem. 4. It follows, from the foregoing rule, that in a cubical vessel, filled with water, the pres on 
the base is equal to the weight of the water; that on each of the four sides, to half the weight or the 
water; aud that on the bottom and the 4 sides together, to 3 times the wt of the water. In a conical 
vessel, formiug au entire cone, the pres oil its hor base is equal to 3 times the wt of the water; and so 
li kewise in a pyramidal vessel; for iu both cases the wt of the water is but % that of a uniform column 
of water of the same height. In a full spherical vessel, the total pres against its entire interior surf, 
is also equal to 3 times the wt of the water, as in a cubical one. 

Since the pres increases with the depth, the props in the dam, Fig 3, 
should be closest together near the bottom ; also the hoops of a tank. 

The following- Table gives the pres to the nearest 

lb per sq ft at diff vert depths ; and also the total pies against a plane one 
foot wide extending vert from the surface to those depths. The first in¬ 
creases as the depths; the last as the squares of the depths. 



For the pres in lbs per sq inch at any given depth, mult the depth in ft 
by .4.14. For tbs per sq ft, mult by 62.5. For tons per sq ft, mult by .0279. For 
the depth in It at which any given pres exists, divide the lbs per sq inch by 

.434; or the lbs per sq ft by 62.5: or the tons per sq ft by .0273. 


D 

Per 

Tot 

D 

Per 

Tot 

D 

Per 

Tot 

D 

Per 

'Tot 

L> 

Per 

Tot 

in 

sq 

p. 

in 

sq 

P. 

in 

sq 

P. 

in 

sq 

P. 

in 

sq 

P. 

Ft. 

Ft. 


Ft. 

Ft. 

Ft. 

Ft. 

Ft. 

Ft. 

Ft. 

Ft. 


i 

62. 

31 

u 

687. 

3781 

21 

1312. 

13781 

31 

1937. 

30031 

41 

2562. 

52531 

2 

125. 

125 

12 

750. 

4500 

22 

1375. 

15125 

32 

2000. 

32000 

42 

2625. 

55125 

3 

1H7. 

281 

13 

812. 

5281 

23 

1437. 

16531 

33 

2062. 

34031 

43 

2687. 

57781 

4 

250. 

500 

14 

875. 

6125 

24 

1500. 

18000 

34 

2125. 

36125 

44 

2750. 

60500 

5 

312. 

781 

15 

937. 

7031 

25 

1562. 

19531 

35 

2187. 

38281 

45 

2812. 

63281 

6 

375. 

1125 

16 

1000. 

8000 

26 

1625. 

21125 

36 

2250. 

40500 

46 

2875. 

66125 

7 

437. 

1531 

17 

1062. 

9031 

27 

1687. 

22781 

37 

2312. 

42781 

47 

2937. 

69031 

8 

500 

2000 

18 

1125. 

10125 

28 

1750. 

24500 

38 

2375. 

45125 

43 

3000. 

72000 

9 

562. 

2531 

19 

1187. 

11281 

29 

1812. 

26281 

39 

2437. 

47531 

49 

3062. 

75031 

10 

625. 

3125 

20 

1250. 

12500 

30 

1875. 

•28125 

40 

2500. 

50000 

50 

3125 

781*25 


! see that at the depth of 36 ft, the pres of water against a single sq ft of surf, whether hor,/ 
lique, is fully 1 ton ; requiring great precaution to prevent leakage, or breaking. At 72 ft, 


Thus we 
vert, or oblique 

it would be 2 tons. &c. A pres of 62}/<j tbs per sq ft, gives a pres of .434 lbs per sq inch. 

Further; let a b, Fig 3J4, be a tube of 36 ft vert height; full of water; with a bore so 
small that the tube would coutain say only one pound of water; and let this tube open at 
its lower eud into a vessel also full of water; the top aud botiom of which are 8 ft apart. 

Then the 1 lb of water in the tube, will cause each sq ft of the top of the vessel, (which 
is 36 ft below the surf of the water iu the tube) to be pressed upward with a force of 2250 
R>s. as per table. Each sq ft of the bottom of ttie vessel (which is 44 ft below the surf 
of the water iu the tube) will be pressed downward with a force of 2750 lbs: aud any par¬ 
ticular sq ft of the sides of the vessel, will be pressed hor outward, with the force given 
in the table, opposite to the depth of the cen of grav of said sq ft below the same water 
surf of the top of the tube, whatever said depth may happen to be. Or. suppose, first 
only the 'ower vessel to be filled with water, and its inner surf to be sustaining the pres 
arising therefrom ; if we then fill the 36 ft tube with its 1 lb of water, this 1 lb will create 
an additional pres or 2250 lbs againsteverv sq ft of said inner surf; so that if the 6 sides of the 
vessel he each 8 ft. squ .re ; or contain fn all 384 sq ft of iuuer surf, this 1 lb of water will produce addi¬ 
tional pres of 864000 lbs. or full 385 tons, against them. If we then press upon the top of the water 
with our thumb to the extent of 1 lb, we shall thereby redouble this enormous pres. This fact, bow- 
ever, belongs to Art. 7, 


a 


Li 


Fi»3i- 


Art. 2. Surfaces pressed on both sides: and immersed. 

When two bodies of water of diff depths, press against two oppo¬ 
site sides of a plane which is completely immersed, whether vert or 
sloping; as, for iustance, against the two sides a b, n o, Fig 4; or 
the two sides d e. c r, then, the total pres against t b. i e , a b, n o, or 
c r. &c, may still be found by the foregoing rule, in Art 1 : but the 
excess of pres against the part a b. or d e. of the immersed plane, 
beynnd the counterpres against the opposite part n o, or c r, will 
be equal to the wt of a column of water whose section is equal 

to the area of the part a b. or d e. (as the case may be;) and 

whose vert height is equal to m n. or x p. the vert diff of level of the 
two bodies of water. Consequently, this excess of outward pres is 
found by mult together, the area of a b or d e, in sq ft; the vert 
height m n or xp. in ft; and the constant 62.5 lbs wt of a. cub ft of 
water. Thus, if a b is 10 ft high, and 20 ft long: and the vert height 
mn, 12 ft; then the excess of pres against a b, over that against« o. will be 10 X 20 X12 X 62.5=-]50000 " 

lbs. The excess will be greater on d c. than on a b, although both are exposed to the same vert depths 

m n. x p : because the area of tie is greater than that of a h. Moreover, this excess of outward pres 
is equally distributed over the entire area of a h or d e ; being no greater at b and e, than at a or d ■ 
in other words, every sq ft of area of a b or d e is pressed outward at right, angles to its surf, by an 
excess of force equal to the wt of a column of water 1 ft sq ; and of a height equal to m n, or x p. 



this pros of the air may. and should be omitted ; because it is counterbalanced by an equal pres of air 
nerninst the opposite side. face, or surf of the pressed body. It becomes necessary, th refnre, to take 

it into consideration only when the opnosit,. f„,-p n f the body is not exposed to a counteroalancin ' 
atmospheric pressure; as when there is a vacuum on that side. " 













































HYDROSTATICS 


225 



This will be understood by means of Fig 5, which may represent five 
plauks. N 2, 3, 4, and 5, forming a dam, and seen endwise; each one 1 ft 



pres against the upper immersed 20 ft area, or that of plank 3. is 3125 lbs • 
while the couuter-pres against it from the other side is 625 lbs; making 
the excess of outward pres equal to 3125 — 625 = 2500 lbs. Again, at the 
lowest plank, number 5, the outward pres exceeds the inward one by 
5625 — 3125 — 2500 lbs, the same as in the upper one. And so of auv other 
equal area of surf,-at auy depth whatever; the excess depending upon the 
vert height of m n, will be equally distributed over a b. It only remains 
to show that the total excess of outward pres against a b, is equal in 
amount to the wt of a uniform column of water with a base equal iu area 
to a b, and with a height equal to inn. Thus, we have seen that m the 
instance before us, the excess amounts to 3 times 2500 lbs, or to 7500 lbs. 
Now, the wt of the column of water will be 60 (or area of a fri X m n (or 
2 ft) X 62.5 lbs = 7500 lbs ; or the same as the excess pres on a b. 

The excess of pres against the eutire side s b, over that against n o, is 
evidently the difif between those two pressures calculated respectively by the rule iu Art 1. 


Art. .3. Surfaces, vert, as b m c s, a n o t, Fig 6, or otherwise, of 
equal widths, b in, a n; commencing’ at the level, b a n in. of 
.the water, but extending to ditf depths, me, no, measured 
• vert; and having the same inclination to the surf of the 
waters sustain total pressures proportional to the squares 

In Fig 6, let the two vert sides, a no t, and b m c s, of a vessel, 
have the same width a n, and b m ; theu if the depth m c, be 2, 3, 
4, 5, &c, times greater than the depth n o, the pres against the surf 
b m cs, will be 4, 0, 16, 25, &c, times greater than that against a no t. 
This will be seen by referring to the pressures figured on the left side 
of Fig 5, where, as stated in Art 2, the surf of plank 1, exposed to 

the pres on the left side, is 20 sq ft; that of planks 1 and 2, 40 sq ft; 

that of planks l, 2, and 3, 60sqft, &c. All these surfs commence at 
the level of the water; and all of them being vert, are of course at 
the same inclination with the water surf; but their depths are re¬ 
spectively 1, 2, and 3 ft. The pres against the surf of 1, is 625 Jbs; 
that against the surf of 1, 2, is 625-j-1875 = 2500; and that against 
the surf of 1, 2, 3, is 625+ 1875 -J- 3125 = 5625. But 2500 is /oitrtimes 
625; and 5625 is nine times 625. Aud the pres against the entire 
surf 8 5, (which is 5 times as deep as plank 1,) is 25 times as great as that against plank 1; or 

625 X 25 = 15625 lbs = the sum of all the pressures marked on the left side of Fig 5. 

This follows, from the Rule iu Art 1; for twice the area of surf, mult by twice the vert depth of the 
ceu of grav below the surf, must give 4 times the pres; three times the area, by three times the depth, 
must give 9 times the pres, <fec. See third columns of table, p 224. 

It follows, also, that at any particular point , or against any given area placed at various depths, the 
pres will increase simply as the vert depth ; thus, if there be three areas, each one sq ft, placed in 
the same positions, but with their centers of grav respectively 8, 16, and 24 ft below the surf, the pres 
against them will be respectively as 8, 16, and 24; or as 1, 2, and 3. See second columns table, p 224. 



Art. 4. The pressure of quiet water, iu auy one given di¬ 
rection, against any given surf, whether vert, hor, inclined, flat, or curved, is equal 
to the wt of a uniform column of water, the area of whose section, parallel to its base, is everywhere 
equal to the area of the projection of the pressed surf taken perp to the given direction ; and the 
height of the column equal to the vert depth of the cen of grav of the pressed surf below the upper 
surf of the water. Hence the 



Fl,, 7 


Rule. To find the pres in lbs, mult together the area 

in sq ft of the projection taken at right angles to the given direction ; the 
vert depth in ft of the cen of grav of the pressed surf below the upper surf 
of the water; and the constant 62.5 lbs wt of a cub ft of water. 

Ex. Let mesa, Fig 7, be an inclined surf, sustaining the pres of water 
which is level with its top m c. Then the total pres against m c s n, and at 
right angles to it, as found by the rule in Art 1, is an illustration of the pres¬ 
ent rule; because the projection of m cs n, taken at right angles to the given 
direction, or parallel to m c s n, is in fact m c s n itself, or equal to it. Hence 
the rule in Art l is merely a simple modification of the present one, appli¬ 
cable to the case of total pres against any surf. 


k 


But if it be rcqil to find only the vert or downward pres 


gainst in c s n, in pounds, mult together the area of the hor projection a n c m 


.or 


q ft; 
pres 



the vert depth in ft of the cen of grav of m c s n below the surf; and 62.5. Or if only the 
against m c s n be sought, mult together the area of the vert projection a o s n\ the vert 
depth of the cen of grav of m c s n; and 62.5. 

In Fig 8 also, the total pres against e f g his found by rule in Art 1 ; while 
the hor and vert pressures against it are found as in Fig 7, by using the projec¬ 
tions e fk i, and high. In Fig 7 the vert pres is downward: while in Fig 8 
it is upward; but this circumstance in no respect affects the rule. 


sloping 


Rkm. 1. At any given depth, the pres, perp to anv given surf, is the same 
in all directions; but Figs 7 and 8 show that the total pres oblique to a given 
surf will be less than the perp one at the same depth ; because an oblique pro¬ 
jection of a surf must be less than the surf itself, which last is the projection 
when the pres is perp to it. Thus, iu a reservoir, the total pres perp to a 
ide, as m ns c, Fig 7, is greater than either the vert or the hor pres upon it. 


1JL 






























226 


HYDROSTATICS 


Again, let Fig 9 represent a conical vessel full of water; 

its base be, 2 ft diam ; its vert height a n. 3 it; theu the circumf of the base will be 
6.2832 ft; the area of the base 3.1416 sq ft; the leugth of its slant side a b or a c, 3.16 

ft; the area of its curved slanting sides will be ^—:— ~ 9.93 sq ft; and the 


vert depth of the cen of grav of the slanting sides will be at two-thirds of the vert 
height a n from the apex u, or 2 ft. 

Here, to hud the total pres against the base, we have by rule ju Art 1, 3.1416 X 3 
X 62.5 = 589.05 0)3. For the total pres against the slant sides, by the same rule, 
9.93 X 2 X 62.5 = 1241.25 lbs. For the vert pres upward'against the entire area of the 
slant sides, we have given the area of the base (which is here the hor projection of 
the slant sides) = 3.1416; and the vert depth of theceu of grav of the slant sides, 2 ft, 
3.1416 X 2 x 62.5 = 392.7 lbs, the upward vert pres. 



Fio 9 

o 


Therefore, 


a 


rr: 



Finally, for the hor pres in any given direction against the slant sides of one half of the cone, we 
have the vert projection of that half, represented by the triangle a b e, with its base 2 ft, and its perp 
height 3 ft; aud consequently, with an area of 3 sq ft. The depth of its cen of grav is 2 ft; theretore, 
3 X 2 X 62.5 = 375 lbs, the reqd bor pres.* 

In Fig 10, which represents a vessel full of water, the total pres 
against the semi-cylindrical surf a v e m d k, and perp to it, must be 
also hor, because the surf is vert; but inasmuch as the surf is curved, 
this total pres, as found by rule in Art 1, acts against it in many di¬ 
rections, which might be represented by an infinite number of radii 
drawn from o as a center. But let it be reqd to find the hor pres in 
lbs, in one direction only, say parallel to o e, or perp toad; which 
would be the force tending to tear the curved surf away from the fiat 
sides a b nv, aud d c s k, by producing fractures along the lines a v 
aud d k; or which would tend to burst a pipe or other cylinder. In 
this case, mult together the area of the vert projection a dkv in sq 
ft; the depth of the cen of grav of the curved surf in ft; (which, in 
the semi-cylinder would be half of e to, or of o i;) and 62.5. Since 
the resulting pres is resisted equally by the strength of the vessel 
along the two lines a v and d k, it is plain that each single thickness 
along those lines need only be sufficient to resist safely one half of it; 
and so in the case of pipes, or other cylinders, such as hooped cisterns 
or tanks. See Art 17. 

Should the pres against only one half of the curved surf, as e d m k 
be sought, and in a direction parallel to o d, tending to produce frac 


m 


' » 

1 1 

1 1 

• • 

1 1 

■ 

...In 

\ 1 

.V ; i 

\ 



EiolO 


s 


tures along the lines e to, aud d k, then use the vert projection oemi; with the same depth : and 62.5 
as before. 

It follows, that if the face of a metallic piston be made concave or convex, no more pres will be reqd 
to force the piston through any dist, than if it were flat; tor the pres against the face of the piston, 

in the direction in which it moves, must be measured by the area of a projection of that face, taken 

at right angles to said direction ; aud the area of said projection will be the same in all three cases. 

Rem. 2. If a bridge pier, or oilier construction. 

Fig 10 3^, be founded on sand or gravel, or on any kind of 

foundation through which water may find its way underneath, even in a very thin 
sheet, then the upward pres of the water will take effect upon the pier; and will tend 
to lift it, with a force equal to the wt of the water displaced by the pier; (Arts 18, 

19. In other words, the effective wt of the submerged portion of the pier, 

will be reduced 62 H fl>s per cub ft; or nearly the half of the ordinary wt of masonry. 

But if the foiludation be on rock, covered with a layer 

of cement to prevent the infiltration of water beneath the masonry, no such effect 
will be produced; but on the contrary, the vert pres downward, afforded by the bat¬ 
tering sides of the pier, and bv its offsets, will tend to hold it down, and thus increase its stability • 
w hich, in quiet water, will then actually be greater than on land. 



Fig 10^ 


Art. 5. To divide a rectangular surf, 
whether vert as a b e d, or inclined as 
m n op, Fig 11, whose lop a b or in n is 
level with I he surf of the water, by a 
hor line x 2, such (hat the total pres 
against the part above said hor line, 
shall equal that against the part be¬ 
low it. 

Rule. Mult one half of the length of b c, or m p, as the case 
may be, by the constant number 1.4142; the prod will be 6 2 
or m x. ’ 

Ex. Let b c= 12 ft. Then 6 X 1.4142 = 8.4852 ft; or b 2. 
Let tap = 16 ft. Then 8 X 1.4142 = 11.3136 ft, or to x. 

Rem. The line x 2, thus found, must not be confounded with 
the cen of pres, which is entirely diff. See Art 8. 



Art. <». In a rectangular surf, whether vert as abed, or in. 
eliue<l as in nop, Fig II. whose top a b or m n coincides with 
the surf of the water, to fin<l any number of points, as 1.2. Av 
through which if hor lines, as 12.r, Arc. be drawn, they\vill 
divide the given surf into smaller rectangles, all of which 
shall sustain equal pressures. 

Rule. First fix on the number of small rectangles reqd. Then for point 1 from the ton molt ti „ 
number 1, by this number of rectangles. Take the sqrt of the prod. Mult this sq i t bv toe entire length 


* In a sphere filled with a fluid the total inside pres = 3 times wt of fluid 























HYDROSTATICS 


227 


J e or m p, as the case may be. Di v the prod by the number of rectangles. The quot will be the dist 
b 1. or n 1, as the case may be. 

For the dist b 2, or n 2, proceed in precisely the same way; only instead of the number 1, use the 
number 2 to be mult by the number of rcctaugles: and so use successively the numbers 3, 4, 5, &o, 
if it be reqd to find that number of poiuts. 

Ex. Let b e — 10 ft; and let it be reqd to fiud 2 points, 1 and 2, for dividiug the rectangular surf 
abed into 3 rectangular parts, which shall sustaiu equal pressures. Here we have for point 1, 


17.32 


3 rectangles 


-=5.773 ft= 6 1. 


24.49 
3 rectangles 


■ =8.163 ft ~b 2. 


1X3 = 3. The sq rt of 3=1.732. And 1.732 X 10 (or 6 c) = 17.32. And 
For point 2, we have 

2X3 = 6. The sq rt of 6= 2.449. And 2.449 X 10 (or 6 c) = 24.49. And 
And so for any number of poiuts. 

Rem. 1. This rule will be found useful in spacing the cross* 
bars of lock^ateN; the hoops around cylindrical cisterflcs; 
and the props to a structure, like Fig; a. 

Rem. 2. For dividing; any surf, as o b c d. Fig; 12. which is not 

rectangular, in the same manner, 

with an accuracy sufficient for most practical purposes, per¬ 
haps the following method is as convenient as any. 

Hulk. First div the surf, as in Fig 12, iuto several small 
hor parts, equal or not, at pleasure. Then by Kule in Art 1, 
find the pres on each part separately, as is supposed to bo 
done in the numbers on the left hand of the fig. The sum of 
these (in this case 15510) Is the total pres against the entire 
surf o b c d. Now suppose we wish to div this surf in 4 parts 
bearing equal pres; first div 15510 by 4 = 3878. Then begin¬ 
ning at the top, add together a number of the separate 
pressures sufficient to amount to 3878; by this means fiud 
point 1. Then proceed with the additio’n until the sum 
amounts to twice 3878, or 7756, which will indicate point 2; 
and in the same manner fiud point 3, by adding up to three 
times3878, or 11634. Then the hor dotted lines ruled through 
points 1. 2, and 3, will give the reqd divisions approximately. 
In this manner the hoops of conical, and other shaped ves- 
Fig. 12. sels, may be spaced nearly enough for practical purposes. 


1620 


2060 

2500. 


Total = 15510 



1180 


c a 


Art. 7. The transmission of pressure through water. Wa¬ 
ter, iu common with other fluids, possesses the important 
property of transmitting pres equally in all directions. TIiuh, 

suppose the vessel, Fig 13, to be entirely closed, and filled with water; 
and suppose the transverse area of T,C, D, aud E, to be each equal to one 
sq inch. Then, if by means of a piston, or otherwise, a pres of 1 lb. 1 
ton, or any other amount, be applied to the one sq inch of area of T, C, 
D, or E. every sq inch of the inner surf of the vessel, and of the pipe a, 
will instautly receive, at right angles to itself, an equal pres of 1 lb, or 
1 ton, &c; in addition to the pres which it before sustained from the 
water itself; and this will occur if the vessel consist of parts even miles 
asunder; as, for instance, if T were miles distant from E; and uuited 
to it by a iong series of tubes. If the vessel were a strong steam boiler 
full of water, a single pres of a few hundred pounds at T, C, &c, would 
burst it. See also fig 3%. 

The hydrostatic press acts on this prin¬ 
ciple. Any body, within the vessel, would also receive 
an equal additional pres on each sq inch of its surf. 

If the top of T be open, the air will press upon the sq inch of the exposed surf of water to the extent 

of nearly 15 lbs ; aud the same degree of pres will also be transmitted to every sq inch of the interior 

surf of the vessel, and its connecting tubes; but no danger of bursting will result from this atmo¬ 
spheric pres, beoause the air also presses every sq inch of the outside of the vessel to the same extent. 

Air, and other gaseous fluids, transmit pres equally in all 
directions, like liquids; but not as rapidly. 




fig. so as to serve us a hinge 


Art. 8. The center of pressure. Let Fig 14 

represent a vessel full of water, and suppose the side P to be perfectly 
loose, so as to be thrown outward by the slightest pres of the water from 
within. Now, there is but one single point, P, in every surf so pressed, 
no matter what its shape may be, to which if we apply a force equal to 
the pres of the water, and in a direction opposite to said pres, the side P 
will be thereby prevented from yielding. Such point is called the cen¬ 
ter of pressure. It must not be understood by this that the actual 
amount of pres of the water against that part of the surface which is 
above the hor dotted line passing through P, is equal to that of the water 
below said line ; but that the sum of the products of the several pressures 
above it, mult by their several leverages, or vert dists from P, is equal 
to the sum of the products of the pressures below, mult by their lever¬ 
ages ; or, in other words, that the sum of the moments around the point 
P, of the pressures above the line, is equal to the sum of the moments 
of those below it; so that if a hor iron rod b b were passed entirely 
through the side P, at the same level as the dotted line, as shown in the 
for the side P to turn on, the side would have no tendency to turn. 










































228 


HYDROSTATICS, 


Art. 9. To find the cen of pres of a quiet 
fluid, against a plane surface. Fig 15. 

1. The center or pressure of a quiet tluid against any plane surface 
whose width is uniform throughout its depth, whether said surface be 
vertical, as eo, or inclined, as ca, (or inclined in the opposite direction:) 
and whose top c, or c, coincides with the hor water surf; is distant vert 
below the water surf, two-thirds of the vert depth, ax, from said water 
surf to the bottom of the plane; as at n and i. inasmuch as a hor line 
at % of the depth of s x, intersects both ca and eo at % of their lengths 
respectively, we might sav at once that the center of pres against a plane 
of uniform width is at two-thirds of its leuglh below the water surface. 

Throughout Art 9 any measure, as yard, foot, or inch 
Ac, may be used. 

2. But if the hor top a, or o. Fig 16, of the rectangular plane ag, or 
o ft, be covered to some depth with water, then the vert depth » m, of the 
cen of pres d, or e, below the surf of the water, will be equal to 

o „ cube of *c — cube of sw 

4 0 f-—-- 

** square of * c — square ot s to 

where sc is the vert depth of the bottom, and s to the vert depth of the 
top, of the pressed surf, below the water surf. Or, in words: From the 
cube of sc, take the cube of sto; aud call the rent a. Then, from the 
square of s c, take the square of s to; and call the rem b. Div a by b, 
aud take two-thirds of the quot for s to. 



3. When a plane surf of any shape whatever, whether 
rectangular, triangular, or circular, &c; whether vert as 
op, Fig 17, or inclined as nin, is entirely immersed, so as to 
be pressed over the entire area of both sides • but by diff 
depths of water on its two sides; then the cen of pres coin¬ 
cides with the cen of grav of the pressed surf. 

In the 3 foregoing figures the supposed surfaces are shown 
edgewise, so that their widths do not appear. 


base a b. 


5. 


cen of pres x, will also be in the line am which bisects the base; 
but ax will be % of am. 



Fig-. 18. 


6 If any plane triangle abc, Fig 19, base up, and hor; have its base 
a b covered’ to some depth nd, with water; then the cen of pres o, will 
be in the line cs which bisects the base ; and no will be equal to 

mx 2 -j- (2 mx X to a) -j- 3 m a 2 
(i»jf 2ma) X 2. 



Fig. 19. 


7. The center of pres against any 
rvano rectangular surface. Fig 20, 
whether vert as mn, or inclined as 
po, or wx-, having its top coinciding 
with the surf of the water; and 
pressed by diff depths of water on 
its opposite sides, as shown in the 
tig: will be vert below the upper 
water surf, a dist equal to 



( 


( . r * 70 / vert \ / . . sq of vert \ / vert area of vert \ 

area of surf m n, x depth a b \ / area of surf * dept h rb - dep , h X p( „. c ' * deptu rb \ 

or p o, or w x A - J \cn. or eo, or sx —\ ur ore u,or s x ~ — J 

, • m “ c ' : / ,1 1'ii/» rtf asiy'f ~ 


area of surf 


.m u, or po, 0 ? 


r / wx X half of a b) - ( c uToTt o'or 1 x * half of r b). 

























































HYDROSTATICS. 


229 


8. To find the cen of pres against either a circular, or an elliptic surf, pressed on 
one side only; whether vert, or inclined; and having its tup either coinciding with 
the surf of the water, or below' it. 

Call the vert depth of the cen of pres below the water surf, h. 

The vert (or inclined, as the case may be/ semi diam of the surf, r. 

The vert dist of the cen of the pressed surf, below the water surf, d. 
r 2 

Then, A = — + d. In a vert circle with top at surf, A = 1% rad. 



Art. 10. Walls for resisting’ the pres of quiet water. A study 

of what we have said on retaining-walls for earth, 
will be of service in this connection. It is of course 
assumed that the water does not find its way under 
the wall; and that the wall cannot slide. In making 
calculations for walls to resist the pres of either earth, 
or water, it is convenient to assume the w all to be but 
one foot in length; (not height, or thickness;) for then 
the number of cub ft contained in it, is equal to that 
of the sq ft of area of its cross-section, or profile; so 
that these sq ft, when mult by the wt of a cub ft of 
the masonry, give the wt of the wall. In ordinary 
cases, it is well for safety to assume that the water 
extends down to the very bottom line of the wall. 
Now, by Art 1, the total pres of quiet water, against 
the rectilineal back of a wall, whether vert or sloping, is found in lbs, by mult to¬ 
gether the area in sq ft of the part actually pressed, (or in contact with the water;) 

half the vert depth of the water, in ft, (being the vert depth of the cen of grav of a 
rectilineal back, below the surf;) and the constant 62.5 lbs; and this total pres is 
alw'ays perp to the pressed area. 

W hen the back of the ivall is vert , as in Fig 20%, this pres p is of course less than 
when it is battered; and is also hor; and it tends to overthrow the wall, by making 
‘it revolve around its outer toe, or edge t. The cen of pres is at c; cs being % the 
jert depth o n; in other words, the entire pres of the water, so far as regards over¬ 
throwing the wall as one mass, (see Art 1, of Force in Rigid Bodies,) may be consid¬ 
ered as concentrated at the point c; where it acts with an overthrowing leverage 11, 
(see Arts 46, 47, 49 of Force). The pres in ii s, mult by this leverage in feet, gives the 

moment in ft-its of the overturning force; (see Art 49, 
Force in Rigid Bodies.) The wall, on the other hand., 
resists in a vert direction go, with a moment equal to its 
wt (supposed to be concentrated at iis cen of grav </,) mult 
by the hor dist a t, which constitutes the leverage of the 

1 )^^ wt with respect to the point t as a fulcrum. If the mo- 

ment of the water is greater than that of the wall, the 
latter will be overthrown; but if less, it will stand. 

Rf.m. 1. Art 49 of Force in Rigid Bodies, will sufficiently 
explain the subjects of moments and leverage; and make 
it evident that the same principle applies also to sloping 
backs, as in Fig 21. Here the overturning moment of the 
water is equal to its calculated pres p X its leverage tl; 
while the moment of stability of the wall is equal to its 
wt X its leverage at. By aid of a drawing to a scale, we may on this principle ascer¬ 
tain whether any proposed wall will stand. For we have only to calculate the pres p ; 
then apply it at c, and at right angles to the back ; prolong it to l; measure 11 by the 
same scale. Then calculate the wt of wall; find its cen of grav q; draw g a vert, and 
measure the leverage at. We then have the data for calculating the two moments. 
For finding the cen of grav, see Cen of Grav, Trapezoid, p 348. 

Ri m. 2. If the water, instead of being quiet, is liable to waves, the wall should 
be made thicker. 

























230 


HYDROSTATICS. 



Fig. 22. 



Fig. 23. 



Fig. 24. 


\rt 11 To find the thickness at base of a wall required to be 

safe against overturning under the pres of quiet wateT level with ns top, and 
pressing against its entire vert back. C aution. See Art. 13, p 231. 


(1st) Vertical wall. Fig 22. 


Thickness _ Height 
in feet = in feet x 


I Factor of safety * Height the proper decimal 
'VT xspgrt.of 'tffl = 1,1 fe,!t X in fuliow " ,|! “ ble 


To change a vert wall into a battered one, see Art. 8, p 691. 


(2d) Right angled triangular wall. Fig 23. 

Thickness Height / Factor of safety * Height the proper decimal 
in feet = in ?eet X \ 2X sp gray of wall ~ iu leet X in followiu « table 
= thickness, mo, of vertical wall X 1.225. 

Notwithstanding their greater thickness at base, such triangular walls con-j 
tain, as seen by the tig, not much more than half the quantity of masonry reqd 
for vert ones of equal stability. This is owing to the fact that their cent of 
grav is thrown farther back; thus increasing the leverage by which the wt of 
the. wall resists overthrow. 


(3d) Wall with vertical back and sloping face. Fig 24. 


Thickness 
at base = 
in feet 


V 


(Ht 2 , ft X factor of safety *) -f (batter A n 2 , ft X sp grav of wall) 
3 X specific gravity of wall 


= Height in feet X the proper decimal in the following table. 


Fi<r OO 

* 

Sp. Gr. 

Lbs per 
Cub Ft. 

Resist = 1.5 pres. 

Resist = 2 pres. 

Resist = 3 pres. 

Dressed Granite... 

2.5 

156 

.447 

.516 

,6"3 

Dressed Sandstone 

2.2 

137 

.477 

.550 

.674 

Mortar Rubble. 

2. 

125 

.500 

.578 

.707 

Brickwork. 

Fig. 23. 

1.8 

112 

.527 

.609 

.746 

Dressed Grauite... 

2.5 

156 

.548 

.633 

.775 

Dressed Sandstone 

2.2 

137 

.584 

.675 

.826 

Mortar Rubble. 

2. 

125 

.613 

.707 

.866 

Brickwork. 

1.8 

112 

.646 

.746 

.913 




U 


Resist = 

~ 1.5 pres. 


Resist = 

r 2 pres. 


Fig. 24. 

c 

c 

o ** 

Batter 

Batter 

Batter 

Batter 

Batter 

Batter 

Batter 

Batter 


2 3 

l in. to 

2 ins. to 

4 ins. to 

6 ins. to 

1 in. to 

2 ins. to 

4 ins. t.. 

6 ins. to 


GO 

Jo 

a foot. 

a foot. 

a foot. 

a foot. 

a foot. 

a foot. 

a font. 

a foot. 

Dressed Granite... 

2.5 

156 

.449 

.458 

.487 

.532 

.519 

.526 

.551 

.593 

Dressed Sandstone 

2.2 

137 

.480 

.488 

.515 

.558 

.552 

.5w) 

.583 

.622 

Mortar Rubble...... 

Brickwork.\ 

2. 

125 

.502 

.510 

.536 

.578 

.571 

.586 

.609 

.646 

1.8 

112 

.530 

.539 

.562 

.602 

.610 

.618 

.640 

.674 


* Factor of safety = 


Required momen t of stability of wall 
overturning moment of water 


See p 229. 


































































HYDROSTATICS. 


231 


Art. 12. Table showing liow the stability of a wall sustain* 
ing water is affected by a change in the form of the wall; 

the quantity of masonry remaining the same. Rem. When the base of a tri¬ 
angular wall, of sp grav 2, is less than § the, ht, the stability is greatest when 
the water presses the vert side; but if the base exceeds J the ht, the stability 
is greatest with the water on the battered side. Caution. See Art. 13. 



All these walls contain precisely the same 
quantity of masonry. The masonry is supposed 

to be mortar rubble, weighing 125 Its percubicfoot; or twice as much 
as water ; or about the same as ordinary rough mortar rubble, if 
the sp gr of the masonry is actually greater or less than this, the 
safety also will be greater or less, in precisely the same proportion. 

Base in 
parts of 
height. 

Approx 
resist of 
wall. 

1 

Vertical wall. 

5 

] 5 

2 

Face vertical; back batters one-tenth height. 

.55 

1.8 

3 

“ “ “ “ one-fifth “ . 

.6 

2.2 

4 

“ “ “ “ one-fourth “ . 

.625 

2.6 

5 

“ “ “ “ one-third “ . 

.667 

3.5 

6 

“ “ “ “ four-tenths “ . 

.7 

4.9 

7 

“ “ “ “ one half “ . 

.75 

14.fi 

8 

Back vertical ; face batters one-tenth height. 

.55 

1.6 

9 

“ “ “ “ one fifth “ . 

.6 

2.1 

10 

“ “ “ “ one-fourth “ . 

.625 

2.2 

11 

“ “ “ “ one-third “ . 

.667 

2.4 

12 

“ “ “ “ four-tenths “ .. 

.7 

2.C 

13 

“ “ “ “ one-half “ . 

.75 

2.9 

14 

Back and face, each batter one-tenth height. 

.6 

2.2 

15 

“ “ “ “ one-fifth “ ... 

.7 

3.4 

16 

“ “ “ “ one-fourth “ . 

.75 

4.6 

17 

“ “ “ “ one-third *' . 

.633 

9.0 

18 

“ “ “ “ four-tenths “ . 

.9 

36.0 


Art. 13. Inability of wall or foundation to crush under 
unequal distribution of pressure. Arts 11 and 12 apply only to 
the stability of a rigid wall resting upon a rigid base, and therefore incapable 
of failure except by overturning as a whole. They show that the stability is 
greatest when the water presses against the sloping side. But in practice the 
point where the resultant of all the pressures on the base of the wall cuts the 
base, must not be so near to either toe as to endanger a crushing of wall or 

of foundation. This consideration often makes it l>est to let the water press 

against the vert back, notwithstanding the consequent loss in stability. 

Thus, Fig 25 represents, to scale, a dam wall at Poona, India, designed by Mr. 

Fife, C. E., of England. It is of mortar rubble, of 150 
lbs per cub ft. Its total vert height is 100 ft; thickness 
uv at base, 60 ft 9 ins; at top, rx, 13 ft 9 ins. The front 

ru slopes 42 ft in 100 ft; and the back xv, 5 ft in 100 ft. 

Its foundation is 7 feet deep; but we here assume that 
the water presses against its entire back xv. Through 
the cen of grav G draw Gs vert. From c, where the 
direction of the pres P of the water strikes Gs, lay off 
cn by scale = 139.6 tons (of 2240 lbs) water pres against 1 
ft in length of xv; and cf = 249.4 tons wt of 1 ft length 
of wall. Complete the parallelogram enmt of forces. 
Its diag e m represents the resultant of all the pressures 
upon the base uv, and cuts the base at a, 20 It back from 
the toe u. Doing the same with the 151.4 tons pres p 
against ru, we get the resultant op, which is greater 
than cm, and cuts the base (at i) only 12.7 ft from the 
toe v, or 7.3 ft less than a is from u. 

Hence, when the water presses against xv the wall is 
less liable to fracture or crushing, and the earth foun¬ 
dation uv is more evenly loaded,and hence less liable to 
vield unequally so as to cause cracks in the wall. On 
this account xv is made the back of the wall, although 
the moment of stability of the wall is then only 2.2 (calling the overturning 
moment of the water 1),'while if the water pressed against ru it would he 3, or 
36 per cent greater. 

For rules governing- distribution of pressure, see Art 14, p 231a. 













































231a 


HYDROSTATICS. 


Art. 14. Distribution of pressure over the base uv, Figs 25, A, 
B, C and D. Let 

uv = the length of the rectangular base, or of any rectangular surface common 
to two bodies which are pressed against each other; or (since the width ot 
the surf is taken as 1) - the area of that surf. . 
p _ t p e resultant of all the extraneous forces pressing one of the two sorts 
against the other. The amount of its pres is represented by the trapezoid 
uosv. Fig A; triangle uov. Fig B; diff of triangles uod — dvs, l;>g C; tri¬ 
angle uod, Fig D; or bv the parallelogram untv in all lour 1-tgs. We 
confine ourselves to cases where P cuts the base in the center ot its width 
measured at right angles tuwt). 

p, = (he resultant of all the resistances of the several points of the other suit. 
It is necessarily equal and opposite to P. 

un — tt - ” — = —— the mean pressure ) 

uv uv V per unit of area u v. 

U o — the maximum pressure l 

vs — (Figs A, B, C) the minimum pressure ' 

This Art. applies equallv whether the surf is hor, vert or inclined, and 
•whether the forces are oblique to it (Figs A,B,C, D); orperp to it (Fig 71, p 357). 
If oblique, a portion of the resistance R is that ot lriction. See Art. ltt, p 31-1. 



The parallelogram untv represents the pres uniformly distributed over uv , 
as it would be if P cut uv at its center, e. The intensity of the pres, or its 
amount per unit of area of uv, would then be everywhere = un. 

But when, as in our Figs, P cuts uv at any other point, a, its pres is unequally 
distributed; the nearer toe, u, receiving the max pres, mo. 

If (Figs A, B) ea does not. exceed one-sixth uv, then 
' • /. . 6 ea\* . 

maximum pressure uo = un II -f J ; and 


minimum pressure vs = (2 un) — uo (See * and Figs.) 

If e a = one-sixth u v (Fig B), this becomes 

maximum pressure uo = 2 un; and minimum pressure vs = 0. 

If an exceeds one-sixth uv (Fig C), vs is less than 0, or minus; i e, the 
toe v has a tendency to rise, and actually does so unless prevented either by the 
rigidity of the pressed part, u d, of the base, or by a tensile resistance (as by the 
adhesion of cement) in the remainder, dv. If it is thus resisted by tension in 
dv (Fig C) we still have 

/ 6 ea\* 

maximum pressure uo = un (1 + J > an( ^ 
minimum pressure vs = (2 un) — uo, 

* Demonstration. See Figs A, B, C. Suppose for a moment that R remains i,t e (so as to be 
nniformlv distributed over uv, ns represented by »» n tv) whileP is at o; then p ai. i: form n couple 
(= one of the forces, as P, X their perpendicular distance from each other). This couple tends 




















HYDROSTATICS. 


231 b 


vs being the tension, (or minus pres) per unit of area, at v. This rarely hap¬ 
pens; for no ordinary mortar or cement can be depended upon to resist such 
tensions as might thus occur. 

To find the neutral axis, d, or point of no pres; lay off uo and vs, and draw 
os. The total pressure, uod, on v d, is — A uo.ud = P plus the tension, dvs, in 
dv = P + ivs.dv. 

But when (as usual) uv is practically incapable of resisting tension, P is sim¬ 
ply concentrated upon a portion, ud Fig D (= 3 ua) of uv: 3 ua is then practi¬ 
cally the base; the remainder, dv, being idle. We then have 

P 

mean pres on 3 u a = -— 

3 ua 


maximum pressure uo = 2 X mean pres on 3 ua = 2-= unX 

iua 

pres at d, and from d to v, — 0. 




In any case, the maximum pressure uo should evidently not exceed the safe 
strength of the masonry or soil. Therefore (if, as usual, no part of the base is 
to be relied upon for tension) ea in feet must not exceed 


uv, in ft X ( .5- 

\ 3 uv 


2 P 




X safe load per sq ft 
Pand the safe load being in the same unit; as both in lbs, or both in tons, etc. 


First class rubble in cement mortar, or good cement concrete, should he safe 
with 8 tons per sq ft, which limit will rarely be reached. Sound earth or gravel 
‘i foundations, sunk to a depth sufficient to protect them from frost, rain, sliding 
A etc, should be safe with from 2 to 4 tons per sq fi. 


to press u downward and raise v. This tendency causes (and is resisted by) a second couple, con¬ 
sisting of an increase, nco. of the pres on u e, and an equal decrease, tes. of the pres on«t); i e, 
tes, instead of pressing on ev as before, would be called upon to help resist the first couple. The 
points, r and x, at which the resultants of the resisting forces nco and (es act, are opposite to 
the cens of grav, G and G', of those triangles. Each is therefore distant % of half uv from e; 
and their dist rar from each other, measd along uv, is twice this, or % u v. The resisting couple 
must (p 351) be equal and opposite to the first one. Hence each of its forces, nco and tes, must 

be=_fir st couple __ P . ea _ t j,e additional pressure, nco, on tt e. (If the dist apart 

leverage of nco and tes %uv 

of the 2 forces is thus measd along u v in both couples, said dists will be in the same proportion to 
each other as the leverages.) The mean additional pres on ue, or the middle ordinate of the triangle 

nco. is = n> . c0 — P -JLg and the max additional pres, no, is = twice this, = — ea 

He %uv 2 %uv 


UV = P X 

4 U V 


eaX — X — = tt n —* ° 
2 u v uv 


^because 


P 

U V 


is = u n 


> 


And 


uo = u 7i + no = un + 


^tt n 


6 e a 
u v ■ 


= un 



6 ea 
u v 


t Fig. D. ua=y-ea; 3 « a = 3 Ay-- e a 


tto = 2 _g_ = 2 ^- =u« 2 


3 it a 3 tt a 


■Or-**) s (- 5 -i-;)' 


1 Fig D. Safe load = uo = it tiX - - - ! (see t) or 3 (.5--®) 

\ uv/ 


V V 


3 (.5-^) 
\ u v / 


3 ( . 

or 3 — = (3 X .5)-—-- I 

it v uv X s afe load 


safe load 


ea _ , 

or ■— = .5 — 

u v 


2 P 


3 u v X safe load 




















232 


HYDROSTATICS. 


Art. 15. The points a and i, Fig 25, are called centers of pressure 
upon the base, or centers of resistance of the base. If similar points, as 
d and z, be found in the same way for other lines, as / h, by treating a part (as 
rxhf) of the wall as if it were an entire wall; a slightly curved line joining 
these points is called the line of pressure. Thus, ba is the line of pres¬ 
sure when the water presses against xv. Each point, as d, in ba, shows where 
any joint, as fh, drawn through that point, is cut by the resultant of all the 
forces acting upon said joint, hi is the line of pres when the water presses 
against ru. These lines do not show the direction of the resultants. Thus, at a, 
the latter is cm, not ba. The angle between the direction of the resultant and a 
line at right angles to the bed or joint, must be less than the angle of friction 
(p355) of the materials forming the joint. 

If from the end to or y of the resultant of the pressures upon any joint, we 
draw to 2 or yl hor, then c2 or ol (as the case may be) measures the entire vert 
pres on that joint: and to 2 or yl measures the hor pres against the back of the 
wall, which tends to cause sliding at the same joint . If the direction of the re¬ 
sultant comes within the limit stated in the preceding paragraph, to 2 or yl will 
he less than the frictional resistance to sliding, which last is = c2 (or ol) X the 
coetf of friction for the surfaces forming the joint. Hence sliding cannot take 
place. 

Such sliding never occurs in the masonry of walls of ordinary forms. Good 
mortar, well set, aids to prevent sliding, but it is better not to rely upon it. See 
Rem 2, p 683. But entire walls have slidden on slippery foundations. See Art. 
9, p 692. 

Art. 16. In California is this (lam of a mining reservoir, built of 

rough stone without mortar, founded on rock. Height, 70 feet; base, 60: top, 6; 

water-slope, 30 feet; outer-slope, 14. To prevent leaking tlie 
water-slope is only covered with 3-inch plank bolted horizon¬ 
tally to 12 by 12 inch strings, built into the stone-work. All 
laid with some care by hand, except a core of about one-fifth of 
the mass, which was roughly thrown in. Cost about S3 per cubic 
yard. It has been in use since 1860. 

Rem. If a dam is compactly hacked with earth 
at its natural slope, and in sufficient quantity to prevent tbe 
water from reaching the dam, the pressure against the dam will 
not be increased. 



Art. 17. To find the thickness of a cylinder to resist safelv the 

pressure of water, steam, Ac, against its interior. If riveted, see next page. 

Where the thickness is less than one-thirtieth of the 
ratlins, as it is in most cases, the usual formula 
... Thickness pressure 

( ] in inches ~ wfe strength X radiuS * 

is employed. It regards the material as being subjected only to a direct tensile 
strain, which is sufficiently correct in such thin shells. 

For somewhat greater pressures and thicknesses. Professor 
F. Reuleaux (Der Konstrukteur, p 52) gives 

Thickness pressure / pressure \ 

in inches - V + 2 X safe strength / X radlu8 -* 

I or very great pressures and thicknesses, as in hydraulic 
presses, cannons, Ac, Professor Reuleaux (Konstrukteur, p 53) gives Lame’s 
formula: 


Thickness 

W in inches 




safe strength + pressure 




X radius.* 


safe strength—pressure 

The three formulae give results as follows, pressures and strengths in lbs per 
square inch: 


Diameter. 

Radius. 

Pressure. 

Safe 

tensile 

strength. 

Thickness, inches. 

Formula (1). 

Formula (2). 

Formula (3). 

20 inches. 

10 inches. 

50 

10000 

.05 

.050125 

05 

U 

U 

500 

it 

.50 

.5125 

5] 3 

U 

it 

5000 

a 

5.00 

6.25 

7.32 


The thicknesses given by the formulae appropriate to the several pressures are 
printed in heavy type. It will be seen that in these cases the results differ 
but slightly, except for very great pressures. ier 


* In all three formulae take the radius in inches, and the prex.su 
in nn-unAx oner Rmifirf. inch. 


re and strength 



































HYDROSTATICS. 


233 


Rem. 2. Want of uniformity in the cooling- of thick castings makes 

them proportionally weaker than thin ones, so that in order to reduce thickness in impellent cases 
we should use ouly best iron remelted 3 or 4 times, by which means an ult cohesion ot' about 30000 
lbs per sq inch may be secured. Hut even with this precaution no rule will 
a PI>ly Hrtlely in practice to cast cylinders whose thickness exceeds either 

about 8 to 10 ins, or the inner rad however small. 

I nder a pres of 8000 lbs per sq iftch, water -will ooze through cast 
iron S or 10 ins thick ; and under but 250 tbs per sq inch, through .5 inch. 

Table of thicknesses »f Single-riveted wrought iron pipes, 

tanks, standpipes, &c, by the above rule, to bear with a safety of 6 a quiet pressure of 1000 ft head 
of water, or 434 lbs re - sq incti ; the ult coll of fair quality plate iron being taken at 4S0u0 tbs per sq 
inch, or at 8000 lbs for a safety of tt; which is farther reduced to 8000 X .56 — 4480 lbs. to allow tor 

weakening by rivet holes; for single-riveted cyls have but about .56 of the 
strength of the solid sheet; and douhle-rivoted ones about .7. With the 

above pres and other data, the rule here leads to thickness = .1016 X inner rad in ins. 

for a similar table for tanks, see p 803; and for cast iron and 
lead pipes, foot of this, and top of next page. (Original.) 


Ths. 

Ins. 


Di. 

Ins. 

Ths. 

Ins. 


Di. 

Ins. 

Ths. 

Ins. 

.025 


5 

.254 


16 

.813 

.051 


6 

.305 


18 

.914 

.076 


8 

.406 


20 

1.016 

.102 


10 

.508 


22 

1.117 

.152 


12 

.600 


24 

1.219 

.203 


14 

.711 


27 

1.371 


Di. 

Ins. 

Ths. 

lliS. 


Di. 

Ins. 

Ths. 

Ins. 


Di. 

Ins. 

Di. 

Ft. 

Ths. 

Ins. 

30 

1.52 


60 

3.05 


120 

10 

6.09 

33 

1 68 


66 

3.35 


132 

11 

6.70 

36 

1.83 


72 

3.66 


144 

12 

7.31 

42 

2.13 


84 

4.27 


192 

16 

9.75 

48 

2.44 


96 

4.88 


240 

20 

12.19 

54 

2.74 


108 

5.49 


288 

24 

14.63 


For a loss head or pressure, or for any safety less than C, it is safe and 

near enough in practice, to reduce the thickness of wrought iron cyls in the same proportion as said 
head, pres, or safety is less than the tabular one. 

Rouble-riveted cylinders. Fairbairn says, are about 1.25 times as strong 
as single-riveted. Hence they may be one-fifth part thinner. Tap-welded 
ones are nearly 1.8 times as strong as single-riveted; and hence may be only 
.56 as thick. 

Many continuous miles of double-riveted pipes in California have 

been in use for years with safetys of but 2 to 2.6. In one case the head is 1720 ft, with a pres of 746 Ibs 
per sq inch ; diam 11.5 ins; thickness, .34 inch ; safety, 2.6 by rule p 232 for such iron as in our table. 

Cast iron city water pipes must be thicker than required by formula 
(1), p 232, in order to endure rough handling and the effects of “ water-ram ” 
(due to sudden stoppage of flow, see second Rem, p 234), and to provide against 
irregularity of casting and the air bubbles or voids to which all castings are 
more or less liable. In the following table the ultimate tensile strength of cast 
iron is taken at 18,000 lbs per square inch. Column A gives thicknesses by Mr. 
J. T. Fanning’s formula (Hydraulic Engineering, p 454). 

Thickness) _ (pres, lbs per sq in + 100) X bore, ins / _ bore, ins \ 

in inches / .4 X ultimate tensile strength ' ° \ 100 /" 

These correspond with average practice. The addition of 100 lbs to the pres is 
made in order to allow for water-ram. Column B gives thicknesses by formula 
(1), p 232, taking coefficient, of safety = 8 (thus making safe tensile strain = 2250 
lbs per square inch) and adding three-tenths of an inch to each thickness given 
by the formula: 


Head in feet 50 


100 

200 


300 

500 

1000 

Pressure, 

21.7 

43.4 

86.8 

130 

217 

434 

Ids per sq in. 












Bo:e, ins. 



Thickness of pipe, in Inches. 




A 

R 

A 

It 

A 

It 

A 

It 

A 

B 

A 

R 

2 

.36 

.31 

.37 

.32 

.38 

.34 

.39 

.36 

.42 

.40 

.48 

.51 

3 

.37 

.81 

.38 

.33 

.40 

.35 

.42 

.40 

.45 

.45 

.54 

.60 

4 

.39 

.32 

.40 

.34 

.42 

.38 

.45 

.42 

.50 

.50 

.61 

.72 

6 

.41 

.33 

.43 

.36 

.47 

.42 

.50 

.48 

.57 

.60 

.75 

.94 

.8 

.45 

.34 

.47 

.38 

.52 

.47 

.57 

.55 

.66 

.70 

.90 

1.15 

10 

.47 

.35 

.50 

.40 

.56 

.50 

.62 

.60 

.74 

.81 

1.04 

1.35 

12 

,49 

.36 

.53 

.42 

.60 

.54 

.67 

.66 

.82 

.91 

1.18 

1.57 

16 

.55 

.38 

.60 

.46 

.70 

.62 

.79 

.77 

.98 

1.10 

1.46 

2.00 

18 

.57 

.39 

.63 

.48 

.74 

.65 

.85 

.84 

1.06 

1.21 

1.60 

2.20 

20 

.61 

.40 

.67 

.50 

.79 

.68 

.91 

.90 

1.15 

1.31 

1.75 

2.50 

24 

.66 

.42 

.73 

.53 

.87 

.77 

1.02 

1.01 

1.30 

1.51 

2.03 

2.84 

30 

.74 

.45 

.83 

.59 

1.01 

.89 

1.19 

1.19 

1.55 

1.82 

2.46 

3.47 

36 

.82 

.47 

.93 

.65 

1.15 

1.01 

1.36 

1.37 

1.80 

2.12 

2.88 

4.11 

48 

.98 

.53 

1.13 

.77 

1.42 

1.24 

1.70 

1.73 

2.28 

2.73 

3.73 

5.38 















































234 


HYDROSTATICS 


Table of thickness of lea<I pipe to bear internal pressures with a 

safety of 6; taking the ultimate cohesion of lead at 1400 tbs per sq inch. By rule on p 232.. 

Rem. Although these thicknesses are safe againstquiet pressures.they might not 
resist shocks caused by too sudden closing of stop cocks against running water. See Service pipes, p 099. 




Heads in Feet. 




Heads in Feet. 


4) 

xa 

o 

a 

100 

200 

300 

400 

500 

0) 

o 

S 

100 

200 

300 

400 

! 500 

►—i 

a 

Pres in lbs per sq inch. 

a 

Pres in lbs per sq inch. 

0) 

u 

43.4 

86.8 

130 

174 

217 

o 

u 

43.4 

86.8 

130 

174 

217 

« 






PQ 







Thickness in Inches. 


Thickness in Inches. 

X 

.026 

.055 

.089 

.128 

.171 

1 

.102 

.221 

.357 

.511 

.682 

% 

.038 

.083 

.134 

.192 

.256 

IX 

.127 

.276 

.447 

.639 

.853 

X 

.051 

.111 

.179 

.256 

.341 

ix 

.153 

.332 

.536 

.767 

1.02 

% 

.064 

.138 

.223 

.320 

.427 

1H 

.178 

.387 

.626 

.895 

1 20 

X 

.076 

.166 

.268 

.383 

.512 

2 

.204 

.442 

.714 

1.02 

1.36 

V, 

.089 

.193 

.313 

.447 

.597 








Rem. The valves of water-pipes must be closed slowly, and 

the necessity for this precaution increases with their diams. Otherwise the sud¬ 
den arresting of the momentum of the running water will create a great pressure against the pipes 
in all directions, and throughout their eutire length behind the gate, even if it be many miles : thus 
endangering their bursting at any point. Hence stop-gates are shut by screws, ’ which pre¬ 
vent any very sudden closing; but in large diams even the screws must be worked very slowly to 
avoid bursting. 


Art. 18. The buoyancy of liquids. When a body is placed in a 

liquid, whether it float or sink, it evidently displaces a bulk of the liquid equal to the bulk of the 
immersed portion of the body ; and the body in both cases, and at any depth, and in any position 
whatever, is buoyed up by the liquid with a force equal to the wt of 
the liquid so displaced. Thus, if we immerse entirely in water a piece 
of cork c. c. Pig 26, or any body of less sp gr than water, the cork will 
by its wt, or force of gravity, tend to descend still deeper; but the 
upward buoyant force of the water, being greater than the downward 
force of gravity of the cork, will compel the latter to rise with a 
force equal to the dirt' betweeu the two. In this case, the cork receives 
a total downward pres equal to the wt of the vert column of water 
above it, shown by the vert lines in vessel 1 ; and a total upward 
pres equal to the wt of the eolumu shown in vessel 2. The dirt' be¬ 
tween these two columns is evidently (from the figs) equal to the 
bulk of the cork itself; therefore the diff between their wts or 
pressures, (or, in other words, the buoyancy of the water,) is equal 
to the wt or pres of the water which would have occupied the place 
of the cork: or, in other words, of the water which is displaced by 
the cork. This diff, or buoyancy, will plainly be the same at any 




.Pi 3 £6 


depth whatever of entire immersion. Now the cork, if left to itself, will continue to rise until a por¬ 
tion of it reaches above the surf, as in vessel 3 ; so that the downward pressing column ceases to 
exist; and the cork is then pressed downward only by its own wt. But as it now remains station¬ 
ary, we know (from the fact that when two opposite forces keep a body at rest, they must be equal to 
one another) that the upward pres of the water must be equal to the wt of the cork' But the upward 
pres of the water arises only from the shaded eolumu shown in vessel 3; and this column is las in 
the case of total immersion) equal to the bulk of water displaced. 


. Therefore, in all cases, the buoy- 

ancy is equal to the wt of water displaced ; and when the body floats on the surf, the buoyancy or 
the wt of water displaced, is also equal to the wt of the body Itself. ’ 


If the immersed body c, c, he of iron, or any other substance spe¬ 
cifically heavier than water, the diff between the upward and downward pres will of course remaia 
the same; or equal to the wt of water displaced. But the wt of the body is now greater than that 
of the water which it displaces; or. in other words, the downward force of gravity of the body is 
greater than the upward buoyant force of the displaced water; and therefore the body descends, or 
sinks, with a force equal to the diff between the two. Thus, if the body be a cub ft of cast iron, 
weighing 430 lbs, while a cub ft of fresh water weighs R'JX lbs, the iron will descend with an effective 
force of only 450 — 62!^ = 387.5 lbs. 


in' the inunerscd body has the same sp gr as the fluid, it will 

neither rise nor sink ; but will remain wherever it is placed; because then the wt of the body, and 

the buoyancy of the water, are equal. 

The air also buoys bodies upward to an extent equal to the 
wt of air displaced: therefore, although a pound of iron, and a pound of 
feathers, weiehed in the air. will balance each other yet in the exhausted bell-glass of an air-pump 
the feathers will outweigh the iron, by as much as the bulk of air which they displaced outweighs 
the bulk of air displaced by the iron. 


A balloon rises in the air on the same principle that cork 
rises in water. Tts ascending force is equal to the diff between its wt when 
full of gas. and the wt of the bulk of air which it displiices. The balloon does not actually tend to 
rise, but to descend; but the atr being, bulk for bulk, heavier than the ball i n r ushes the latter 
upward with more force than the gravity, or the wt of the balloon, exerts to ririia; it down. So also 
warm smoke has no tendency in itself to rise. It Is punned tip by the heavi cold air No suhstanoe 
tends to rise ; but all tend downward toward the center of the earth. no 8UDSta “ce 











































BUOYANCY, FLOTATION, METACENTER, ETC. 235 


The downwd force of gr;vv may be regarded (p 347) as concentrated at the cen of 
gray G of a floating body. The upwd pres, or buoyancy,f of the water may similarly 
be regarded as acting at the cen of gr W of the displaced water.* W is also called 
the center of pressure, or of buoyancy, of the water; and a vert line 
drawn through it is called the axis, or vertical, of buoyancy, or of flo¬ 
atation. Ordinarily,J W shifts its position with every change in that of the body. 
Tims in L it is at the ceu of gr of the rectangle o o b b; and in N at that of the tri- 


jl- 

G 

0 / 

wj 



N, 


AW/ 




7 0 i l 

f . 

e/ 

/ P 

»/ /_ 

/ 1 

f 

AW/ 4 

7 fw / 




angle a a v. 

When a floating 

body, L. P or R, is at j T F 

rest, and undisturbed 
by any third force, 
as F, it is said to be 

in equilibrium, 

and G and W are then 

in the same vert line YI w t /iw/o, : 

11 Figs L and R, or 
eeFigP; which line 
is called the axis, 
or vertical, of 
equilibrium.^ n 

When a third force, as F, causes the axis of equilib to lean, as in Figs N, 0 and S, 
then if a vert line be drawn upwd from the cen W of buoy, the point M where said 
line cuts said axis, is called the metacenter of the body. || G and W are then no 
longer in the same vert line;J and the two opp and vert forces, grnv and buoy, act¬ 
ing upon those points respectively, form a “couple” (Case 3, p 351); and, when the 
third force F is removed, they no longer hold the body in equilib, but cause it to 
rotate. If (as in Figs 0 and S) the positions of G and W are then such that the 
metacenter M is above the cen of gr G, this rotation will tend to restore, the body to 
its former position, and the body is said to have been (before the application of the 
third force F) in stable equilibrium.^ But if(asinN)M is below G the direc¬ 
tion of rotation is such as to upset the body, by causing it to depart further from its 
fminer position, and the body is said to have been in unstable equilibrium.^ 


R 


t 


G 

t w 



The tendency or moment in ft-lbs of a floating body either to upset or to right itself, is,(Case 3, p351,) 

_ the wt of the body (or the equal .. the hor dist between W M and G H, 
upwd pres of the water) in lbs * Figs N, O and S, in ft. 

The third force F may of course be so great as to overpower the tendency of the body to right it¬ 
self. Thus, a ship may upset in a hurricane, although judiciously loaded and ballasted for ordinary 
winds. A hor section »f a body at water-line is called its plane of flotation. 


* The body is in fact stcfed upon by other forces, such as the hor 

pressures of the water against its immersed portions; but as all of these in any one given direction 
are balanced by equal ones in the opposite direction, they have no effect upon the forces G and \V. 
It is also acted upon by the air, which presses it downwards with a force of 14.75 lbs per sq inch : hut 
this is balanced by an equal pres Qf the surrounding air upon the surface of the water, and which is 
transmitted tart-7, vert upwards against the immersed bottom-of the floating body. 

f This buoyancy is made up of the parallel upward pressures of the 
innumerable vert filaments of the displaced water as shown by Fig 26, and 

the axis of flotation is their resultant, as in the case of parallel forces. 

t The shape of a body (as that of a sphere or cylinder TJ) may be such that the position of its cen of 
buov W, relatively to that of its cen of gr G, is not changed bv the rotation of the body about a given 
axis (as any axis of the sphere or the longitudinal axis of the cyl), but remains constantly in the 
same vert line with G, so that the body, in rotating, remains in equilib. Such a body is said to be 
in indifferent equilibrium about said axis. But if a cyl U be made to 
rotate about its transverse axis x x, it plainly comes under the remarks on Figs R and S, and may 
(before rotating) be in either stable or unstable equilib about that axis according to the way in which 
its wt is distributed. 

II This metacenter shifts its position on the line t t according to the inclination of the latter. 

'4 Uneven loading, instead of a third force, may cause a vessel at rest to 
lean as at P; and yet the vessel so leaning may be in equilib; for its axis e e of equilib may be vert, 
although not coinciding with the axis «1‘ symmetry of the vessel, as it does at 

t t ill T j. 

H In floating bodies, this may sometimes (as in Figs R and S) be the case even when the cen of 
buoy W ( not the metacenter ) is helow the cen of gr G ; because, when the body is forced to lean, W 
moves to another point in it, and this point may be such as to bring M above G. W is always below 
O in bodies nf uniform density, floating at rest if any part of the body is above water. When sucb 
bodies are entirely submerged, W and G coincide. 







































236 


HYDRAULICS. 


Art. 19. A body lis'bte'r than water, if placed a* 
the bottom of a Vessel containing- water, wall not 
rise unless the water can get under it. to buoy it, 
or press it upward, sis the air presses a balloon or 
smoke upward. Thus, if one side of a block of light wood, 

perfectly flat ami smooth, be placed upon 'be similarly flat aud smooth bottom of a 
vessel, aud held there until the vessel is filled with water, the downward pres will 
keep it in its place, until water insinuates itself beneath through the pores of the 
wood. But if the wood be smoothly varnished, to exolude water from its pores, it 
will remain at the bottom. 

On the other hand, a piece of metal may be pre¬ 
vented from sinking in water, by subjecting it to a suffi¬ 
cient upward pres only, while the downward pres is excluded. Thus, if the bottom 

I °f an open glass tube, t, Fig 27, and a plate of iron to, be made smooth enough to be 

1 Jr. 07 water tight when placed as iu the fig; and if in this positiou they be placed in a 
' vessel of water to a depth greater than about 8 times the thickness of the iron. the 
upward pres of the water will hold the irou in its place, and prevent its sinking,’ 
because it is pressed upward by a column of water heavier than both the column of air, aud its own 
weight, which press it downward. On this principle iron ships boat. 

Rf.m. 1. A retaining-wall, as in Fig 2S, 
founded on piles, may be strong enough to re¬ 
sist the pres of the earth e behind it, in case water does not fiud 
its way underneath; and yet may be overthrown if it does; or 
even if the earth ss around the heads of the piles becomes satu¬ 
rated with water so as to form a fluid mud. In either case, the 
upward pres of the water p.gaiust the bottom of the wall will vir¬ 
tually reduce the wt of all such parts as are below the water surf, 
to the extent of 62*4 ®> s per cub ft; or nearly one-half of the or¬ 
dinary wt of rubble masonry in mortar. 

Rem. 2. Although the piles under a wall, as in Fig 28, may be 
abundantly sufficient to sustain the wt of the wall; and the wall 
equally strong in itself to resist the pres of the backing e; yet if 
the soil ss around the piles be soft, both tiiey and the wall may be pushed outward, aud the latter 
overthrown by the pres of the backing e. From this cause tile wing-walls of bridges, when built 
on piles in very soft soil, are frequently bulged outward and disfigured. In such cases, the piling, 
and the wooden platform on top of it, should extend over the whole space between the walls; or else 
some other remedy be applied. 




Art.20. Draught of vessels. Since a. floating body displaces a wtof liquid 

equal to the wt of the body, we may determine the wt of a vessel and its cargo, by ascertaining how 
many cub ft of water they displace. The cub ft, mult bv 62*4, will give the reqd wt in lbs. Suppose, 
for instance, a flat-boat, with vert sides, 60 ft long. 15 ft wide, and drawing unloaded 6 ins, or .5 of 
a ft. In this case it displaces 60 X Id X .5 = 150 cub ft of water; which weighs 450 X 62J4 = 28125 
lbs; which consequently is the wt of the boat also. If the cargo then be put in. and found to sink 
the boat. 2 ft more, we have for the wt of water displaced by the cargo alone, 60 X 15 X 2 X 62*4 = 
112500 lbs : which is also the wt of the cargo. So also, knowing beforehand the wt of the boat and 
cargo, and the dimensions of the boat, we can find what the draught will be. Thus, if the wt as before 

140625 

be 140625 lbs, and the boat 60 X 15, we have 60 X 15 X 62*4 — 56260; and-— 2 5 ft the required 

56250 

draught. In vessels of more complex shapes, as in ordinary sailing vessels, the calculation of the 
amount of displacement becomes more tedious; but the principle remains the same. 


Art. 21. C'omprc.HKi Utility of liquidM* Liquids are not entirely in¬ 
i’- inpressible ; hut for most engineering purposes they may be so considered. The bulk of water is 
diminished but about one-thousandth part by a pres of 324 lbs per sq inch, or 22 atmospheres ; vary¬ 
ing very slightly with its temperature. It is perfectly elastic; regaining its original bulk when the 
p es is removed. 


HYDRAULICS. 


Art. 1. Hydraulics treats of the flow or motion of water through 

pipes, aqueducts, rivers, and other channels; also through orifices or openings of various kinds; of 
machinery for raising water; as well as that in which water furnishes the moving power. The science 
of hydraulics, in many of its departments, is but imperfectly understood; therefore, some of the rules 
given on the subject are to be regarded merely as furnishing close approximations to the truth. 

On the flow of water through pipes. 

Inasmuch as the experiments on which the following rules are based, were made with pipes care¬ 
fully laid in straight lines; and perfectly free from all obstructions to the flow of the water, some 
allowance must in practice be made for this circumstance. Workmen do not lay long lines of pipes 
in perfectly straight lines; it is almost impossible to avoid very numerous, although slight devia¬ 
tions, both vert and hor; the soil itself, in which the pipes are imbedded, especially when in embkt, 
will settle unequally ; especially in streets liable to heavy traffic, which not otilv fre tenth deranges,* 
hut, occasionally breaks water pipes whose tops are 3 or 4 ft below the surf. The material used for 
calking the joints, may be oarelessly left projecting into the interior, and thus can obstructions - the 
water is frequently muddy, or is impregnated with certain salts, or gases, which form deposits, or 
incrustations, which materially impede the flow. . Moreover, the pit ; themselves are not 

cast perfectly straight, or smooth, or of uniform diam ; and irregular swelling ... by producing eddies. 


















HYDRAULICS, 


237 


if 




' 


v 


retard the flow as well as contractions ; and accumulations of air do the same. Under the most favor¬ 
able circumstances, therefore, it is expedient to make the diams of pipes, eveu for temporary pur¬ 
poses, sufficiently large to discharge at least 20 per ct more than the quantity actually needed ; and 
if there is occasion to anticipate deposit, or incrustation, a still larger allowance should be made in 
permanent pipes, especially in those of small diam ; because in them the same thickness of incrusta¬ 
tion occupies a greater comparative portion of the area. Perhaps it would be best to allow an equal 
increase, of say from to 1inch, to each diam, whether great or small; inasmuch as the thickness 
of incrustation will be the same for all diams, or nearly so. The cost of pipes does not increase as 
rapidly as their discharging capacities; thus, if the diam be increased only yL part, the disch will 
be increased about 25 per cent; if % part, nearly 50 per cent; if part, the disch will be doubled. 

Within these limits, the increase of thickness for the larger diams, and the increased 
expense of laying, will add but little to the cost; which will therefore augment only a little more 
rapidly than the diams. 

The increased diam involves no waste of water; since the disch may be regulated by stopcocks. 



The term HEAD or TOTAL HEAD of water, as applied to the flowage of 

water through canals, pipes, or openings in reservoirs, &c, means the vert dist iv or p o, Fig 1, from 
the level surf, mi, of the water in the reservoir, or source of supply, to the center (or more properly to 
the cen of grav) o, of the orifice (whether the end of a pipe, r o, t o, v o, z o, l o ; or any other kind of 
opening) through which the disch takes place freely, into the air; or the vert dist a u, or f g, from 
the same surf, m i, to the level surf, g u, of the water in the lower reservoir; when the disch takes 
place under water. Thus, in the case of disch into the air, the vert dist i v or po, is the total head 
for either of the pipes r o, t o, v o, z o, or l o; and i k is the head for the orifice, k, in the side of the 
reservoir. And for disch under water, au, or f g, is the head for either the pipe j, or the opeuiug n; 
without any regard whatever to their depths below the surf of the lower water; which, according to 
the older authorities, do not at all affect their disch. 

A portion of a pipe may have a head greater than the total head of the entire pipe. Thus the 
point 6 in the pipe l o, has a head 6 1; while the entire pipe has only the head p o. 

Both in theory and in practice it is immaterial as regards 
the vel, and the quantity of water discharged. whether the 
pipe is inclined downward, as ro, Fig 1; or hor, as vo ; or in¬ 
clined upward, as lo% provided the total head po , and also 
the length of the pipe, remain unchanged. If one pipe is longer 
than another, its sides will evidently present more friction against the water, and thus diminish the 
vel and the quantity of disch. The inclined pipes, r o, lo, being of course a little longer than the 
hor one uo, will therefore each disch a trifle less water; but if the hor one were extended slightly 
beyond o, so as to give it the same length as the others, then each of the three would disch the same 
quantity in the same time. 


Art. 1 n. Divisions of the Total Head. In any pipe, as so.ro , 

to. v o, z o, or l o, Fig 1 , the total head has three distinct duties to perform; 1st, to overcome the 
resistance to entry at s, r, t. v, z. or /; 2d, to overcome the resistances within the pipe; and, 3d, to 
give to the water, entering the pipe, the uniform velocity with which it actually flows. 

For convenience, we regard the total head as divided into three portions, corresponding to these 
duties; namely, 1st, the entry head; 2d, the resistance, or friction, head; and, 3d. the velocity head. 

Art. 1 tt. The velocity head is the height through which a body must 
fall, in vacuo, to acquire the vel with which the water actually flows into the pipe. It is therefore — 


v 2 

—. io which v is the vel in ft per sec ; and g is the acceleration of gravity, or 32.2 
This head will be found in the table, p 258, opposite to the actual vel. 

Art. 1 c. Experiment shows that, with the usual sharp-edged entry, the en¬ 
try head is, near enough for practice, = half the vel head. If the entry is shaped 
like' Fig 7, scarcely any entry head will be required. But, in pipes louger than about 1000 

diameters, the entry head bears so slight a proportion to the total head, that this advantage is of but 
little importance. It becomes more apparent in shorter pipes. 

Art. 1 fl. In Fig 1 we will assume that for any of the pipes, i s represents 
the sum of the vel and "entry heads. Then the remainder s v, or w o, of the total head, is the 
friction head; or the head which is just sufficient to balance the friction and 
other resistances within the pipe, and thus leave lor the vel head nothing to do hut to impart vel lo 




































238 


HYDRAULICS. 


the unresisting water. If. hr shortening the pipe, or by smoothing its inner surf, we diminish the 
total friction, then a less friction head will he required; but the vel will, at the same time, be 
increased, and this will require a greater vel head, and entry head, so that the three together make 
up the total head, as before. Since the friction is equal to the force or head reqd to overcome it, it 
also is represenied by w o. 

Art. 1 e. The friction head may as in v o, 2 o,and l o, Fig 1, be all above the entrance 
to the pipe, and therefore outside of the pipe : or, as in a pipe laid from s to o, it may be all below 
the entrance, and within the pipe : or. as in r o and t o. it may be partly above, and partly below, the ; 
entrance; and therefore partlv within, and partly without, "the pipe. The vel and disch, after the 
pipe is filled, are not afflbeted by this difference in position of the entry end ; but the pressures in the 
pipe, aud the vels while the water is filling an empty pipe, are affected by it, as explained in Arts 1 l 
and 1 o. 

Art. 1/*. But it is necessary that the entry end of the pipe 
should be placed so far below the surf m i, that there shall he left, 

above the cen of grav of the entry end, at least a head, t s, sufficient to perform the duties of the entry 
and vel heads. If the entry end of any of the pipes be raised above «, a portion of the vel head will 
be in the pipe. In other words, the head ?n the pipe will be more than sufficient to overcome the 
resistances in the pipe: and the surplus will act as vel head, and will give greater vel to the water 
in the pipe. The reduced head thus left above the entry end will plainly be insufficient to maintain 
the supply for the greater vel, aud the pipe will run only partly full. 

In ordinary cases of pipes of considerable length, the sum of the entry and vel heads theoretically 
required, is but a small portion of the total head, and rarely exceeds a foot. Indeed, in a pipe of 
considerable diameter, the upper half of its cross section at the entry end may often be more than 
enough to provide sufficient entry and vel heads above the cen of grav of said cross section : so that 
the top of the entry end might, so far as these considerations alone are concerned, project above the 
surf of the water in the reservoir. But the end of the pipe should in practice always be entirely be¬ 
low the surf; otherwise air and floating impurities will be drawn into it, and cause obstructions. 
Moreover, the water surf of reservoirs is always liable to considerable changes of height; and the 
entry end of the pipe must be placed at such a depth that the water can flow into it with sufficient 
vel w hen at its lowest stages. As before stated, this will cause no diminution or increase of disch,. 

Art. 1 ff. To find flic friction licad reqd for any part of 
a pipe; Knowing the fric head reqd for the whole pipe. Since the friction, in a 
pipe of uniform diam, is (other things being equal) in proportion to its length; aud since w o, Fig 1, 
represents the total friction, or reqd friction head, we have 

Total length . Length of the . . . The friction head reqd 

of the pipe • given portion • • • for that portiou. 

Or, having drawn w o by scale, « w hor, and s o; 

Total length . Length of the . . . A dist. as s c, to be laid 

of the pipe • given portion • • • off from s on s o. 


Or 


A dist, as g 6, to be laid off 
from g on g to. 



b-r 


A 

\ 

i 

\ 

\ 

\ , 


V 


\ i 


Then a vert line, as 5 c, drawn from b or c, and joining g to and s o, gives by scale the friction head 
reqd. 

Art. 1 h. If the pipe is straight, as r o, v o, l o, the friction in any part begin¬ 
ning at the reservoir, as l 6 in the pipe l o, may be found at once by drawing a line 6 *1 vert upward 
from the axis of the pipe at 6 . The line 2 3 will then give the friction in 1 6. It also gives the fric- 
tion in r 4, or in that part of v o which lies between v and the dotted line 1 6 . It must be remem¬ 
bered that all the pipes in Fig 1 are supposed to be of the 
same actual length. T hey would thus end at different points 
o, and strictly, a separate diagram must be drawn for each 
pipe. In a part of the pipe not beginning at the reservoir, 
as in r o, v o. or l o, betweeu points vertically under c and 
a-, the amount of friction is giveu by the line d x , for it is 
plainly — y x — be. 

Art. 1 j. If tlio pipe is vert, as v o. 

Fig 1 A ; let ts(on its axis to) represent, as before, the sum 
of the vel and entry heads. From s, v, and o, respectively, 
draw hor lines s w, v k, and o y. making o y — v o. Draw 
the oblique line s y. Then, to find the friction in any part, 
as vq, begiumn? at the reservoir, from q lay off qd hor, aud 
equal to v q, and draw the vert line o d. crossing s y at g. 
Then b g will give the friction in v q. 

Art. 1 te. If the pipe is curved,and 

if the curvature is uniformly distributed along its length, or 
so slight that it may he neglected; the friction heads reqd 
for the several portions of the pipe, may be found in the 
same way as for straight pipes, as in Art 1 H. Otherwise 
they must be found by proportion, as in Art 1 O. 

Art. 1 l. While water is filling- 
ail empty pipe, the excess of the total head 
above the requirements of friction. &c, gives to the water a 

greater vel than it has after the pipe is filled- 

but this gradually decreases as the advancing water encoun¬ 
ters the friction along the increased lengths of pipe tilled; and finally becomes least when the water 
fills the w hole length, and begius to flow from the disch end, o. But if only the vei and entrv 
heads are left nbove the entry end, as in a pipe laid from s to o, there will plainlv t„. Q0 suc u v 
of total head, and, consequently, no such change of vel during the filling of the i in- , 

When a pipe of uniform diameter is flowing full, and is entirely open at its discharge i 

tbe vel throughout the pipe is equal to that at the out How 


9 


0 


* • \ 

\! \ 

- ] d 


\ 

\ 

M 


Fig.l 




















HYDRAULICS 


239 


Rut if the opening o he contracted, but so shaped (see Fig 7, p 260) that the resistance of its edges 
may be neglected ; then 

Area of , Area of . . vel of . vel throughout 

cross-section of pipe • cross-section of outflow • • outflow • the pipe. 

Art. 1 lit. Of the outward, or bursting, pressure of water 
ill pipes. When any pipe is full of water at rest , the entire head acts as pressure 
head, and the pres is greater than when the water is flowing through it. Thus, at the point 4 iu the 
pipe r o, Fig 1, it is that due to the head 4 1; at the point6 in the pipe l o, it is that due to the head 


Therefore its amount in tbs per sq inch is 


6 1 ; at the point o, in any of the pipes, that due to o p. 

= Total head in ft X .434, as per rule p 224. 

But if an opening be made in any part of the pipe, or of the reservoir, the water will of course be 
•J3t in motion, and although the level in the reservoir be maintained, the pres in all parts of the pipe 
and of the reservoir will be reduced. The greater the area, and number, of such openings, the 
greater will be the vel, and the greater will be the reduction of pres. A part of this reduction is, in 
all cases, due to the vel of the water in the pipe, which requires the consumption of a part of the 
head (the vel head) for its maintenance. Another part is due to resistance to entry, and the remain¬ 
der is due to friction within the pipe. 

Art. 1 n. The foregoing is true of all other vessels, as well as of pipes. Thus, 
if an opening o be made anywhere in a vessel V, Fig 1 B, the pres throughout will be reduced, aud 
the water in the pipes p and g, will no longer stand at the same level i as that in 
the vessel, although the vessel be kept full; but will fall to some lower one, l, 
which will depend for its height upon the relative areas of cross section of the 
orifice o aud of the vessel V. For this reason, two equal openings in a reser¬ 
voir, at equal heights, will always give less than double the disch that one of 
them alone would give: but where the area of the reservoir is very large in 
comparison with that of the openings, the discrepancy is inappreciable. It 
follows, also, from the above, that a vessel V, Fig 1 C, although kept con¬ 
stantly full of water, will weigh, less when the orifice o is open, than when it 
is closed; because the pres on the bottom of the vessel, as well as that on the 
sides, is reduced by the opening of the orifice. 

Art. 1 ©. The pressure head of running water upon 

any point in a pipe between the orifice and the reservoir, is 

the head .. ‘ he head consumed 

due to the , th ® , *“ overcoming re- 
vel at ' entT 5' + sistances in tbe pipe 
♦ head between the reservoir 

thatpomt and the point. 

Thus, at the point 6 in the pipetoFig 1, thepres head is = 3 6 = 1 6 minus (1 


f the total 
~ head on > 
t that point) 


) 



2 + 2 3 ); 1 


2 being 

4 


0 . 


V 


s 



Fiir.l C 


= i «, or = the'sum of the vel and entry heads. At 4 in the pipe r o, the pres head is only 3 4=1 
minus (1 2 + 2 3.) Iu a straight inclined or hor pine, the pres bead at 
any point is thus given by the length of a vert line drawn from the poiut to the 
line s o. 

Art. 1 p. If the pipe has gentle curves, or if the cur¬ 
vature is uniformly distributed along the length of the pipe, the pres head may be 
found in the same wav as for a straight pipe in Art 1 o. But if the curves are of 
considerable extent, aind unevenly distributed along the pipe, first find, by Art l g, 
the friction head reqd for that part of the pipe between the reservoir and the poiut in 
question. Then find the pres head by the above formula. 

Art. 1 Q. For a \ r ert pipe v o Fig 1 A, draw the diagram as 

directed in Art 1 j. Then g d (= t q or a d — (a b + b g) ) gives the pres head at q. 

At the point o in any of the pipes, Fig 1 or Fig 1 A, the pres head is zero, sup¬ 
posing the pipe to be entirely open at that point. 

In a pipe laid along the line s o, Fig 1, the pres head will be zero at all points. 

Art. It*. In a pipe r m or r in’, Fig 1 D, closed at its end, m or 
m', hut having an orifice o between its end aud the reservoir, 

the pres at any point, x, between the orifice and the closed end, is equal to the pres iu the tube 
opposite the orifice, pins that arising from the head o' x 
between the orifice aud the point. If the point, as x’, is 
higher than o. this head is negative, and the pres at x’ 
will be less than that opposite o. If the area of the ori¬ 
fice o is such as to pass the entire flow of the pipe with¬ 
out obstruction, the pres opposite o is zero. Otherwise 
there will be a pres at o varying with the amount of ob¬ 
struction to outflow at that point. 

Art. 1 &. If a vert or oblique pipe be in¬ 
serted into one containing water under pres, the water 
will rise in the first, and the vert ht to which it rises is 
the head producing the pres at the point where the tube is attached. Such tubes are called 
niezometers, or pressure-measurers. In order that the ht of the water in them 
may be known, they are made of glass, at least in that part or their length where the surface of the 
column is likely to be, or eise thev are provided with a floating index. 

The piezometer is used for detecting the positions of ob¬ 
structions in a line of pipes. If the water iu the piezometer is found at any time 
to fall below its proper level, it shows that the pressure in the main pipe a,t that point has become 
diminished by some obstruction in the interval between it and the reservoir; but if the water rises 
above its proper level, it indicates that the pressure there has been increased by an obstruction 

be,mtul the piezometer. By having? several piezometers, ihe pomt at 

which an obstruction has taken place can be approximately ascertained, aud thus much of the labor 
or searching for it, avoided. 

16 























240 


HYDRAULICS, 




Fig.l E 

-4-f-i-! 


Art If If we imagine any pipe, full of water, to be supplied with a number 
of piei.n.etT.nC then a linejoining the tops of the columns of water in the several 

throtighoot, a, ro, . c or l o Fig 

th If'thSp^orific'e 1 aVo be^COMtracted, the hyd grade line must be drawn 

from a to some point, as e, immediately over o, and depending, is “case 

• the points will also be higher than 

-i ft before, because the vel in the pipe is 

*- -i xi reduced by the contraction ; and the 

sum i s of the vel and entry heads 
will be less. 

If the disch at o is 
under water, the effect 
upon the position of the grade liue 
will be the same as that of a con¬ 
traction of the orifice at o. The 
point e will be on tlje surf of the 
lower water, and immediately over o. 

Art. 1 v. If the pipe, of uniform 

diam, (whether discharging freely or through a con¬ 
tract ed opening at o, whether into the air or under 
water), is bent or curved, the hyd grade 
line will still be straight, provided the 
resistances are equal in each equal division of the hor 
length of the pipe, as in Fig 1 E, where equal divisions 
v w, w x, kc. of the total length, correspond with equal 
divisions v a, a b, kc, of the hor length. 

Rut in Fig 1 F, the hyd grade line will take the 
shape s a o. For if, in accordance with Art. 1 G. we 
divide s o into two equal parts, s m, m o. correspond¬ 
ing with the two equal parts vr.ro. of the length of the 
pipe, we obtaiu m c — a e for the head consumed in the 
resistances in vr, leaving only r a for the pres head at r. 

Art. 1 tr. Fig 1 Q shows the effect upon the Iiyd grade line, 
caused by a contraction or other partial obstruction, as at o. Here, a 




Fig.l F 


3 ? : 

v 


(i jp 

<?k 1 


r 7n/ , , 


sk 4A w ' 



btese retard the flow. 



vert line a G drawn from the axis of the wider pipe lo. shows 1st, 
the sum ab (=r i s) of the vel and entry heads for l o; 2d, the total 
friction head b m (— to e) for lo: of which be has been used in 
overcoming the resists between I aud 6, leaving cm as a part of 
the pres head at 6 (which part decreases from s v at l to zero at o); 
and, 3d, a pres head 7716 (= e o) uniform throughout l o. and 
caused by the contraction at o. This last is the total head for 
o o'. Thiis, the line 6' a' shows, 1st, the sum a' b' of the vel and 
entry beads for 00 '; and 2d, its total friction head b'6 ', of which 
c' 6' is the pres head at 6'. 

In practice, owing to pnrtinl obstructions, such as curves, re¬ 
ductions of diameter, etc, the hyd grade line is tarely straight. 

To find tile vel in a pipe like l o'. Fig 1 G, bee 

P 254. 

Art. 1 ar. If the pipe agnyo. Fig 1^, of uni¬ 
form diam, is first filled with water, which is then allowed to flow' 

out at o ; the pipe becomes a siphon. 

The pres at any point, g or y, in the pipe below the 
hyd grade line s o, is given bv vert liDes p u or y v drawn from the 
point to s o. Rut at every point, as >», above s o. there is a nega¬ 
tive pres, or tendency to vacuum, the amount of which is given 
by a vert line, as n r, drawn from the point in questiou to s o. 

This tendency to vacuum at n causes an accumulation there of 
particles of air that have been carried along by the water or which 
find their way into the pipe through imperfect joints, etc; aud 
If this could be prevented, the vel and disch through agnyo would he the 
. same as they would be if the pipe 

|) ^ were revolved about a straight 

line joining its two ends, so as 
to bring each part of it below so. 

But if the water be admitted? 
to the empty pipe at a. while the 
orifice at o is open, t he part agn 
will run full, and will have sn 
for its hvd grade liue, and zn 
for its friction head. But the 
shorter part ?i o has a greater 
friction head t 0 . Consequently, 
the water m it will move more 
rapn.ly tha.ii that in agn. which 
will thus he unable to maintain 
a sum -i.-ut supply at n; aud no 


will be only part full, and will carry off the water as in an open gutter. 





























HYDRAULICS 


241 


The sy phon, or siphon. If one leg a 6 of a bent tube or pipe a be, 
Fig M, of any diam, filled with water, and with both its ends stopped, 
be placed in a reservoir of water, as in the fig; and if the stoppers be 

b then removed, the water in the reservoir will begin to flow out at c, and 
will continue to do so until its level is reduced to t, which is the same as 
that of the highest end c of the pipe or syphon. The flow will then stop. 
The parts a b and b c are called the legs of the syphon, b being its high¬ 
est point; and this is correct so far as relates to it merely as a piece of 
tube; but considering it purely with regard to its character as a hydrau¬ 
lic machine, the part t a below the level of the highest end c. may be en¬ 
tirely neglected; for the water in the reservoir will not be drawn down 
below the level of the highest end, whether that be the inner or the outer 
one. Therefore, if the disch end be above the water in the reservoir, as, 
for instance, at w, no flow will take place. The vert height b o, from the 
highest part of the syphon, to the lowest level t, to which the reservoir 
is to be drawn down, must not, theoretically, exceed about 33 or 34 ft; 
or that at which the pres of the air will sustain a column of water. 
Practically it must be less, to allow for the friction of the flowing water, 
and for air which forces its way in. And still less at places far above sea 
level; for at such the reduced weight of the atmospheric column will not 
balance so great a height of water. In order readily to understand, or 
at any time to recall the principle on which the syphon acts, bear in 
mind that we may theoretically consider the end of the inner leg to be 
not actually immersed below the water surf, but only to be kept precisely 
at it, as the surf descends while the water is flowing out: but may re¬ 
gard the vert dist b o as the length of the outer leg; and a varying dist, which at first is b s, and finally 
b o (as the surf of the reservoir descends) as the length of the inner leg; and that the flow continues 
only while this outer leg is longer than tins inner one. The books are wrong in saying that the outer 
leg b c must be longer than the inner one b a, in order that the water mAy run at all. The principle 
then is simply this: that both these legs b c, and bi, being first filled with water, (the part ia being 
considered at first as a portion of the reservoir , and not of the syphon,) it follows that when the stop¬ 
pers are removed from the ends c and a, the air presses equally against these ends; but the great vert 
head of water b o in the outer leg b c, presses against the air at c, with more force than the small head 
of water bs in the inner leg bi, does against the air at a or i.* Consequently, the water in b c will 
tend to fall out more rapidly than that in b i ; and as it commences to fall, would produce a vacuum at 
b, were it not that the pres of the air against the other end a or i, forces the water up i b, to supply 
the place of that which flows out at c. In this manner the flow continues until the surf of the water 
in the reservoir descends to t. on the same level as c. The pressures of the vert heads bo, bo, in the 
two legs be, bt, being then equal, it ceases. 

The syphon principle may be employed for draining ponds into lower ground at a considerable dist, 
even though an elevation of several feet (in practice perhaps not exceeding about 28 ft above the level 
to which the pond is to be reduced) may intervene. In snch a case an escape must be provided 
at the summit (or summits, if there are more than one) of the bends, for the disch of free air, which 
will inevitably enter, and soon stop the flow, unless this precaution be taken. The air-valve p.297, 



will not answer for this, because as soon as the valve v opens, the syphon becomes 
in effect two separate tubes open at top; and the water will fall in both. An ori¬ 
fice at the escape will be needed for filling the syphon at the start; and to pre¬ 
vent the water thus introduced, from running out. stopcocks must be provided at 
the ends, and kept closed until the filling is completed. 

The greatest pains must be taken to make all the joints perfectly air-tight. 

The motive power or head which causes the flow in a syphon, is the 
vert dist y o, from the surf of the reservoir, to the disch end c; or in other words, 
it is the diff, s o, between the theoretical lengths b s and b o, of the two legs. Con¬ 
sequently, the farther c is below y the more rapid will be the flow ; and it is plain 
that as the surf gradually sinks below y, the less rapid will the flow become. Hav¬ 
ing this head, the entire length ab c of the syphon, and its diam, all in ft, the 
disch may be found approximately by either of the rules given in Art 2 for straight 
pipes. These rules gi ve 55% galls per min, instead of the 45% galls actually dischd 
by Col Crozet’s syphon, with a head of 20 ft, as slated on p242, which see. 


* Said pressure of the air is of course not exerted directly at a or i; but is transmitted to a through 
the water in the vessel; and thence upward to t through the water in the siphon. 














242 


HYDRAULICS 


At Bine Ridge Tunnel, Virginia, Col. C. Crozet constructed a drainage 
svphon 1792 ft long of cast iron faucet pipes 3 ins bore, 9 ft long. Its summit was 
9*ft above the surface of the water to be drained ; and its discharge end was 20 ft 
below said surface, thus giving it a head of 20 ft. At the summit 570 ft from the 
inlet, was an ordinary cast iron air-vessel with a chamber 3 ft high and 15 ins 
inner diam. In the stem connecting it with the syphon was a cut-off stop¬ 
cock ; and at its top was an opening G ins diam, closed by an air tight screw lid. 
At each end of the syphon was a stopcock. To start the flow these end 
cocks are closed, and the entire syphon and air-vessel are filled with water through 
the opening at top of air-vessel. This opening is then closed airtight, and the two 
end cocks afterwards opened; the cut-off' cock remaining open. The flow then 
begins, and theoretically it should continue without diminution, except so 
far as the head diminishes by the lowering of the surface level of the pond. But 
in practice with very long syphons this is not the case, for air begins at once 
to disengage itself from the water, and to travel up the syphon to the summit, 
where it enters the air-vessel, and rising to the top of the chamber gradually 
drives out the water. If this is allowed to continue the air would first fill the en¬ 
tire chamber, and then the summit of the syphon itself, where it would act as a 
wad completely stopping the flow. The water-level in the air chamber 
can be detected by the sound made by tapping against the outside with a hammer. 

To prevent this stoppage, the cut-off at the foot of the chamber is 
closed before the water is all driven out; and the lid on top being removed the 
chamber is refilled with water, the lid replaced, and the cut-off again opened. 
The flow in the meantime continues uninterrupted, but still gradually diminish¬ 
ing notwithstanding the refilling of the chamber; and after a number of refill¬ 
ings it will cease altogether, and the whole operation must then be repeated by 
filling the whole syphon and air chamber with water as at the start. 

At Col. Crozet’s syphon at first owing to the porosity of the joint-caulking, 
which was nothing but oakum and pitch, air entered the pipes so rapidly as to 
drive all the water from the chamber and thus require it to be refilled every 5 or 
10 minutes; but still in two hours the syphon would run dry. The joints were 
then thoroughly recaulked with lead, and protected by a covering of white and 
red lead made into a putty with Japan varnish and boiled linseed oil. But even 
then the chamber had to be refilled with water about every two hours; and after 
six hours the syphon ran dry, and the whole had to be refilled. In this way it 
continued to work. 

In the writer’s opinion an inside, and probably an outside coating of the pipe* 
and air-vessel with the coal pitch varnish, Art 33, p 291, would effect a great im¬ 
provement. 



HYDRAULICS. 


243 


Art. 2. Approximate formula' for the velocity of water in 
straight, smooth, cylindrical iron pipes, as ro, vo, lo, Fig 1, p,237. Having the 
total head po, and the length and diameter of the pipe. 

Approx I coefficient I diam in ft X total head in ft 
mean vel > = m X a 'i — n —— ; —:— tt- — ■ ■ ,. - : — r 

in ft per sec J as below \ total length in It -f- 54 dtams in It 


Table of coefficients 44 m ”, 


/ — r; - t --- 



diameter* of pipe, 

in feet 

• 

/ diam X head 









\ length + 54 diams 

.05 

.10 

.50 

1 

1.5 

2 

3 

4 


m 

ill 

in 

ill 

ni 

in 

in 

ill 

.005 

29 

31 

33 

35 

37 

46 

44 

47 

.010 

34 

35 

37 

39 

42 

45 

49 

53 

.020 

39 

40 

42 

45 

49 

52 

56 

59 

.030 

41 

43 

47 

50 

54 

57 

60 

63 

.050 

44 

47 

52 

54 

56 

60 

64 

67 

.100 

47 

50 

54 

56 

58 

62 

66 

70 

.200 ) 
and over j 

48 

51 

55 

58 

60 

64 

67 

70 


The above coefficients are approximate averages deduced from a large number 
of experiments. In most cases of pipes in fair condition, carefully laid, and 
straight or nearly so, they should give results within say from 5 to 10 per cent 
of the truth. But slight differences as to roughness etc, may cause much 
greater variations, especially in small pipes, for in such a given roughness of 
surface bears a greater proportion to the whole area of cross section than in a 
pipe of large diameter. Extreme accuracy is not to be expected in such matters. 

As in a river the velocity half way across it, and at the surface, is usually 
greater than at the bottom and sides, so in a pipe the velocity is greater at the 
center of its cross section than at its circumf. The mean velocity 
referred to in our rules is an assumed uniform one which would give the same 
discharge that the actual ununiform one does. 

Hence 

Discharge _ Mean velocity y Area of cross section 

in cub ft per sec in It per sec A of pipe in sq ft. 

See tables pp 125 to 140, 157, 247. 

1 cubic foot = 7.48052 U. S. gallons 
1 U. S. gallon = .13368 cubic foot = 231 cubic inches. 

For K litter’s formula, as applied to pipes, see p 244. 


* For intermediate diameters, etc, take intermediate coefficients from the table by simple pro 
portion. 


























244 


HYDRAULICS. 


In the case of loop pipes with low heads, the sum of the velocity and entry 
heads (see pp 237,238) is frequently so small that it may be neglected. Where 
this is the case, or where their amount can be approximately ascertained, Kut¬ 
ter’s formula, although designed for open channels, may be used. This 
formula is the joint production of two eminent Sw iss engineers, E. Ganguillet 
and W. R. Kutter, but for convenience it is usually called by the name of the 

latter. , _ . . „ 

It is, properly speaking, a formula for finding the coefiicient c in the well 

known formula, 


/diameter 


JHeaii velocity = c i/mean radius X~slope 

’•« <>• i tu ( i 

According to Kutter, 


X slope 


For English measure. 

C = 


41 .6 + + -M 1 ! 

slope n 


„ .00281 \ 
( 41 - 6 +r 

\/mean rad in feet 


1 + 


For metric measure. 

23 + + 1 
slope n 
~~ 7~” .00155 \ 

1+ _1 

iXnieau rad in metres 


See also tables of c, pp 275 to 278. 


The mean radius is the quotient, in feet or in metres, obtained by divid¬ 
ing the area of wet cross section, in square feet or in square metres, by the wet 
perimeter (see below) in feet or in metres. In pipes running full, or exactly half 
lull, and in semicircular open channels running full, it is equal to one fourth 
of the inner diameter. 


The wet perimeter is the sum, a b co Figs 28,29,30, p 271, of the lengths, 
aft, be, co, in feet or in metres, found by measuring(at right angles to the length 
of the channel) such parts of its sides and bottom as are in contact with the 
water. In pipes running full, it is of course equal to the inner circumference. 


The slope is = 


friction head wo Fig 1, p 237 

length of pipe measured in a straight line from end to end. 


= sine of angle wso, Fig 1. 

In open channels, this becomes 

slop** = fall of water surface in any portion of the length of the channel 

length of that portion 

= fall of water surface per unit of length of channel 

= sine of the angle formed between the sloping surface and the horizon. 

The number indicating the slope in any given case is plainly the same for 
English, metric and all other measures. 


“ ** ” is a “ coefficient of roughness ” of w*et perimeter, and of course 
depends chiefly upon the character of the inner surface of the pipe. For iron 
pipes in good order and from 1 inch to 4 feet diameter, n may be taken at from 
.010 to .012; the lower figures being used where the pipe is in exceptionally good 
condition. 

If the diameter, or the mean radius, is in feet, metres etc, the velocity will be 
in feet, metres etc, per second. 


















rv 


HYDRAULICS. 


245 


TEbc diameter or tbe slope, required lor a given velocity, 

inay Ik; found by trial as follows: assume a diameter, or a slope, as the case may 
be; take the corresponding c from tables, pp 275, etc. Then say 

Approx Din in required 

for the given vel 

Approx Slope required 

in r the given vel 

With the approximate diameter 

v'— cj /'mean radius X slope. If t/is near enough to the given velocitv, the 
assumed diameter (or slope) is the proper one. If not, try again, assuming a 
greater diameter or slope than before if v' is less than the required velocity, and 
vice versa. 




mean 

radius 


X 4 = 


velocity 


. c l/mean rad 


( velocity \ ‘ 
c 7/slope/ 

HIS/ \ 


X 4 

velocity \ 2 
c\/\ diam / 


(or slope) and c, thus obtained, say 


To reduce cub ft to IT. S. gallons, mult by 7.48. Since, therefore, 8 cub ft are equal to 60 gals, (very 
nearly,) if we divide the cub ft per 24 hours, by 8, we get the number of persons that may be 
daily supplied with 60 gals each, by a pipe constantly running full, and at the vel given in the third 
col. This condition does not exist in city water-pipes; the water in them being comparatively stag¬ 
nant. Therefore, the results of the rule and table do not at all applj' to them. 

Rem. If the pipe, instead of being straight, has easy curves. 

(say with radii not less than 5 diams of the pipe,) either hor or vert, the disch will not be materially 
diminished, so long as the total heads, and total actual lengths of pipes remain the same; provided 
the tops of all the curves are kept below the hydraulic grade line ; and provision be made for the 
escape of air accumulating at the tops of the curves. See Fig 44 A, p 297. 

Notwithstanding’ w hat is said about bends on pages 255, 256, we 

advise to make the radius as much more than 5 diams as can conveniently be done. 

1 To find either the area of pipe, opening, or channel-way ; 
or the mean vel; or the quantity discharged, when the other two 
are given. This applies to openings in the sides of vessels, to rivers, and to all other channels as 
well as to pipes. 

Disch in cub ft Disch in cub ft 

Area in _ P er secoa d Mean vel _ per secoud 

sq feet. mean vel in * n per sec area in 

feet per sec. sq feet. 

Disch in cub ft _ area in ^ mean vel in 
per second sq feet A ft per second. 

Or all the terms may be in inches instead of feet; and minutes or hours instead of seconds. 















246 


HYDRAULICS 


TABLE 2. Weight of water (at 62*4 lbs per cub foot) con¬ 
tained in one foot length of pipes of different bores. (Original.) 


Bore. 

Ins. 

Water. 

Lbs. 

Bore. 

Ins. 

Water. 

Lbs. 

Bore. 

Ins. 

Water. 

Lbs. 

Bore. 

Ins. 

Water. 

Lbs. 

Bore. 

Ins. 

Water. 

Lbs. 

Bore. 

Ins. 

Water. 

Lbs. 

X 

.00331 

2. 

1 3581 

3X 

5.0980 

IX 

19.098 

V*X 

61.877 

22 

164.33 

X 

.02122 

X 

1.5331 

4. 

5.4323 

X 

20.392 

14. 

66.545 

23 

179.60 

¥» 

,04775 

X 

1.7188 

X 

6.1325 

8. 

21.729 

X 

71.384 

24 

195.56 

X 

.08488 

% 

1.9150 

X 

6.8750 

X 

23.109 

15. 

76.392 

25 

212.20 

% 

.13263 

X 

2.1220 

X 

7.6601 

X 

24.530 

X 

81.568 

26 

229.51 

* X 

,10098 

X 

2.3395 

5. 

8.4880 

X 

25.993 

16. 

86.916 

27 

247.51 

% 

.25994 

X 

2.5676 

X 

9.3580 

9. 

27.501 

X 

92.434 

28 

266.18 

1. 

.33952 

v» 

2.8063 

X 

10.270 

X 

30.641 

17. 

98.121 

29 

285.53 

X 

.42969 

3. 

3.0557 

X 

11.225 

10. 

33.952 

X 

103.97 

30 

305.57 

X 

.53050 

X 

3.3156 

6. 

12.223 

X 

37.432 

18. 

110.00 

31 

326.27 

% 

.64190 

X 

3.5862 

X 

13.262 

11. 

41.082 

X 

116.20 

32 

347.66 

X 

,76392 

% 

3.8673 

X 

14.345 

X 

44.901 

19. 

122.56 

33 

369.74 

% 

.89654 

X 

4.1591 

X 

15.469 

12. 

48.891 

X 

129.10 

34 

392.48 

% 

1.0398 

% 

4.4615 

7. 

16.636 

X 

53.049 

20. 

135.81 

35 

415.90 

y» 

1.1936 

X 

4.7745 

X 

17.846 

13. 

57.379 

21. 

149.73 

36 

440.00 


And in larger pipes, as the squares of their bores. Thus a pipe of 40 or « 

CO ins bore, will contain 4 times as much as one of 20 or 30 ins bore; and one of y 3 ,, X as much as 
one of % inch. At 62X Bis per cuh ft, a sq inch of water 1 ft high weighs .432292 of a lb. 


I 3 






. 

I 


































HYDRAULICS 


247 


TABLE 3. Areas and Contents of Pipes; and square roots 

of Itianis. (Original ) Correct. 


Diam. 

in 

Ius. 

Diam. 

in 

Feet. 

Area in 
sqft, also 
cub ft, 
in 1 foot 
length of 
Pipe. 

Sq. rt. 
of 

Diam. 
in Ft. 

Diam. 

in 

Ins. 

Diam. 

in 

Feet. 

Area in 
sq ft, also 
cub ft, 
in 1 foot 
length of 
Pipe. 

Sq. rt. 
of 

Diam. 
in Ft. 

Diam- 

in 

Ins. 

Diam. 

in 

Feet. 

Area in 
sq ft, also 
cub ft, 
in 1 foot 
length of 
Pipe. 

Sq. rt. 
of 

Diam. 
in Ft. 

X 

.0208 

.0003 

.145 

4. 

.3333 

.0873 

.579 

15. 

1.250 

1.227 

1.118 

5-16 

.0260 

.0005 

.161 

X 

.3438 

.0928 

.588 

x 

1.271 

1.268 

1.127 

Vs 

.0313 

.0008 

.177 

X 

.3542 

.0985 

.596 

X 

1.292 

1.310 

1.136 

7-16 

.0365 

.0010 

.191 

Vs 

.3646 

.1040 

.604 

X 

1.313 

1 353 

1.146 

X 

.0417 

.0014 

.204 

X 

.3750 

.1104 

.612 

16. 

1.333 

1.396 

1.155 

9-16 

.0469 

.0017 

.217 

Vs 

.3854 

.1167 

.621 

X 

1.354 

1.440 

1.163 

% 

.0521 

.0021 

.228 

X 

.3958 

.1231 

.629 

X 

1.375 

1.485 

1.172 

11-16 

.0573 

.0026 

.239 

Ys 

.4063 

.1296 

.637 

X 

1.396 

1.530 

1.181 

X 

.0625 

.0031 

.250 

5. 

.4167 

.1363 

.645 

17. 

1.417 

1.576 

1.190 

13-16 

.0677 

.0036 

.260 

X 

.4271 

.1433 

.653 

y* 

1.437 

1.623 

1.199 

% 

.0729 

.0042 

.270 

X 

.4375 

.1503 

.660 

X 

1.458 

1.670 

1.207 

15-16 

.0781 

.0048 

.280 

Vs 

.4479 

.1576 

.669 

X 

1.479 

1.718 

1.216 

1 . 

.0833 

.0055 

.289 

X 

.4583 

.1650 

.677 

18. 

1.5 

1.767 

1.224 

1-16 

.0885 

.0062 

.297 

Vs 

.4688 

.1725 

.685 

X 

1.542 

1.867 

1.241 

X 

.C938 

.0069 

.305 

X 

.4792 

.1803 

.693 

19. 

1.583 

1.969 

1.258 

3-16 

.0990 

.0077 

.314 

Ys 

.4896 

.1878 

.700 


1.625 

2.074 

1.274 

X 

.1042 

.0085 

.322 

6. 

.5 

.1964 

.707 

20. 

1.667 

2.182 

1.291 

5-16 

.1094 

.0094 

.330 

X 

.5208 

.2131 

.722 

X 

1.708 

2.292 

1.307 

Vs 

.1146 

.0103 

.338 

X 

.5417 

.2304 

.736 

21. 

1.750 

2.405 

1.323 

7-16 

.1198 

.0113 

.346 

X 

.5625 

.2485 

.750 

X 

1.791 

2.521 

1.339 

X 

.1250 

.0123 

.354 

7. 

.5833 

.2673 

.764 

22. 

1.833 

2.640 

1.354 

9-16 

.1302 

.0133 

.361 

X 

.6042 

.2867 

.777 

X 

1.875 

2.761 

1.369 

% 

.1354 

.0144 

.368 

X 

.6250 

.3068 

.791 

23. 

1.917 

2.885 

1.384 

11-16 

.1406 

.0155 

.375 

X 

.6458 

.3276 

.803 

X 

1.958 

3.012 

1.399 

% 

.1458 

.0167 

.382 

8. 

.6667 

.3491 

.817 

24. 

2.000 

3.142 

1.414 

13-16 

.1510 

.0179 

.389 

X 

.6875 

.3712 

.829 

25, 

2.083 

3.409 

1.443 

Ys 

.1563 

.0192 

.395 

X 

.7083 

.3941 

.841 

26. 

2.166 

3.687 

1.472 

15-16 

.1615 

.0205 

.402 

X 

.7292 

.4176 

.854 

27. 

2.250 

3.976 

1.500 

2 . 

.1667 

.0218 

.408 

9. 

.75 

.4418 

.866 

28. 

2 333 

4.276 

1.528 

1-16 

.1719 

.0232 

.414 

X 

.7708 

.4667 

.879 

29. 

2 416 

4.587 

1.555 

X 

.1771 

.0246 

.420 

X 

.7917 

.4922 

.890 

30. 

2.500 

4.909 

1.581 

3-16 

.1823 

.0260 

.427 

X 

.8125 

.5185 

.902 

31. 

2.584 

5.241 

1.607 

X 

.1875 

.0276 

.433 

10. 

.8333 

.5454 

.913 

32. 

2.666 

5.585 

1.633 

5^16 

.1927 

.0291 

.440 

X. 

.8542 

.5730 

.924 

33. 

2.750 

5.940 

1.658 

Vs 

.1979 

.0308 

.445 

X 

.8750 

.6013 

.935 

34. 

2.834 

6.305 

1.683 

7-16 

.2031 

.0324 

.451 

X 

.8958 

.6303 

.946 

35. 

2.916 

6.681 

1.708 

X 

.2083 

.0341 

.457 

11. 

.9167 

.6600 

.957 

36. 

3.000 

7.069 

1.732 

9-16 

.2135 

.0358 

.462 

X 

.9375 

.6903 

.968 

38. 

3.166 

7.876 

1.779 

Vs 

.2188 

.0375 

.467 

X 

.9583 

.7213 

.979 

40. 

3.333 

8.727 

1.825 

11-16 

.2240 

.0394 

.473 

X 

.9792 

.7530 

.990 

42. 

3.500 

9.621 

1.871 

X 

.2292 

.0412 

.478 

12. 

1. 

.7854 

1.000 

44. 

3.666 

10.56 

1.914 

13-16 

.2344 

.8432 

.484 

X 

1.021 

.8184 

1.010 

48. 

4.000 

12.57 

2.000 

Ys 

.2396 

.0451 

.489 

X 

1.042 

.8522 

1.020 

54. 

4.500 

15.90 

2.121 

15-16 

.2448 

.0471 

.495 

X 

1.063 

.8866 

1.031 

60. 

5.000 

19 63 

2.236 

S. 

.2500 

.0491 

.500 

13. 

1.083 

.9218 

1.041 

66. 

5.500 

23.76 

2.345 

X 

.2604 

.0532 

.510 

X 

1.104 

.9576 

1.051 

72. 

6.000 

28.27 

2.449 

X 

.2708 

.0576 

.520 

X 

1.125 

.9940 

1.060 

78. 

6.500 

33.18 

2.550 

Vs 

.2813 

.0621 

.530 

X 

1.146 

1.031 

1.070 

84. 

7.000 

38.48 

2.646 

X 

.2917 

.0668 

.540 

14. 

1.167 

1.069 

1.080 

90. 

7.500 

44.18 

2.739 

Vs 

.3021 

.0716 

.550 

X 

1.187 

1.108 

1.090 

96. 

8.000 

50.27 

2.828 

X 

.3125 

.0767 

.560 

X 

1.208 

1.147 

1.099 





Ys 

.3229 

.0819 

.570 

X 

1.229 

1.187 

1.110 



1 



For contents in gallons, see p 157. 





































248 


HYDRAULICS. 


Art. 3. To find the total head required for a given velocity, or 
given discharge, through a straight, smooth, cylindrical iron pipe of 
known diarn and length. 

If the pipe is curved, see Rem p 245. 


If the discharge is given, first find 

mean velocity discharge in cubic feet per second 


in feet per second area of cross section of pipe in square feet 


Then 


diam X head _ mean velocity in feet per second 
approx -yj i en gth + 54 diams the proper divisor as follows 


diam of pipe in ft .05 .10 .50 1 1.5 2 3 4 

divisor 40 43 46 48 51 54 58 61 

(for intermediate diams, take intermediate divisors by guess.) 


From table Art 2, p 243, take the coefficient m corresponding to this value of 


I diam X head 
\ length + 54 diams 


, and to the given diam. 


Then 


Total X (length in ft + 54 diams ip ft) 

head =--- _ w .. - :—7 - 

j a f eet in 2 X diam in leet 


To find the Friction head. Weisbach’s formula. 

( .01716 \ Length Vel 2 in 

.0144 + -is. in leet y, ft per sec 

l X -D7aS- X ~64X" 

persec / in feet 

For the total head, we have only to add together, the friction head 
so found, the velocity head, taken from the next table, or from Table 10, p 258, 
opposite the given velocity, and the entry head (= say half the velocity head)! 
The sum of the velocity head and entry head rarely amounts to a foot. 

TABLE 4%. Of the vel, and discharge of water through straight, smooth 
cylindrical cast-iron pipes; with the friction head required for each 100 feet in 
length; and also the velocity head. Calculated by means of Weisbach’s formula by 
James Thompson, A M; and George Fuller, C E, Belfast, Ireland. The vel head 
remains the same for any length of pipe; being dependent only on the velocity of the 
water in the pipe. 

The entry head is equal to about half the vel bead. 














HYDRAULICS 


249 


TABLE 4 y v 


Vel. in 

Feet, 

per Sec. 

Vel- 
head in 
Feet. 





Diam. in Inches. 





3 

3^ 

4 

* l A 


5 

Frhead 
Ft per 
100 ft. 

Cub ft 
per Min 

i 

Frhead 
Ft per 
100 ft. 

i 

Cub ft 
Iper Min 

Fr head 
Ft per 
100 ft. 

Cub ft 
per Min 

Frhead 
Ft per 
100 ft. 

Cub ft 
per Min 

Frhead 
Ft per 
100 ft. 

Cub ft 
per Min 

2.0 

.062 

.659 

5.89 

.565 

8.02 

.494 

10.4 

.439 

13.2 

.395 

163 

2.2 

.075 

.780 

6.48 

.669 

8.82 

' .585 

11.5 

.520 

14.6 

.463 

18.0 

2.4 

.090 

.911 

7.07 

.781 

9.62 

.683 

12.5 

.607 

15.9 

.547 

19.6 

2.6 

.105 

1.05 

7.65 

.901 

10.4 

.788 

13.6 

.701 

17.2 

.631 

21.3 

2.8 

.122 

1.20 

8.24 

1.03 

11.2 

.900 

14.6 

.800 

18.5 

.720 

22.9 

3.0 

.140 

1.35 

8.83 

1.16 

12.0 

1.02 

15.7 

.905 

19.8 

.815 

24.5 

3.2 

.160 

1.52 

9.42 

1.31 

12.8 

1.14 

16.7 

1.02 

21.2 

.915 

26.2 

3.4 

.180 

1.70 

10.0 

1.46 

13.6 

1.27 

178 

1.13 

22.5 

1.02 

27.8 

3.6 

.202 

1.89 

10.6 

1.62 

14.4 

1.41 

18.8 

1.26 

23.8 

1.13 

29.4 

3.8 

.225 

2.08 

11.2 

1.78 

15.2 

1.56 

19.9 

1.39 

25.2 

1.25 

31.0 

4.0 

.250 

2.28 

11.8 

1.96 

16.0 

1.71 

20.9 

1.52 

26.5 

1.37 

32.7 

4.2 

.275 

2.49 

123 

2.14 

16.8 

1.87 

22.0 

1.66 

27.8 

1.50 

34.3 

4.4 

.302 

2.71 

12.9 

2.33 

17 6 

2.03 

23.0 

1.81 

29.1 

1.63 

36.0 

4.6 

.330 

2.94 

13.5 

2.52 

18.4 

2.21 

24.0 

1.96 

30 4 

1.76 

37.6 

4.8 

.360 

3.18 

14.1 

2.72 

19.2 

238 

25.1 

2.12 

31.8 

1.91 

39.2 

5.0 

.390 

3.43 

14.7 

2.94 

20.0 

2.57 

26.2 

2.28 

33.1 

2.05 

40.9 

5.2 

.422 

3 68 

15.3 

3.15 

20.8 

2.76 

27.2 

2.45 

34.4 

2.21 

42.5 

5.4 

.455 

3.94 

15.9 

3.38 

21.6 

2.96 

28.2 

2.63 

35.8 

2.37 

44.2 

5.6 

.490 

4.22 

16.5 

3.61 

22.4 

3.16 

29.3 

2.81 

37.1 

2.53 

45.8 

58 

.525 

4.50 

17.1 

3.85 

23.2 

3.37 

30.3 

3.00 

38.4 

2.70 

47.4 

6.0 

.562 

4 78 

17.7 

4.10 

24.0 

3.59 

31.4 

3.19 

39.7 

•2.87 

49.1 

6.2 

.600 

5.08 

18.2 

4.36 

24.8 

3.81 

32.4 

3.39 ' 

41.0 

3.05 

50.7 

6.4 

.640 

5.39 

18.8 

4.62 

25.6 

4.04 

33.5 

3.59 

42.4 

3.23 

52.3 

6.6 

.680 

5.70 

19.4 

4.89 

26.4 

4.28 

34.5 

3.80 

43.7 

3.42 

54.0 

6.8 

.722 

6.02 

20.0 

5.16 

27.3 

4.52 

35.6 

4.01 

45.0 

3 61 

55.6 

7.0 

.765 

635 

20.6 

5.45 

28.0 

4.77 

36.6 

4.24 

46 4 

3.81 

57.2 


Diam. in Inches. 


Vel. in 
Feet 
per Sec. 

Vel- 
head in 
Feet. 

6 

7 

8 

9 

10 

Frhead 
Ft per 
100 ft. 

Cub ft 
per Min 

Frhead 
Ft per 
100 ft. 

Cub ft 
per Miu 

Frhead 
Ft per 
100 ft. 

Cub ft 
per Min 

Frhead 
Ft per 
100 ft. 

Cub ft 
per Min 

Frhead 
Ft per 
100 ft. 

Cub ft 
per Min 

20 

.062 

.329 

23 5 

.2S2 

32.0 

.247 

41.9 

.220 

53.0 

.198 

65.4 

2.2 

.075 

.390 

25.9 

.334 

35.3 

.293 

46.1 

.260 

58.3 

.234 

72.0 

2.4 

.090 

.456 

28.2 

.300 

38.5 

.342 

50.2 

.304 

63.6 

.273 

78.5 

2.6 

.105 

.526 

30.6 

.450 

41.7 

.394 

54.4 

.350 

68.9 

.315 

85.1 

2.8 

.122 

.600 

32.9 

.514 

44 9 

.450 

58.6 

.400 

74.2 

.360 

91.6 

30 

.140 

.679 

35.3 

.582 

48.1 

.509 

62.8 

.453 

79.5 

.407 

98.2 

3.2 

.160 

.763 

37.7 

.654 

51.3 

.572 

67.0 

.508 

84.8 

.458 

105 

3.4 

.180 

.851 

40 0 

.729 

54.5 

.638 

71.2 

.567 

90.1 

.510 

111 

3.6 

.202 

.943 

42.4 

.808 

57.7 

.707 

75.4 

.629 

95.4 

.566 

118 

3.8 

.225 

1.04 

44.7 

.892 

60.9 

.780 

79.6 

.693 

101 

.624 

124 

4.0 

.250 

1.14 

47.1 

.979 

64.1 

.856 

83.7 

.761 

106 

.685 

131 

4.2 

.275 

1.25 

49.5 

1.07 

67.3 

.935 

87.9 

.832 

111 

.748 

137 

4.4 

.302 

1.35 

51.8 

1.16 

70.5 

1.02 

92.1 

.905 

116 

.814 

144 

4.6 

.330 

1.47 

54.1 

1.26 

73.7 

1.10 

96.3 

.981 

122 

.883 

150 

4.8 

.360 

1.59 

56.5 

1.36 

76.9 

1.19 

100 

1.06 

127 

.954 

157 

5.0 

.390 

1.71 

58 9 

1.47 

80.2 

1.28 

105 

1.14 

132 

1.03 

163 

5.2 

.422 

1 84 

61.2 

1.58 

83.3 

1.38 

109 

1.23 

138 

1.10 

170 

5.4 

.455 

1.97 

63.6 

1.69 

86.6 

1.48 

113 

1.31 

143 

1.18 

177 

5.6 

.490 

2.11 

65.9 

1.81 

89.8 

1.58 

117 

1.40 

148 

1.26 

183 

5.8 

.525 

2 25 

68.3 

1.93 

93.0 

1.68 

121 

1 50 

154 

1.35 

190 

60 

.562 

2.39 

70.7 

2 05 

96.2 

1.79 

125 

1.59 

159 

1.43 

196 

6 2 

.600 

2.54 

73.0 

2.18 

99.4 

1.90 

130 

1.69 

164 

1.52 

203 

64 

.640 

2.69 

75.4 

2.31 

102 

2.02 

134 

1.79 

169 

1.61 

209 

6 6 

.680 

2.85 

77-7 

2.44 

106 

2.14 

138 

1.90 

175 

1.71 

216 

68 

.722 

3.01 

80.1 

2 58 

109 

2.26 

142 

2.01 

180 

1.81 

222 

7.0 

.765 

3.18 

82.4 

2.72 

112 

2.38 

146 

2.12 

185 

1.90 

229 






















































































250 


HYDRAULICS, 


V 


TABLE 4% — (Continued.) 


Yet. in 
Feet 
per Sec. 

Vel- 
head in 
Feet. 

Diam. in Inches. 

11 

1 

O 

13 

14 

15 

Frhead 
Ft per 
100 ft. 

Cub ft 
per Min 

Frhead 
Ft per 
100 ft. 

Cub ft 
per Min 

Fr head 
Ft per 
100 ft. 

Cub ft 
per Min 

Frhead 
Ft per 
100 ft. 

Cub ft 
per Min 

Frhead 
Ft per 
100 ft. 

Cub ft 
per Min 

2.0 

.062 

.180 

79.2 

.165 

94.2 

.152 

110 

.141 

128 

.132 

147 

2.2 

.075 

.213 

87.1 

.195 

lu3 * 

.180 

121 

.167 

141 

.156 

162 

2.1 

.090 

.248 

95.0 

.228 

113 

.210 

133 

.195 

154 

.182 

176 

2.0 

.105 

.287 

103 

.263 

122 

.242 

144 

.225 

167 

.210 

191 

2.8 

.122 

.327 

111 

.300 

132 

.277 

156 

.257 

179 

.240 

206 

3.0 

.140 

.370 

119 

.339 

141 

.313 

166 

.291 

192 

.271 

221 

32 

.160 

.416 

127 

.381 

151 

.352 

177 

.327 

205 

.305 

235 

3.4 

.180 

.464 

134 

.425 

160 

.393 

188 

.365 

218 

.340 

250 

3.6 

.202 

.514 

142 

.472 

169 

.435 

199 

.404 

231 

.377 

265 

3.8 

.225 

.567 

150 

.520 

179 

.480 

210 

.446 

243 

.416 

2S0 

4.0 

.250 

.623 

158 

.571 

188 

.527 

221 

.489 

256 

.457 

294 

4.2 

.275 

.680 

166 

.62 4 

198 

.576 

232 

.534 

269 

.499 

309 

4.4 

.302 

.740 

174 

.679 

207 

.626 

243 

.582 

282 

.543 

324 

4.6 

.330 

.803 

182 

.736 

217 

.679 

254 

.631 

295 

.589 

339 

4.8 

.360 

.867 

190 

.795 

226 

.734 

265 

.682 

308 

.636 

353 

5.0 

.390 

.935 

198 

.857 

235 

.791 

276 

.734 

321 

.685 

368 

5.2 

.422 

1.00 

206 

.920 

245 

.850 

287 

.789 

333 

.736 

383 

5.4 

.455 

1.07 

214 

.986 

254 

.910 

298 

.845 

346 

.789 

397 

5.6 

.490 

1.15 

222 

1.05 

264 

.973 

309 

.903 

359 

.843 

412 

5.8 

.525 

1.22 

229 

1.12 

273 

1.04 

321 

.964 

372 

.899 

427 

6.0 

.562 

1.30 

237 

1.19 

283 

1.10 

332 

1.02 

385 

.957 

442 

6.2 

.600 

1.38 

245 

1.27 

292 

1.17 

343 

1.09 

397 

1.01 

456 

6.4 

.640 

1.47 

253 

1.35 

301 

1.24 

354 

1.15 

410 

1.08 

471 

6.6 

.680 

1.55 

261 

1.42 

311 

1.31 

365 

1.22 

423 

1.14 

486 

6.8 

.72 1 

1.64 

269 

1.50 

320 

1.39 

376 

1.29 

436 

1.20 

500 

7.0 

.765 

1.73 

277 

1.59 

330 

1.46 

387 

1.36 

449 

1.27 

515 


Vel. in 
Feet 
perSec. 

Vel- 
head in 
Feet. 

Diam. in Inches. 

16 

17 

18 

19 

20 

Frhead 
Ft per 
100 ft. 

Cub ft 
per Min 

Frhead 
Ft per 
100 ft. 

Cub ft 
per Min 

Fr head 
Ft per 
100 ft. 

Cub ft 
per Min 

Frhead 
Ft per 
100 ft. 

Cub ft 
per Min 

Fr head 
Ft per 
100 ft. 

Cub ft 
per Min 

2.0 

.062 

.123 

167 

.116 

189 

.110 

212 

.104 

236 

.099 

262 

2.2 

.075 

.146 

184 

.138 

208 

.130 

233 

.123 

260 

.117 

288 

2.4 

.090 

.171 

201 

.161 

227 

.152 

254 

.144 

283 

.137 

314 

2.6 

.105 

.197 

218 

.185 

246 

.175 

275 

.166 

307 

.158 

340 

2.8 

.122 

.225 

234 

.212 

265 

.200 

297 

.189 

331 

.180 

366 

3.0 

.110 

.255 

251 

.240 

284 

.226 

318 

.214 

354 

.204 

393 

3.2 

.160 

.286 

268 

.269 

302 

.254 

339 

.241 

378 

.229 

419 

3.4 

.180 

.319 

284 

.300 

321 

.283 

360 

.269 

401 

.255 

445 

3.6 

.202 

.354 

301 

.333 

340 

.314 

382 

.298 

425 

.283 

471 

3.8 

.225 

.390 

318 

.367 

359 

.347 

403 

.328 

449 

.312 

497 

4.0 

.250 

.428 

335 

.403 

378 

.380 

424 

.360 

472 

.342 

523 

4.2 

.275 

.468 

352 

.440 

397 

.416 

445 

.394 

496 

.374 

550 

4.4 

.302 

.509 

368 

.479 

416 

.452 

466 

.429 

519 

.407 

576 

4.6 

.330 

.552 

385 

.519 

435 

.490 

488 

.465 

543 

.441 

602 

4.8 

.360 

.596 

402 

.561 

454 

.530 

509 

.502 

567 

.477 

628 

5.0 

.390 

.642 

419 

.605 

473 

.571 

530 

.541 

590 

.514 

654 

5.2 

.422 

.690 

435 

.650 

492 

.614 

551 

.581 

614 

.552 

680 

5.4 

.455 

.740 

452 

.696 

511 

.657 

572 

.623 

638 

.592 

707 

5.6 

.490 

.791 

469 

.744 

529 

.703 

594 

.666 

661 

.632 

733 

5.8 

.525 

.843 

486 

.793 

548 

.749 

615 

.710 

685 

.674 

759 

6.0 

.562 

.897 

502 

.844 

567 

.798 

636 

.755 

709 

.718 

785 

6.2 

.600 

.953 

519 

.897 

586 

.847 

657 

.802 

732 

.762 

811 

64 

.640 

1.01 

536 

.951 

605 

.898 

678 

.851 

756 

.808 

838 

6.6 

.680 

1.07 

553 

1.01 

624 

.950 

700 

.900 

780 

.855 

864 

6.8 

.722 

1.13 

569 

1.06 

643 

1.00 

721 

.951 

803 

.904 

896 

7.0 

.765 

1.19 

586 

1.12 

662 

1.06 

742 

1.00 

827 

.953 

916 





































































HYDRAULICS, 


251 


TABLE 4l%. —(Continued.) 


Diam. in Inches. 


Vel. in 
Feet 
perSec. 

Vel- 
heaii in 
Feet. 

22 

24 

26 

28 

30 

Frhead 
Ft per 
100 ft. 

Cub ft 
per Min 

Frhead 
Ft per 
100 ft. 

Cub ft 
per Min 

Frhead 
Ft per 
100 ft. 

Cub ft 
per Min 

Frhead 
Ft per 
100 ft. 

Cub ft 
per Min 

Fr head 
Ft per 
100 ft. 

Cub ft 
per Min 

2.0 

.062 

.090 

316 

.082 

377 

.076 

442 

.070 

513 

.066 

589 

2.2 

.075 

.106 

348 

.097 

414 

.090 

486 

.083 

564 

.078 

648 

2.4 

.090 

.124 

380 

.114 

452 

.105 

531 

.097 

616 

.091 

707 

2.6 

.105 

.143 

412 

.131 

490 

.121 

575 

.112 

667 

.105 

766 

2.8 

.122 

.164 

443 

.150 

528 

.138 

619 

.128 

718 

.120 

824 

3.0 

.140 

.185 

475 

.170 

565 

.157 

663 

.145 

770 

.136 

883 

3.2 

.160 

.208 

507 

.191 

603 

.176 

708 

.163 

821 

.152 

942 

3.4 

.180 

.232 

538 

•2L3 

641 

.196 

752 

.182 

872 

.170 

1001 

3.6 

.202 

.257 

570 

.236 

678 

.218 

796 

.202 

923 

.189 

1060 

3.8 

.225 

.284 

601 

.260 

716 

.240 

840 

.223 

974 

.208 

1119 

4.0 

.250 

.311 

633 

.285 

754 

.263 

885 

.244 

1026 

.228 

1178 

4.2 

.275 

.340 

665 

.312 

791 

.288 

929 

.267 

1077 

.249 

1237 

4.4 

.302 

.370 

697 

.339 

829 

.313 

973 

.290 

1129 

.271 

1296 

4.6 

.330 

.401 

728 

.368 

867 

.339 

1017 

.315 

1180 

.294 

1355 

4.8 

.360 

.434 

760 

.397 

905 

.367 

1062 

.341 

1231 

.318 

1414 

5.0 

.390 

.467 

792 

.428 

942 

.395 

1106 

.367 

12S3 

.343 

1472 

5.2 

.422 

.502 

823 

.460 

980 

.425 

1150 

.394 

1334 

.368 

1531 

5.4 

.455 

.538 

855 

.493 

1018 

.455 

1194 

.423 

13S5 

.394 

1590 

5.6 

.490 

.575 

887 

.527 

1055 

.486 

1239 

.452 

1437 

.422 

1649 

5.8 

.525 

.613 

918 

.562 

1093 

.519 

1283 

.482 

1488 

.450 

1708 

6.0 

.562 

.652 

950 

.598 

1131 

.552 

1327 

.513 

1539 

.478 

1767 

6.2 

.600 

.693 

982 

.635 

1168 

.586 

1371 

.544 

1590 

.508 

1826 

6.4 

.610 

.735 

1013 

.673 

1206 

.622 

1416 

.577 

1641 

.539 

1885 

6.6 

.680 

.778 

1045 

.713 

1244 

.658 

1460 

.611 

1693 

.570 

1943 

6.8 

.722 

.821 

1077 

.753 

1282 

.695 

1504 

.645 

1744 

.602 

2003 

7.0 

.765 

.867 

1109 

.794 

1319 

.733 

1548 

.6S1 

1796 

.635 

2061 


TABLE 5. Of fifth roots and fifth powers. 


Power. 

No. or 
Root. 

Power. 

No. or 
Root. 

Power. 

No. or 
Root. 

Power. 

No. or 
Root. 

Power. 

No. or 
Root. 

Power. 

No. or 
Root. 

.0000100 

.1 

.000142 

.170 

.004219 

.335 

.077760 

.60 

.695688 

.93 

8.11368 

1.52 



.000164 

.175 

.004544 

.340 

.084460 

.61 

.733904 

.94 

8.66171 

1.54 

.0000110 

.102 

.000189 

.180 

.004888 

.345 

.091613 

.62 

.773781 

.95 

9.23896 

1.56 



.000217 

.185 

.005252 

.350 

.099244 

.63 

.815373 

.96 

9.84658 

1.58 

.0000122 

.104 

.000248 

.190 

.005638 

.355 

.107374 

.64 

.858734 

.97 

10.4858 

1.60 



.000282 

.195 

.006047 

.360 

.116029 

.65 

.903921 

.98 

11.1577 

1.62 

.0000134 

.106 

.000320 

.200 

.006478 

.365 

.125233 

.66 

.950990 

.99 

11.8637 

1.64 



.000362 

.205 

.006934 

.370 

.135012 

.67 

1. 

1 . 

12.6049 

1.66 

.0000147 

.108 

000408 

.210 

.007416 

.375 

.145393 

.68 

1.10108 

1.02 

13.3828 

1.68 

.00001(51 

.110 

.000459 

.215 

.007924 

.380 

,156403 

.69 

1.21665 

1.04 

14.1986 

1.70 

.0000176 

.112 

.000515 

.220 

.008459 

.385 

.168070 

.70 

1.33823 

1.06 

15.0537 

1.72 

.0000193 

.114 

.000577 

.225 

.009022 

.390 

.180423 

.71 

1.46933 

1.08 

15.9495 

1.74 

.0000210 

.116 

.000644 

.230 

.009616 

.395 

.193492 

.72 

1.61051 

1.10 

16.8874 

1.76 

.0000229 

.118 

.000717 

.235 

.010240 

.400 

.207307 

.73 

1.76234 

1.12 

17.8690 

1.78 

.0000219 

.120 

000796 

.240 

.011586 

.41 

.221901 

.74 

1.92541 

1.14 

18.8957 

1.80 

.0000270 

.122 

.000883 

.245 

.013069 

.42 

.237305 

.75 

2.10034 

1.16 

19.9690 

1.82 

.0000293 

.124 

.000977 

.250 

.014701 

.43 

.253553 

.76 

2.28775 

1.18 

21.0906 

1.84 

.0000318 

.126 

.001078 

.255 

.016492 

.44 

.270678 

.77 

2.48832 

1.20 

22.2620 

1.86 

.0000314 

.128 

.001188 

.260 

.018453 

.45 

.288717 

.78 

2.70271 

1.22 

23.4849 

1.88 

.0000371 

.130 

.001307 

.265 

.020596 

.46 

.307706 

.79 

2.93163 

1.24 

24.7610 

1.90 

.0000401 

.132 

.001435 

.270 

.022935 

.47 

.327680 

.80 

3.17580 

1.26 

26 0919 

1.92 

.0000432 

.134 

.001573 

.275 

•025480 

.48 

.348678 

.81 

3.43597 

1.28 

27.4795 

1.94 

.0000465 

.136 

.001721 

.280 

.028248 

.49 

.370740 

.82 

3.71293 

1.30 

28.9255 

1.96 

.0000500 

.138 

.001880 

.285 

.031250 

.50 

.393904 

.83 

4.00746 

1.32 

30.4317 

1.98 

.0000538 

.140 

.002051 

.290 

.034503 

.51 

.418212 

.84 

4.32040 

1.34 

32.0000 

2.00 

.0000577 

.142 

.002234 

.295 

.038020 

.52 

.443705 

.85 

4.65259 

1.36 

36.2051 

2.05 

.0000619 

.144 

.002430 

.300 

.041820 

.53 

.470427 

.86 

5.00490 

1.38 

40.8410 

2.10 

.0000663 

.146 

.002639 

.305 

.045917 

.54 

.498421 

.87 

5.37824 

1.40 

45.9401 

2.15 

.0000710 

.148 

.002863 

.310 

.050328 

.55 

.527732 

.88 

5.77353 

1.42 

51.5363 

2.20 

.0000754 

.150 

.003101 

.315 

.055073 

.56 

.558406 

.89 

6.19174 

1.44 

57.WS50 

2.25 

.0000895 

. 155 

.003355 

.320 

.060169 

.57 

.590 490 

.90 

6.63383 

1.46 

64.3634 

2.30 

.000105 

.160 

.003626 

.325 

.065636 

.58 

.624032 

.91 

7.10082 

1.48 

71.6703 

2.85 

.000122 

.165 

.003914 

.330 

.071492 

.59 

.659082 

.92 

7.59375 

1.50 

79.6262 

2.40 




























































252 


HYDRAULICS 


TABLE 5. Of fifth roots anti fifth powers — (Continued.) 


Power. 

No. or 
Koot. 

Power. 

No. or 
Bool. 

Power. 

No. or 
Root. 

Power. 

No. or 
Koot. 

Power. 

No. or 
Hoot. 

Power. 

No. o 
Root 

88.2735 

2.45 

2824.75 

4.90 

85873 

9.70 

2609193 

19.2 

20511149 

29.0 

459165024 

54. 

97.6562 

2.50 

2971.84 

4.95 

90392 

9.80 

2747949 

19.4 

21228253 

29.2 

503284375 

55. 

107.820 

2.55 

3125.00 

5-00 

95099 

9.90 

2892547 

19.6 

21965275 

29.4 

550731776 

56. 

118 814 

2.60 

3450.25 

5-10 

100000 

10.0 

3043168 

19.8 

22722628 

29.6 

601692057 

57. 

130.686 

2.65 

3802.04 

5-20 

110408 

10.2 

3200000 

20.0 

23500728 

29.8 

65635076b 

58. 

143.489 

2.70 

4181.95 

5.30 

121665 

10.4 

3363232 

20.2 

24300000 

30.0 

71492429! 

59. 

157.276 

2.75 

4591.65 

5-40 

133823 

10.6 

3533059 

20.4 

26393634 

30.5 

777600000 

60. 

172.104 

2.80 

5032.84 

5-50 

146933 

10.8 

3709677 

20.6 

28629151 

31.0 

844596301 

61. 

188.029 

2.85 

5507.32 

5.60 

161051 

11.0 

3893289 

20.8 

31013642 

31.5 

916132832 

62 

205.111 

2.90 

6016.92 

5-70 

176234 

11.2 

4064101 

21.0 

33554432 

32.0 

992436543 

63. 

223.414 

2.95 

6563.57 

5-80 

192541 

11.4 

4282322 

21.2 

36259082 

32.5 

10737418-24 

64. 

213.000 

3.00 

7149.24 

5-90 

210034 

11.6 

4488166 

21.4 

39135393 

33.0 

1160290625 

65. 

263.936 

3.05 

7776.00 

6-00 

228776 

11.8 

4701850 

21.6 

42191410 

33.5 

1252332576 

66. 

286.292 

3.10 

8445.96 

6-10 

248832 

12.0 

4923597 

21.8 

45435424 

34.0 

1350125107 

67. 

310.136 

3.15 

9161.33 

6-20 

270271 

12.2 

5153632 

22.0 

48875980 

34.5 

14 53933568 

68. 

335.544 

3.20 

9924.37 

6.30 

293163 

12.4 

5392186 

22.2 

52521875 

35.0 

1564031349 

69. 

362.591 

3.25 

10737 

6.40 

317580 

12.6 

5639493 

22.4 

56382167 

35.5 

1680700000 

70. 

391.354 

3.30 

11603 

6-50 

343597 

12.8 

5895793 

22.6 

(1046(5176 

36 0 

1804229351 

71. 

421.419 

3.35 

12523 

6.60 

371293 

13.0 

6161327 

22.8 

64783487 

36 5 

1934917632 

72. 

454.354 

3.40 

13501 

6.70 

400746 

13.2 

6436343 

23.0 

69343957 

37.0 

2073071593 

73. 

488.760 

3.45 

14539 

6.80 

432040 

13.4 

6721093 

23.2 

74157715 

37.5 

2219006624 

74. 

525.219 

3.50 

15640 

6.90 

465259 

13.6 

7015834 

23.4 

79235168 

38.0 

2373046875 

75. 

563.822 

3.55 

16807 

7.00 

500490 

13.8 

7320825 

23.6 

84587005 

38.5 

2o3o5253 i fi 

76. 

604.662 

3.60 

18042 

7.10 

537824 

14.0 

7636332 

23.8 

90224199 

39.0 

2706784157 

77. 

647.835 

3.65 

19349 

7.20 

577353 

14.2 

7962624 

24.0 

96158012 

39.5 

2887174368 

78. 

693.440 

3.70 

20731 

7.30 

619174 

14.4 

8299976 

24.2 

102400000 

40.0 

3077056399 

79. 

741.577 

3.75 

22190 

7.40 

663383 

14.6 

8648666 

24.4 

108962013 

40.5 

3276800000 

80 

792.352 

3.80 

23730 

7.50 

710082 

14.8 

9008978 

24.6 

115856201 

41.0 

3486784401 

81. 

845.870 

3.85 

25355 

7.60 

759375 

15.0 

9381200 

24.8 

123095020 

41.5 

3707398432 

82. 

902.242 

3.90 

27068 

7.70 

811368 

15.2 

9765625 

25.0 

130691232 

42.0 

3939040643 

83. 

961.580 

3.95 

28872 

7.80 

866171 

15.4 

10162550 

25.2 

138657910 

42.5 

4182119424 

84. 

1024.00 

4.00 

30771 

7.90 

923896 

15.6 

10572278 

25.4 

147008443 

43.0 

4437053125 

85. 

1089.62 

4.05 

32768 

8.00 

984658 

15.8 

10995116 

25.6 

155756538 

43.5 

4704270176 

86. 

1158.56 

4.10 

34868 

8.10 

1048576 

16.0 

11431377 

25.8 

164916224 

44 0 

4984209207 

87. 

1230.95 

4.15 

37074 

8.20 

1115771 

16.2 

11881376 

26.0 

174501858 

44.5 

5277319168 

88. 

1306.91 

4.20 

89390 

8.30 

1186367 

16.4 

12345437 

26.2 

184528125 

45.0 

5584059449 

89. 

1386.58 

4.25 

41821 

8.40 

1260493 

16.6 

12823886 

26.4 

I950I0045 

45.5 

5904900000 

90. 

1470.08 

4.30 

44371 

8.50 

1338278 

16.8 

13317055 

26.6 

>05962976 

46.0 

6240321451 

91. 

1557.57 

4.35 

47043 

8.60 

1419857 

17.0 

13825281 

26.8 

217402615 

46.5 

6590815232 

92. 

1649.16 

4.40 

49842 

8.70 

1505366 

17.2 

14348907 

27.0 

229345007 

47 0 

6956883693 

93. 

1745.02 

4.45 

52773 

8.80 

1594947 

17.4 

14888280 

27.2 

241806543 

47.5 

7339040224 

94. 

1845.28 

4.50 

55841 

8.90 

1688742 

17.6 

15443752 

27.4 

254803968 

48.0 

7737809375 

95. 

1950.10 

4.55 

59049 

9.00 

1786899 

17.8 

16015681 

27.6 

268354383 

48.5 

8153726976 

96. 

2059.63 

4.60 

62403 

9.10 

1889568 

18.0 

16604430 

27.8 

282475249 

49.0 

8587340257 

97. 

2174.03 

4.65 

65908 

9.20 

1996903 

18.2 

17210368 

28.0 

297184391 

49.5 

9039207968 

98. 

2293.45 

4.70 

69569 

9.30 

2109061 

18.4 

17833868 

28.2 

! 12500000 

50.0 

9509900499 

99. 

2418.07 

4.75 

73390 

9.40 

2226203 

18.6 

18475309 

28.4 

145025251 

51. 



2548.04 

4.80 

77378 

9.50 

2348493 

18.8 

19135075 

‘i8.6 

180204032 

52. 



2683.54 

4.85 

81537 | 

9.60 

2476099 

19.0 

19813557 

28.8 

418195493 

63. 




j 



































HYDRAULICS 


253 


TABliE 6. Of the square roots of the fifth powers of mini- 
hers. In this table the numbers and the roots are supposed to be in the same di¬ 
mensions ; that is, both in inches, or both in feet, &c. See the next table. 


No. 

Sq. Rt. 
of 5th 
Rower. 

No. 

Sq. Rt. 
of 5th 
Power. 

No. 

Sq. Rt. 
of 5th 
Power. 

No. 

Sq. Rt. 
of 5th 
Power. 

No. 

Sq. Rt. 
of 5th 
Power. 

No. 

Sq. Rt. 
of 5ih 
Power. 

.25 

.031 

7. 

129.64 

17.5 

1281.1 

31. 

5351 

49 

16807 

76 

50354 

.5 

.177 

7.25 

141.53 

18. 

1374.6 

31.5 

5569 

50 

17678 

77 

52027 

.75 

.485 

7.5 

154.05 

18.5 

1472.1 

32. 

5793 

51 

18575 

78 

53732 

1 . 

1. 

7.75 

167.21 

19. 

1573.6 

32.5 

6022 

52 

19499 

79 

55471 

1.25 

1.747 

8. 

181.02 

19.5 

1679.1 

33. 

6256 

53 

20450 

60 

57243 

1.5 

2.756 

8.25 

195.50 

20. 

1788.9 

33.5 

6496 

54 

21428 

81 

59049 

1.75 

4.051 

8.5 

210.64 

20.5 

1902.8 

34. 

6741 

55 

22434 

82 

60888 

2 . 

5.657 

8.75 

226.48 

21. 

2020.9 

34.5 

6991 

58 

23468 

83 

62762 

2.25 

7.594 

9. 

243. 

21.5 

2143.4 

35. 

7247 

57 

24529 

84 

64669 

2.5 

9.882 

9.25 

260.23 

22. 

2270.2 

35.5 

7509 

58 

25620 

85 

66611 

2.75 

12.541 

9.5 

278.17 

22.5 

2401.4 

36. 

7776 

59 

26738 

86 • 

68588 

3 . 

15.588 

9.75 

296.83 

- 23. 

2537. 

36.5 

8049 

60 

27886 

87 

70599 

3.25 

19.042 

10. 

316.23 

23.5 

2677.1 

37. 

8327 

61 

29062 

88 

72646 

3.5 

22 918 

10.5 

357.2 

24. 

2821.8 

37.5 

8611 

62 

30268 

89 

74727 

3.75 

27 232 

11. 

401.3 

24.5 

2971.1 

38. 

8901 

63 

31503 

90 

76843 

4. 

32. 

11.5 

448.5 

25. 

3125. 

38.5 

9197 

64 

32768 

91 

78996 

4.25 

37.24 

12. 

498.8 

25.5 

3283.6 

39. 

9498 

65 

31063 

92 

81184 

4.5 

42.96 

12.5 

552.4 

26. 

3146.9 

39.5 

9806 

66 

35388 

93 

83408 

4.75 

49 17 

13. 

609.3 

26.5 

3615.1 

40. 

10119 

67 

36744 

94 

85668 

5 . 

55.90 

13.5 

669.6 

27. 

3788. 

41. 

10764 

68 

38131 

95 

87965 

5.25 

63.15 

14. 

733.4 

27.5 

3965.8 

42. 

11432 

69 

39548 

96 

90298 

5.5 

70.94 

14 5 

800.6 

28. 

4148.5 

43. 

12125 

70 

40996 

97 

92668 

5.75 

79.28 

15. 

871.4 

28.5 

4336.2 

44. 

12842 

71 

42476 

98 

95075 

6. 

88.18 

15.5 

945.9 

29. 

4528.9 

45. 

13584 

72 

43988 

99 

97519 

6.25 

97.66 

16. 

1024. 

29.5 

4726.7 

46. 

14351 

73 

4553] 

100 

100000 

f>.5 

107.72 

16.5 

1105.9 

30. 

4929.5 

47. 

15144 

74 

47106 



6.75 

118.38 

17. 

1191.6 

30.5 

5138. 

48. 

15963 

75 

48714 




TABLE 634. Numbers, in inches. Square roots of fifth powers, in feet. 



Sq. Rt. of 
5th Pow. 


Sq. Rt. of 
5th Pow. 


Sq. Rt. of 
5th Pow. 


Sq. Rt. of 
5th Pow. 


Sq. Rt. of 

5th Pow. 

Ins. 

Feet. 

Ins. 

Feet. 

Ins. 

Feet. 

Ins. 

Feet. 

Ins. 

Feet. 

X 

.00006 

m 

.0547 

12. 

1.000 

TlX 

4.813 

42 

22.92 

H 

.00017 

4. 

.0041 

X 

1.108 

23 

5.086 

43 

24.31 

X 

.00035 

X 

.0731 

13. 

1.221 

X 

5.365 

44 

25.74 

% 

.00062 


.0827 


1.342 

24 

5.657 

45 

27.23 

X 

.00098 

H 

.0971 

14. 

1.470 

25 

6.264 

46 

28.77 

% 

.00144 

5. 

.1120 

X 

1.605 

26 

6.909 

47 

30.36 

1. 

.0020 

X 

.1271 

15. 

1.747 

27 

7.593 

48 

32.00 

X 

.0027 

X 

.1428 

X 

1.896 

28 

8.316 

49 

33.69 

V\ 

.0035 

% 

.1590 

16. 

2.053 

29 

9.079 

50 

35.44 

X 

.0044 

6 

.1768 

X 

2.217 

30 

9.882 

51 

37.25 

X 

.0055 

X 

.2160 

17. 

2.389 

31 

10.73 

52 

39.13 

% 

.0067 

7. 

.2599 


2.567 

32 

11.61 

53 

41.02 

¥< 

.0081 

X 

.3088 

18. 

2.756 

33 

12.54 

54 

42.96 

yi 

.0096 

8. 

.3628 

X 

2.950 

34 

13.51 

55 

44.97 

2 . 

.0113 

X 

.4228 

19. 

3.155 

35 

14.53 

56 

47.05 

X. 

.0152 

9. 

.4871 

X 

3.365 

36 

15.59 

57 

49.17 

X 

.0198 

X 

.5577 

20. 

3.586 

37 

16.69 

58 

51.35 

X 

.0252 

10. 

.6339 

X 

3.813 

38 

17.84 

59 

53.60 

3. 

.0312 

X 

.7162 

21. 

4.051 

39 

19.04 

60 

55.90 

X 

.0383 

11 . 

.8043 

X 

4.297 

40 

20.29 

61 

58.27 

X 

.0459 

X 

.8990 

22. 

4.551, 

41 

21.58 

























































254 


HYDRAULICS 


Arf. 4 a. To find the discharge through a compound pipe 

v n , Fig 1 H, composed of any number of pipes, a, b, c, z , of different diam¬ 
eters, which decrease from the reservoir toward the outflow o. 




Fig.lH 


First find what part of the total head II is employed in forcing the water through 
the last pipe z alone , thus: 

Lot L a, L 6, L c, and L z, be the lengths in ft of the pipes a, b, c, and z, respectively; 
D a, D 6; D c, and D z their diameters in ft; and A a, A b, A c, and A z the areas of 
their cross sections in sq ft. Then 


The head In 
ft employed 
in forcing the _. 
waterthrough 


The total head H in feet 


the last pipe 
z alone 


1_ *~ X (iaSXDtt A 62 X D 6 A c! X Dc 


A z 2 x D z 


/La+(5iXDu) , L6-H54XD6) , Lc-f(54XDc) 


)] 


However many divisions the pipe may have, proceed in the same way as above, using 


Area 2 X Diarn 


Length + 54 Diams 


for the last or narrowest division; and 


Length + 54 Diams 


Area 2 X Diam 


for each of the others. 

Then, by the formulae, Art. 2, find the velocity in ft per second, and discharge in 
cub ft per second, of the last pipe z, using its actual diameter, length, and cross 
sectional area, and the head just found. Said discharge is evidently the discharge 1 
for the compound pipe. 

For the velocity in any portion, as b, say 


area of cross sectiou . area of cross . . velocity . velocity in the 

of the given portion • section of z • • in z • given portion. 


For the above rule and formula, we are indebted to Mr. Howard Murphy, C E, 
of Phila; and for the opportunity of testing it experimentally, to Messrs Morris, 
Tasker & Co, Limited, Pascal Iron Works, Philadelphia, 




























HYDRAULICS, 


255 


Art. f>. On the resistance wtiicti cnrved bends oppose to tbe 
How of water through round pipes. Well-rounded bends of large rad, 

whether vert, or hor, produce but little resistauce; except so 
far as the first may cause accumulations of sediment, or of 
air. According to Weisbach, the rules of DuBuat. Navier, 
and other authorities, are erroneous; and he gives the follow¬ 
ing one for ascertaining the additional head reqd to overcome 
the resistauce produced in a circular pipe, by a bend formed 
by an arc of a circle : Kuowing the rad r e of the pipe, (or, in 
other words, half its diatn,) in feet; the rad rs, of the axis 
r no of the bend, in feet; tbe ceutral augle rso in degrees; 

(which is equal to the angle dbx, or c6o,)and the reqd vel 
of the water in the pipe, in ft per sec. 

Rule. Div the central anglersoin deg, by 180. 

Call the quot a . Next, square the reqd vel. Div this sq by 
the constant number 64.4. Call the quot b. Div the inner rad 
r e of the pipe, in ft, by the rad rs of the axis mo of the 
bend, in ft. Call the quot c. Take from the following Table 
7. the number in columu d , which corresponds to c; (unless 

c be less than .1, in which case always take .13 as d.) Finally, mult together this number d. the 
quot a, and the quot b. The prod will be the reqd head in feet; which must either be added to the 
head previously calculated for the straight pipe, if the original vel is required to be maintained; or 
must be subtracted from it. in case the head does not admit of increase, and a new calculation made 
to ascertain the diminished vel under the head thus reduced. If there is more than one bend of the 
same dimensions, an equal alteration of head must be made for each ; or, if they are of diff radii, 
and with diff central angles r 8 o. a separate calculation must be made for each. Rennie’s experi¬ 
ments, at the end of this Art, seem to prove that this is by no means the case. So far as the writer 
is aware, we have no reliable data for calculating the effects of a succession of bends. 

In shape of a formula, Weisbaeh's rule stands thus: R being 

the rad of the axis of the bend, and r the rad of the pipe : 



Additional _ ,, , q, — 

head in feet ~ >131 + 1 ‘ 847 


fr in ft\ 
\ R in ft/ 


square of vel 
X in ft per sec 


central angle 
X in degrees 


64.4 180. 

The expression fj means the sq root of the 7th power. When the rad of the bend exceeds 5 diams 
of the pipe, then 1.847 ^ ^ ^ becomes inappreciable in practice, and may be omitted from the for¬ 
mula. When the pipe is square, instead of circular, the formula becomes 


..... .. . . . _ / r \ •? vel2 central angle 

Additional head = .124 4- 3.104 ( — X - X- 

^ V R/ 2 64.4 180. 


TABLE 7. 


c. 

d. 

c. 

d. 

c. 

d. 

c. 

d. 

c. 

d. 

.1 

.131 

.325 

.17 

.5 

.29 

.675 

.60 

.85 

1.18 

.15 

.135 

.35 

.18 

.525 

.32 

.7 

.66 

.875 

1.29 

.2 

.138 

.375 

.195 

.55 

.35 

.Tib 

.73 

.9 

1.41 

.225 

.145 

.4 

.206 

.575 

.39 

.75 

.80 

.925 

1.54 

.25 

.15 

.425 

.225 

.6 

.44 

.775 

.88 

.95 

1.68 

.275 

.155 

.45 

.24 

.625 

.49 

.8 

.98 

.975 

1.83 

.3 

.16 

.475 

.264 

.65 

.54 

.825 

1.08 

1. 

2. 


Ex. A straight pipe 1 mile long, and 18ins diam, with a total head of 20 ft, will disch water with 
a vel of 4 ft per sec; but it has been found necessary to introduce a circular bend of 90°, with a rad 
s r. Rig 2, of 3 feet. What addition mu<t be made to the 20 ft head, to compensate tor the additional 
resistance caused by the bend ; so that the reqd vel of 4 ft per sec may still be maintained? 

Here, 90° -r 180 = .5 = a. Next, tbe square of the reqd vel in ft per sec, is 4 X 4 == 16. A_nd 16 -r 
64.4 — .2484 = b. The rad re of the pipe (.75 ft), div by the rad r s of the bend (5 ft), — .75-r 5 = .15 
= c : and opposite this .15 in the column c of the foregoing table, we fitid d = .135. Fiuallj’, 
a X b X d = 5 X .2484 X -135 = .0168 ft, or about oue-fifth inch only, the additional head reqd. See 
next table, No. 8. 


Du limit's rule for the additional head required to over¬ 
come the resistance of circular bends in water pipes. Having 

diam of pipe, in ft; rad of bend, in ft, central angle w s o, Fig 2; and vel in ft per sec. Div er, Fig 
2, or half the diam of the pipe, by the rad s w of the outer side of the bend. The quot will be the 
nat versed sine of Du Buat’s angle Of reflexion. Take this versed sine from unity, or 
1. The Rem will be the nat cosine of the same angle. From the Table of Nat Sin and Tang, take 
both the augle and the nat sine corresponding to this nat cosine. Call the angle R. Also, square the 
nat sine, and call this square S. Take the angle tv s o from 180°. Div the rem by twice the angle R 
of reflexion just found. Call the quot T. Finally, mult together the constant dec .00375, the square 
of the vel iu ft per sec, the quot T, and the square S. The prod will be the reqd extra head in feet, 


17 






































256 


HYDRAULICS. 


EX. The same as the foregoing one for Weisbaoh's rule; that is, a hVnV^ufre'in 

rad s w of outer side of bend, 5.75 ft; vel 4 ft per sec. What extra head will the bend require, in 

order that this vel may not be diminished 1 


— .13043 = nat versed sine of angle of reflexion. And 1 — .13043 — .86957 - nat cos of 


Here, r^r - .13043 : 

same angle. In the Table of Nat bin &c, we find, opposite the nat cos .86957, tlm angle R - 

29° 35'; and its nat siue .4937. The square of .4937 = .2437 - S. Again, 180 — 90° — 90°. An 

90°__ _ 5400 min __ ^ ^ ^ T f'i na uy ) t ) le square of the vel is 16 : heuce, we have .00375 X 

16 X 1 52 X .2437 = .0222 ft. the reqd extra head; or about 34 of au inch. Weisbach s rule gave 
.0168 ft, or about one-fifth of an inch. Heuce we see that the resistance produced by well-rounded 
bends is not great. 



Fit?. 3. 


When the rad r s, Fisc 2, of the bend, is less 
than about two diains of the pipe, which will rarely hap¬ 
pen the resistance to the How of the water increases very rapidly; while, ou 
the’other hand, by Weisbach’s rule, as we understand it, no advantage ap¬ 
pears to be gained by using a rad greater than 5 diams of the pipe.* Employ¬ 
ing Weisbach’s formula, the writer has drawn up the following table of beads 
reqd to overcome the resistance of one bend of 90°, for diff \els in ft per sec, 
and for anv diam whatever. This table extends from a rad of 5 diams down 
to one of % diam ; which is the smallest possible, inasmuch as it leads to a 
bend like Fig 3. 

A vel. of 12 ft per sec is equal to 8.18 miles per hour; one which will rarely 
occur, inasmuch as it requires a head of about 330 feet per mile. 


> 


f 


TABEE 8. Heads required to overcome the resistance in 
circular bends of 90°. Original. 


Velocity in feet per Second. 

1 ft. j 2 ft. j 3 ft. j 4 ft. | 5 ft. j 6 ft. | 7 ft. | 8 ft. j 9 ft. j 10 ft. | 12 ft. 
HEADS IN FEET. 


Rad = 5 diams of 
the pipe. 

.001 

.004 

.009 

.016 

.025 

.036 

.050 

.065 

.082 

.101 

.145 

Rad - 3 diams... 

.001 

.004 

.010 

.017 

.027 

.038 

.052 

.069 

.086 

.106 

.153 

Rad = 2 diams... 

.1)01 

.005 

.011 

.019 

.029 

.042 

.057 

.074 

.094 

.116 

.167 

Rad — 1% diams... 

.001 

.005 

.012 

.021 

.033 

.048 

.066 

.086 

.108 

.134 

.192 

Rad = 134 diam... 

.002 

.007 

.015 

.026 

.041 

.059 

.080 

.104 

.132 

.163 

.235 

Rad = l diam... 

.002 

.009 

.020 

.036 

.056 

.081 

.110 

.144 

.182 

.225 

.324 

Rad = % diam... 

.005 

.018 

.041 

.072 

.113 

.162 

.221 

.288 

.365 

.450 

.649 

Rad = 34 diam... 

.0)6 

.062 

.140 

.248 

.388 

.559 

.761 

.994 

1.26 

1.55 

2.24 


If the central angle r s o, Fig 2, should be either greater, or 
less than 90°, then the heads given in the table, must be increased, or dimin¬ 
ished directly in the same proportion. 

Experiments by Rennie, with a pipe 15 ft long; and *4 inch bore ; with 
4 ft head, gave the following disch in cub ft per sec: 

Straight.00699 cub ft. I One bend at right angles near end.00556 

15 semicircular bends.00617 •• | 24 bends at right angles.00253 

The mean of many careful experiments tried at Rivorpool, 
England, with a leaden pipe, 75 ft long, % inch bore, under 8 ft head, gave the ( 
following number of secs to discharge one gallon of water: 

75 ft pipe, straight and horizontal.. 81.56 sec. 4 vertical bends near discharge end 85.00 sec. 

2 hor bends near discharge end ... 83.33 “ 4 vertical bends near supply end... 84.00 “ 

2 hor bends near supply end.81.80 “ 























































































































HYDRAULICS. 


257 


TABLE 9. 


Ang. of 


Ang .of 


Ang. of 


Def. 

Constant 

Def. 

Constant. 

Def. 

Constant. 

iu Degs. 


iu Deg. 


in Degs. 


140° 

2.431 

70° 

.533 

25° 

.049 

130 

2.158 

60 

.364 

20 

.030 

1-20 

1.861 

50 

.234 

15 

.016 

110 

1.556 

40 

.139 

10 

.007 

100 

1.260 

35 

.102 

5 

.002 

90 

.984 

30 

.073 



80 

.740 






Coustants intermediate of those in the table may be obtained near enough by simple proportion. 



n a m 


Fig. 5. 


Tiled isoh is diminished by swellings, or enlargements in pipes, 

as well as by contractions, beuds, and knees; on account ot' the eddies which they produce, &c. 

Art. 6. Inasmuch as the pres of quiet water against, and perp to, any given 
surf, is (other things beiug equal) in proportion to the vert height of the water above the cen of grav 
of the pressed surf, (see Art 1, Hydrostatics,) it follows that in two pipes of the same diams, as a 6, 
and c h, Fig 5, the pres against, and at right 
augles to, the equal bases, mn of the vert pipe, 
and op of the inclined one, are equal; because 
the vert heights, a b and hg, of the water above 
the cen of grav a and c. of the equal bases, are 
equal in the two pipes. If the base of the inclined 
pipe be cut so that ty becomes the base, then the 
base is no longer a circle, but an ellipse; the 
area of which will always be greater than that 
of the circular one; and since the vert height h g 
remains unchanged, the pres against the base t y, 
and perp to it, will be greater than that against 
op, in the same proportion as the two areas. 

The upright pipe may be but 1 ft long; and the 
inclined one 1 mile, or 10 miles long, still the pres 
f at the base ran will be the same as that at the 
base op, so long as the vert height a 6 is equal to 

the vert height gh. The greater weight of the water in c h. does not increase the pres at its lower end 
op ; said weight being sustained by the under part pw of the inclined pipe. 

If, therefore, two steam pumps, with plungers of equal diameter, were 
employed ; one to force the water up the one foot long vertical pipe, and the 
other to force it up the ten miles of inclined pipe; both engines would have to 
exert the same force to balance their respective columns of water ; i e, to uphold 
them. In other words, the static pressureof the water is the same in both cases; 
being equal to the weight of a cylindrical column of water of the same diameter 
as the plunger ,* and as long as the vertical stretch, a b or g h, of the pipes. 

But in order to move the water in either pipe at a given velocity, an additional 
force is required, equal to the weight of a cylindrical column oi water, the 
diameter of which is equal to that of the plunger , and the length of which is 
equal to the total head (calculated by Art 3, p 248) required to force water at 
the given velocity through a pipe of the given dimensions. The weight of this 
second column is the pressure, or motive/orce, necessary to give the required veloc¬ 
ity to the water, to overcome its friction in the pipe, and to put water into the 
pipe at its lowerend as fast as it passes out at the upper end. See Art 1 a, p 237. 

The total force, or total steam pressure required in the cylinder, is the sum of 
these two pressures ; namely, of the static pressure and the motive lorce. 

In the foregoing we have assumed that, the resistance to entry is only such as 
would be encountered by water when forced by any means from a reservoir into 
the open end of an ordinary pipe ; so that the “entry head” may be taken as 
equal to about the velocity head. In practice, a much greater resistance to 
entry is offered by valves, by sudden bends in pipes leading to stud Irom air 
chambers, etc. Still, in most cases the entry head, even as thus increased, is 
but trifling in comparison with the total head. 

Art. 7. The flow of water throngli openings, or apertures, 
in the sides or bottom of the containing vessel, or reservoir. 

Theoretically, the vel with which water should flow through such an opening, is equal to that which 
would be acquired by a heavy bodv falling freely through a height equal to the head, or depth or 
water measured vert from the level surf of the water in the reservoir, to the center of the opening; 
.... correctly, to its cen of grav. This theoretical vel is found in ft per sec, by mult the sq 

i t o! said head, or vert depth in ft, by the constant number 8.03; or, mult the head itself in ft, by 

* Because the pluuger now forms practically the bottom, n m or op, ot the pipe. See Kent 't 
p 'll 3. 











































258 


HYDRAULICS 


64.4. and take the sq rt of the prod. In practice, we may use 8 and 64. as near enough. The theo¬ 
retical, as well as the actual ditch, or the quantity in cub ft, whioh Hows out per sec. is evidently 
equal in all cases to the prod of the theoretical, or of the actual vel, (as the case may be,) in ft per 
sec. mult by the area of the opeuing in sq ft. 

These theoretical laws apply equally to all Huids, whatever may be their sp grav ; thus, theoretically, 
mercury, water, air, &c. will all How with equal vels from openings of equal sizes, uuder equal heads. 

Practically, however, only the mean vel, and the disch through the VCIia CODtPftCtfl, or 
contracted vein, (see Kig 11,) which forms itself just outside of certain kinds of opeuings, 
(and which is smaller thau the opeuiugs themselves,) are actually very nearly equal to the theoret¬ 
ical ones; but through the very opening itself they are usually less. The discrepancy is greater iu I 
some cases than in others: depending chiefly on the shape of the openiug. 

On this account, the theoretical vel and disch found by the foregoiug rule, must usually be dimin¬ 
ished by mult them bv certain decimal numbers corresponding to the various kinds of openings • and 
called coefficients of discharge. These coetfs have in many cases been determined by experimi--it 
very approximately ,- and will be found in the following articles. It will be seen iu Remark 5, p 260, 
that, by the use of a peculiarly formed adjutage, or attachment, to small opeuings, the actual disuil 
mav even be increased beyond the theoretical one. 

The following table will save the trouble of calculating the theoretical vel, previously to mult it bv - 
the corresponding coed' of disoh, for obtaining the actual vel. The ooeffs for diff kiuils of opeuings 
will be found further ou. 

TABLE lO. Of the theoretical velocities in feet per sec, 

with which water should How out into the air. under diff heads, through opeuiugs iu the bottom or 
sides of the containing reservoir; the surf level of which remains constantly at the same height. 

Weisbach says (see third footnote to Art 9) that when water flows out of an opening under water, 
as at n. Fig 1, the vel and disch are about iXr part less than when it flows into the open air, under 
equal heads. When the disch is made under water, the vert dist a u. Fig l. between the surf levels 
of the two reservoirs, must be taken as the head. These theoretical vels are very nearly the actual 
mean ones at the contracted vein; see Art 9. Calling the head, H, then 

^ ^*fn per^sec ^ 2 ff = l/ 64-4 H ~ 8-03 times the sq rt of the head in ft. 

Theoret ical head _ _ vel 2 _ (square of theoret vel\ v nlM 

in feet ~ g g ~ V in ft per sec ) ~ 


Head 

Vel. 

Head 

Vel. 

Head Vel. 

Head 

Vel. 

Head 

Vel. 

Head 

Vel 

Head 

Vel. 

Feet. 

Ft per 

Feet. 

Ft pet 

Feet. 

Ft per 

Feet. 

Ft per 

Feet. 

Ft per 

Feet. 

Ft pei 

Feet. 

Ft per 

sec. 


sec. 


sec. 


sec. 

sec. 

sec. 

sec. 

.005 

.57 

.29 

4.32 

.77 

7.04 

1.50 

9.83 

7. 

21.2 

28 

42.5 

76 

63.9 

.010 

.80 

.30 

4.39 

.78 

7.09 

1.52 

9.90 

.2 

21.5 

29 

43.2 

77 

70.4 

.015 

.98 

.31 

4.47 

.79 

7.13 

1.54 

9.96 

.4 

21.8 

30 

43.9 

78 

70.9 

.020 

1.13 

.32 

4.54 

.80 

7.18 

1.56 

10.0 

.6 

22.1 

31 

41.7 

79 

71.3 

.025 

1.27 

.33 

4.61 

.81 

7.22 

1.58 

10.1 

.8 

22.4 

32 

45.4 

80 

71.8 

.030 

1.39 

.34 

4.68 

.82 

7 26 

1.60 

10.2 

8. 

22.7 

33 

46.1 

81 

72.2 

.035 

1.50 

.35 

4.75 

.83 

7.31 

1.65 

10.3 

.2 

23.0 

34 

46.7 

82 

72.6 

.040 

1.60 

.36 

4.81 

.84 

7.35 

1.70 

10 5 

.4 

23.3 

35 

47.4 

83 

73.1 

.045 

1.70 

.37 

4.87 

.85 

7.40 

1.75 

10.6 

.6 

23.5 

36 

48.1 

81 

73.5 

.050 

1.79 

.38 

4.94 

.86 

7.44 

1.80 

10.8 

.8 

23.8 

37 

48.8 

85 

74.0 

.055 

1.88 

.39 

5.01 

.87 

7.48 

1.85 

10.9 

9. 

24.1 

38 

49.5 

86 

74.4 

.060 

1.97 

.40 

5.07 

.88 

7.53 

1.90 

11.1 

.2 

24.3 

39 

50.1 

87 

74.8 

.065 

2.04 

.41 

5.14 

.89 

7.57 

1.95 

11.2 

.4 

24.6 

40 

50.7 

88 

75.3 

.070 

2.12 

.42 

5.20 

.90 

7.61 

2. 

11.4 

.6 

24 8 

41 

51 3 

83 


.075 

2.20 

.43 

5.26 

.91 

7.65 

2.1 

11.7 

.8 

25.1 

42 

52.0 

90 

76.1 

.080 

2.27 

.44 

5.32 

.92 

7.70 

2.2 

11.9 

10. 

25.4 

43 

52.6 

91 

76.5 

.085 

2.34 

.45 

5 38 

.93 

7.74 

2.3 

12.2 

.5 

26.0 

44 

53.2 

92 

9.3 

76.9 

.000 

2.41 

.46 

5.44 

.94 

7.78 

2.4 

12.4 

11. 

26.6 

45 

53.8 

77.4 

.095 

2.47 

.47 

5 50 

.95 

7.82 

2.5 

12.6 

.5 

27.2 

46 

51.4 

91 

77.8 

.100 

2.54 

.48 

5.56 

.96 

7.86 

2.6 

12.9 

12 . 

27.8 

47 

55.0 

95 

78.2 

.105 

2.60 

.49 

5.62 

.97 

7.90 

2.7 

13.2 

.5 

28.4 

48 

55.6 

96 

78.6 

.110 

2.66 

.50 

5.67 

.98 

7.94 

2.8 

13.4 

13. 

28.9 

49 

56.2 

97 

79.0 

.115 

2.72 

.51 

5.73 

.99 

7.98 

2.9 

13.7 

.5 

29.5 

50 

56-7 

98 

79.4 

.120 

2.78 

.52 

5.79 

1 Ft. 

8.03 

3 . 

13.9 

14. 

30.0 

51 

57.3 

99 

79 8 

.125 

2.84 

.53 

5.85 

1.02 

8.10 

3.1 

14.1 

.5 

30.5 

52 

57.8 

100 

80.3 

.130 

2.89 

.54 

5.90 

1 04 

8.18 

3.2 

14.3- 

15. 

31.1 

53 

58.4 

125 

89.7 

.135 

2.95 

.55 

5.95 

1 06 

8.26 

3.3 

14.5 

.5 

31.6 

54 

59.0 

150 

98.3 

.140 

3.00 

.56 

6.00 

1.08 

8.34 

3.4 

14.8 

16. 

32.1 

55 

59.5 

175 

106 

.145 

3.05 

.57 

6.06 

1.10 

8.41 

3.5 

15. 

.5 

32.6 

56 

60.0 

200 

114 

.150 

3.11 

.58 

6.11 

1.12 

8.49 

3.6 

15.2 

17. 

33.1 

57 

60.6 

225 

120 

.155 

3.16 

.59 

6.17 

1 14 

8.57 

3.7 

15.4 

.5 

33.6 

58 

61.1 

250 

126 

.160 

3.21 

.60 

6.22 

1.16 

8.64 

3.8 

15.6 

18. 

34.0 

59 

61.6 

275 

133 

.165 

3.26 

.61 

6.28 

1.18 

8.72 

3.9 

15.8 

.5 

34.5 

60 

62.1 

300 

139 

.170 

3.31 

62 

6.32 

1 20 

8.79 

4 . 

16.0 

19. 

35.0 

61 

62.7 

350 

150 

.175 

3.36 

.63 

6 37 

1.22 

8.87 

.2 

16.4 

.5 

35.4 

62 

63.2 

400 

160 

.180 

3.40 

.64 

6.42 

1.24 

8.94 

.4 

16.8 

20. 

35.9 

63 

63.7 

450 

170 

.185 

3.45 

.65 

6.47 

1.26 

9.01 

.6 

17.2 

.5 

36.3 

64 

61.2 

500 

179 

.190 

3.50 

.Oft 

6.52 

1.28 

9.08 

.8 

17.6 

21. 

36.8 

65 

64.7 

550 

188 

.195 

3.55 

.67 

6 57 

1.30 

9.15 

5. 

179 

.5 

37.2 

66 

65.2 

600 

197 

.200 

3.59 

.68 

6.61 

1.32 

9.21 

.2 

18.3 

22. 

37.6 

67 

65.7 

700 

212 

.21 

3.68 

.69 

6.66 

1.34 

9.29 

.4 

18.7 

.5 

38.1 

68 

66.2 

800 

227 

.22 

3.76 

.70 

6.71 

1.36 

9.36 

.6 

19. 

23. 

38.5 

69 

66.7 

900 

241 

.23 

3.85 

.71 

6.76 

1.38 

9.43 

.8 

19 3 

.5 

38.9 

70 

67.1 

1000 

254 

.24 

3.93 

.72 

6.81 

1.40 

9.49 

6. 

19.7 

24. 

39.3 

71 

67.6 

.25 

4.01 

.73 

6.86 

1.42 

9.57 

.2 

20.0 

.5 

39.7 

72 

68.1 



.26 

4.09 

.74 

6.91 

1.44 

9.63 

.4 

20.3 

25 

40.1 

73 

68.5 



.27 

4.17 

.75 

6 95 

1.46 

9.70 

.6 

20.6 

26 

40.9 

74 

69.0 



.28 

4.25 

.76 

6.99 

1.48 

9 77 

.8 

20.9 

27 

41.7 

75 

69.5 









































HYDRAULICS. 259 


Art. 8. On tlie flow of wafer 
through vertical openings fur¬ 
nished with short tubes. When water 

flows from a reservoir, Fig 6, through a vert partition 
m m a a, the thickness a m of which is about 2H or 8 times 
the least transverse dimension of the opening, (whether 
that dimension be its breadth, or its height;) or when, if 
the partition be very thin, as n n , the water flows through 
a tube as at t, the length of which is about 2 or 3 times its 
least transverse dimension, then the eflluent stream will 
eutirely fill the opening, or the tube, as shown in Fig 6; or, 
in technical language, will run with a full flow; or a full 
bore ; and will disch more water in a given time, than if 
the tube were either materially longer or shorter. For if 
longer than 3 times the least transverse dimension, the 
flow will be impeded by the increased friction against the 
sides of the tube ; and if shorter than about twice the least 
transverse dimension, the water will not flow in a full stream, but in a contracted one. as shown by 
Fig 11. This will be the case whether the tube be circular, or rectilinear, in its cross-section. 

To find approximately the actual vel. and disch into tlie 
nir. through a tithe, or opening;, either circular or recti¬ 
linear in its outline, or cross-section; and whose length e i, 
or c e. in tlie direction of the flow, is about 2)4 or It times its 
least transverse dimension ; when the surface-level, .v, Fig;' (», 
remains constantly at the same height; anil which height 
must not be below the upper edge of the tube, or opening. 

Rule 1. Take out the theoretical vel from Table 10, corresponding to the head measured vert 

from tiie center (or more properly, the cen of grav) c, of the opening, to the level water surf s. Mult 
it by the coeff of disch .81. The prod will be the reqd vel, in ft per sec. Mult this actual vel by tlie 
transverse area of the opening, in sq ft. If circular, knowing its diam. this area will be found in 
Table 3. The prod will be the quantity of water dischd, in cub ft per sec; within, probubly, 3 

or 4 per cent. 

Rule 2. Find the sq rt of the head in ft. Mult this sq rt by 6.5. The prod will be the nctual 
vel in ft per sec. 

Ex. An opening c o ; or box-shaped tube c t, Fig 6. is 3 feet wide, by .25 of a ft high ; and its length 
in the direction cl or c e in w'hich the water flows is nbout .62 of a ft, or about 2 hi times its least 
transverse dimension, or its height. The head from the cen of grav c, of the opening, to the constant 
surf-level s, is 4 feet. What will be the vel of the water: and how much will be dischd per sec? 

By Rule 1. The theoretical vel (Table 10. ) corresponding to a head of 4 ft is 16 ft per sec. 

And 16 X .81 — 12.96 ft per sec, the actual vel reqd. Again the transverse area of the opening, or of 
the tube, is 3 ft X -25 ft = .75 sq ft. And .75 X 12.96 = 9.72 cub ft; the quantity dischd per sec. 

By Rule 2. The sq rt of 4 is 2. A nd 2 X 6.5 = 13 ft per sec. the reqd vel, as before; the very slight 
diff being owing to the omission of small decimals in the coefTs. 

Rf.m. 1. If the short tube t projects partly iusiile of the vert 

partition n n, the disch will be diminished about x /s P a rt. In that case, use .71 

or .7 instead of the .81 of Rule 1 ; or 5.7 instead of the 6.5 of Rule 2. 

Rem. 2. When the thickness a m of the vert partition m m a a ; or the length c e of the tube t. Fig 
6. is increased to about 4 times the least transverse dimension of the opeuing; or of the diam, when 
circular; then the additional friction against its sides begins appreciably to lessen the vel and disch. 
In that case, or for still greater lengths, tip to 100 diams, they may be found approximately, by using 
instead of the coeff of disoh .81 in Rule 1, the following coeffs, by whioh to mult the theoretical vels 
of Table 10. Or use Rule, p 243. 

TABLE 11. 



Length of 
Pipe 

in Diams. 

Coeff. 

Length of 
Pipe 

in Diams. 

Coeff. 

4 

.80 

40 

.62 

6... 

... .76 

50... 

... .60 

10 

.74 

60 

.57 

15... 

... .71 

70... 

... .55 

20 

.69 

80 

.52 

25... 

... .67 

90... 

... .50 

30 

.65 

100 

.48 


Rem. 3. When the length of the opening or tube, in the direction in which the wateir fh 

*s than about twice its least transverse dimension, the disoh is diminished , so that for lengths from 

4 times, down to openings in a very thin plate, we may use .61, instead of the .81 of Rule 1. lor 

REIT'S ;r,h«"'.E K? !&>h lhro.,1, .hort r nlng. »« I'M .. ™ .ho.. I. 

ig 6 mav be increased to nearly the theoretloal ones of Table 10, by merely rounding off neatly the 
Iges of the entrance end or mouth, as in Fig 7; which is the shape, and half actuabs ze of onei with 
hich Weisbaoh obtained .975 of the theoretical vel and discharge, when the head was 10 ft, and .958 


































* 


HYDRAULICS 


260 


with a head of oue foot; so that in similar cases, .975, and .958 may be used instead or the eoeff .81 
in Rule 1. 



As much as .92 to .9+ mav be obtained by widening the opening, m n, toward its onter mouth, o *, 
Fig, 8, making the divergence, or angle a. about 5°: or by widening it toward its iunor mouth, as at 
t c, Fig 9; but increasing the augie of divergence, at 6. to from 11° to 16°. In all cases, we cousider 
the small end as being the opening whose area must be multiplied by the vel to get the discharge. 

Iii some experiments made witli large pyramidal wooden 
troughs 9.5 ft long, with an inner mouth of 3.'J X 2.4 it, and a discharging one 
of .62 X .11 ft; and under a head of 9J4 feet, the discharge was .98 of the theoretical oue, due to the 
smaller end. Therefore, .98 may be used in such cases, instead of the .81 of Rule l. 

Rem. 5. The discharge through a short opening* of small 
transverse section may even be made 50 per cent greater 
than the theoretical one, by adopting the shape, Fig 10; where m n is sup¬ 
posed to he the diameter of the opening. 
The best proportions appear to be about as 
follows: o y = 9iuches; to n= Much; be 
— 1.8 inch ; o s~ % iuch ; ad = 2 ius; the 
curves, a m, aud dn, being quadrants; 
the angle, x, of divergence, about 5° 6'; 
and the tube of polished metal. In this 
case use 1.55, or more safely, 1.5. instead 
of the .81 of Rule 1. The only experi¬ 
ments with this form have been on a very 
small scale. To what extent it may be 

applicable is unknown. 

So far as regards the ordinary operations of the engineer, this subject is perhaps more curious than 
useful; for he will rarely have any difficulty in making his openiugs large enough, without resorting to 
such aids; except, perhaps, that of rounding off the inner edges, as in Fig 7; which is usually done. 


a 



Fig. 10. 




Art. 9. On the disch of wafer through openings in thin 
vert partitions, with plane or flat faces, ee, or tin, Fig 11.* If the 

face e e, or n n, instead of beiug plane, and vert, should be curved, 
or inclining in diff directions toward the opening, then the disch 
will he altered. When water Hows from a reservoir. Fig 11, through 
a vert plane plate or partition nn, which is uot thicker than about 
the least transversedimensiou of the opening, whetherthatdimension 
be its breadth, or its height o o ; t or w’hen. if the partition e e itself 
is much thicker, we give the opening the shape shown at b, (which 
evidently amounts to the same thing,) then the effluent stream will 
not pass out with a full flow, as in Fig 6, but will assume the shape 
shown in Fig 11; forming, just outside of the opening, what is 
called the vena contractu, or contracted vein. In order that this 
contraction may take place to its fullest extent, or become complete, 1 
the inner sharp edges of the opening must not approach either the 
surf of the water, or the bottom or sides of the reservoir, nearer 
than about 1 times the least transverse dimension of the opening. 
The contracted vein occurs at. a dist of about half the smallest Hi- 
mension of the orirtce. from the orifice Itself. In a circular orifice, 
at about half the diam dist; and ordinarily its area is about .62 or nearly % that of the orifice itself. 

At this poult the actual mean vel of the stream is very nearly (about .971 the theoretical vel given by 
I able 10, and hence the actual diache are but .62, or nearlv % of the theoretical ones. 

Case 1. To find the actual disch into air.]; Ilirougli either a 
circular or rectilinear^ opening in a thin vert plane parti- 




D 

Fig. 12. 


* We believe that these rules for thin plate are also sufficiently approximate 
for most practical purposes, if the opening be in the bottom of the reservoir • 
or in an inclined, instead of a vert side. 

t When the side of a reservoir, or the edge of a plank. Ac. over which water 
flows, has no greater thickness than this, the water is said to flow through, 
or over, thin plate, or thin partition. 

J Should the disch take place under water, as in Fig 12, both surf-levels re- 
maiming constant, then the head to he used is the vert diff n o, of the two 
levels. After making the calculation with this head, we should, according to 

Weisbaeh, deduct the part; inasmuch as he states that the disch is that 
much less when under water, than when it takes place freelv into the air 
Other experimenters, however, assert that it is precisely the same in both cases 
§ If the shape of the opening is oval, triangular, or irregular the head 
must be measured vert from its cen of grav. ’ 















































HYDRAULICS. 


261 


tion, when the contraction is complete; ami when the surf- 
level, .v, remains constantly at the same height; water being 

supplied to the reservoir as fast as it runs out at the open¬ 
ing.* 


Rule 1. When the head, measured vert from the center (or rather from the cen of grav) c, of the 
opening, to the surt level s of the reservoir, is not less than 1 ft. uor more thau 10 ft; and when the 
least transverse dimension of the opening is not less than an inch, mult the theoretical vel in ft per 
sec due to the head, (Table 10, ) by the coefficient of disch .62. The prod will be the actual 

mean vel of the water through the opening. Mult this vel by the area of the opening iu sii ft - the 
prod will be the disch in cub ft per sec, approximately. 

When the head is greater thau 10 ft, use .6, instead of .62. 

Rule 2. Piud the sq rt of the head iu ft. Mult this sq rt by 5; the prod will be the vel in ft per 
sec ; which mult by the area as before for the disch. 

Ex. What will be the disch through an opening in complete contraction, whose dimensions are 6 
ins, or .5 ft vert; and 4 ft hor ; the vert head above the cen of grav of the opening being constantly 
6 feet? 

By Rule 1. The theoretical vel (Table 10, ) corresponding to 6 ft head, is 19.7 ft per sec. And 

19.7 X <>2 — 12.214 ft, the reqd vel. Again, the area of the opening = .5 X 4 — 2 sq ft; and 12 214 X 
2 — 24.428 cub ft per sec; the disch. 

By Rule 2. The sq rt of 6 = 2.45 ; and 2.45 X 5 -12.25 ft per sec, the reqd vel; and 12.25 X 2- 
24.5 cub ft per sec, the disch. 

Both very approx even if the orifice reaches to the surface of the issuing water. 

Kent. 1. The coef .62 is a mean of results of many old experimenters. 

In 1874 Genl. T. G. Ellis of Massachusetts conducted an elaborate series (Trans Am Soc C E, Feb 
1876) on a large scale, the general results of which, within less than l per ct, are given in the follow¬ 
ing table. See also Rem 3. The sharp edged orifices were in iron plates .25 to .5 inch thick. 


Orifice. 

Head above Center. 

Coef. 

2 ft sq. 

2. to 3.5 ft. 

.60 to .61 

2 “ long, 1 ft high 

1.8 to 11.3 “ 

.60 to .61 

2 “ long. .5 high 

1.4 to 17.0 “ 

.61 to .60 

2 “ diam. 

1.8 to 9.6 “ 

.59 to .61 


Rem. 2. Extreme care is reqd to obtain correct results; but for many 

purposes of the engiueer au error of 5 to 10 per ct is uuimpurtaui. 

It will rarely happen that greater accuracy is required than may be obtained by the foregoing 
rules; but when such does occur, aid may be derived from the following tallle deduced 

from llic experiments of Lesbros and Poneelef. on openings 8 ins 

wide, ol dirt' heights, and with diff heads. Use that coeff iu the table which applies to the case, in¬ 
stead of the .62 of Rule 1. In some of the cases in this table, the upper edge of the opening is 
nearer the surf-level of the reservoir than times its least transverse dimension. 


TABLE 12. Coefficients for rectangular openings in tbisi 
vertical partitions in full contraction.* 


Head 
abovecen. 
of grav. of 
opening 
in Feet. 

Head 
abovecen. 
of grav. of 
opening 
in Inches. 

Ins. 

8 

The bre- 

II 

Ins. 

6 

idt.h in a 

EIGHT 

Ins. 

4 

1 the ope 

OF O] 

In 8. 

3 

nings — 

PEN IN 

Ins. 

2 

inches. 

a. 

Ins. 

.033 

.4 








.8 







.0833 

1 






.64 

.1*25 

I % 





.61 

.64 

1666 

2 




-60 

.6*2 

.64 

.*2083 




59 

-61 

.62 

54 

250 

3 



.60 

.61 

.62 

.64 

.2917 



.57 

.60 

.61 

.62 

.64 

.3333 

4 

........ 

.58 

.60 

.61 

53 

.64 

.3750 


.56 

.59 

.60 

51 

.63 

.64 

.4167 

5 

57 

.59 

.61 

.62 

53 

54 

.6666 

8 

.59 

.60 

.61 

.62 

.63 

.64 

1 

12 

.60 

.60 

.61 

.62 

.63 

.63 

3 

36 

.60 

.60 

.61 

.62 

52 

53 

5 

60 

.60 

.60 

.61 

.61 

52 

52 

10 

120 

.60 

.60 

.60 

.60 

.60 

51 


las. 

.4 


.70 

.69 

.68 

.68 

.68 

.67 

57 

56 

j6S 

.66 

.66 

.65 

.64 

.63 

.62 

.61 


Rf.m. 3. Careful experiments on openings 4^ ft wide, and It* 
ins high, under heads of from 6 to 15 ft, show that the coeff .62 will give results 
correct within -^L- part, for openings of that size also, under large heads; although the thickness of 
Mie partition varied on its diff sides, from 12 to 20 ins. It must be recollected, however, that nothing 
more than close approximations are to be attained in such matters. 

Rem. 4. It has been asserted by some writers, that wlien two or more 
contifjuous openings are discharging at the same time from the same reser¬ 
voir. they disch less in proportion than when only one of them is open. Other experiments, how¬ 
ever. seem to show that this is not the case; it is therefore probable, at least, that the diff, if any, 
is but trifling. See Art 1 N, p. 239. 


* See first footnote on preceding page. 












































262 


HYDRAULICS 


Case 2. The discharge thron^h thin vert partitions in coin* 
pletecontraction, when the surface-level, i'ijf IS, descends 
as the water tlows out into the air. In this case, if the reservoir is 

prismatic that is, if its hor sections are everywhere equal; ami if no water is flowing iuto the reser¬ 
voir. to supply the plac of that which tlows out, theu, to liuil the time reqd to disch the reservoir. 

Rout. Inasmuch as the lime in which such a reservoir entirely discharges itself, is twice that in 
which the same quantity would flow out under a constant head, as in Case 1, therefore, cal¬ 

culate the disoh in cub ft per sec by Rule 1, Art H; div the number of cub It cou- 
3tl 1 tained in the reservoir, above the level g of the bottom of the opening. Fig 13, by 


TV 


this disch ; the quot w ill be the number of sec in which a volume equal to that in 
the reservoir, to the depth y. would run out in Case 1. of a constant head. Aud 
hvice this number will be the seconds reqd to empty the reservoir in Case 2, of a 
varying head. 

Rem. If it should be reqd to find the time in which such a prismatic reservoir 
would partly empty itself, as, for instance, from m to n, Fig 13, first calculate, by 
the above rule, the secs necessary to empty it if it had only been filled to n: aud 
afterward calculate as if it had been filled to m. The diff between the two times 
will evidently be the time reqd to empty it from m to n. If the opening is not iu 
complete contraction, see Arts 11, &c. 

If ttiediscli is into a lower reservoir, whose 
surf-level remains constant, proceed in the same manner; 
only use the diff of level of the two surfs as the head, and afterward (accordiug 
to Weisbach) increase the time -A part. 

Art. 10. Disci* from a reservoir R, Fi^ 14, the surf-level, s, 
of which remains constantly at the same fteiglit,; tlirougl* 
an opening. <>, in thin vert partition; and iu complete con¬ 
traction; but entirely under water; and into a prismatic 
reservoir, m. 

Seconds required 

to discharge a quantity = 
c d a, the level c remaining 
constant. 


Fig. 13. 



./height a c .. hor area of 
in ft x m in sq ft 


area of opening v v w 
o iu sq ft x - b - x 


./height a c .. hor area of y „ 

Seconds required _ iu ft_ x m i u sq ft x 

to raise level iu m from clou ~ area of opening fio v 8 03 

o in sq ft x x 8,w 


Fig. 14. 


Seconds require*] (' a ^ tll x 

m a in m from y- to - ' 


to raise level in m from c to — 
any other level, d. 


~j\ vtior area of 

m in sq ft 


X 2 


Area of opening v .... v R 
o in sq ft x x 


Rem. 1. If it slionld be reqd to find the time of filling m, from 
its bottom e, up to d. we may do so very approximately by calculating by 
the first rule in Art H. the time reqd from e to the center of the opeuiug o, us if all that portion of 
the disch took place iuto air; and afterward, from the center of the opening to d, by the rule just 
given. This case is similar to that of filling a lock from the canal reach above, in which the surf- 
level may be considered coustant. 

Rem. 2. If the bottom of the opening o. slionld coincide with 
the bottom of the reservoir, then the coeff will become greater than .62. 

See Art 11, for obtaiuing eoeffs for imperfect contraction. 

Hem. 3. Of the opening, instead of being in complete con¬ 
traction. ia of any of the shapes Figs 6 to 9, then a reference to Art 8 will show 
what coeff must be substituted for .62. 

Case 3. Disch from one prismatic reservoir. Fig 15, W. into 
another, X. of any comparative sizes whatever, through an 
opening o, in a plane thin vert partition, and in complete 
contraction; when the water rises in X. while it falls in W. 

To find the time in which the water, flowing from W into X, through 
o, will fall through the diet as, so as to stand at the same level sc, in 
both reservoirs. 

In this case, the water reqd to fill X from e to d, (d being the bottom 
of the opening o.) flows out into the air; and the time necessary for it 
to do so, must be calculated separately from that reqd above d, which 
flows into water. 

Rui.b. First from e to d. Find the hor area of each reservoir, in 
sq ft. Muit the hor area of X, by the vert depth d e in ft, for the cub 
ft contained in that portion. Div these cub ft by the hor area of W. 
The quot will be the dist am, in feet, through which the water iu W 
must descend, in order to fill X to d. 



Seconds re¬ 
quired to low- _ 
er from a to m, and — 
raise from e to d. 


Twice the / / 

hor area of X l w 
W iu sq ft 


head 

in 


an head m n ^ 

ft in ft / 


Area of opening 

© iu Bq ft * * b.O 6 















































HYDRAULICS 


263 


Seconds required 

to lower from m to s, and raise 
from d to c. (Very approx) 


Hor area of v twice the hnr area v ./head m n 
X in sq ft A of W in sq ft X v j n 

Area of /hor area hor area\ 

opening X of W -|- of X I X -62 X 8.03 

oin sq ft \in sq ft in sq ft/ 


Ex. Let the hor area of W be 100 sq ft: and that of X, 60 sq ft. Let a n be 20 ft; and m n 16 ft; 
and the area of the opening o, 3 sq ft. In what time will the water descend from a to «, and rise 
from e to c ? 

inasmuch as the method of finding the time for filling from e to d, by the water failing from a to 
m, requires uo further exemplification, we will coufine ourselves to the additional time necessary for 
filling from d to c, by the water falling from m to s. To find this, we have, the sq rt of the head 

m u = 4 ft; and the sum of the 2 areas = 100 4- 60 = 160. Hence, — * * * -— = . 4S( ^ > - = 

160 X 8.03 X 3 X .62 2389.73 

20.1 sec; the additional time reqd, very approximately. 


Note 1. If tlie opening 1 , as rf, Fig 16, reaches 
to the very bottom of the reservoirs, we may 

consider all the water flowing from R into T, as flowing into water. 
Therefore, usiug the head am, we at once calculate the time necessary 
for the water iu the two reservoirs to arrive at the same level s c, by 
the last process of the preceding rule; or, in other words, by the pro¬ 
cess given iu the preceding example. But in this case it must be borne 
in mind that the opeuiug o is uo longer in complete contraction, inas¬ 
much as the contraction along its lower edge is suppressed. 

The disch will consequently be somewhat increased; and a coeff 
greater than .62 becomes necessary. The method of finding this, is 
given in the following Case 4. A reference to Art 8 will give the coeff 
in case the opeuing is shaped as Figs 6 to 9. 



Art. 11. fuse 4. The discharge through openings in plane 
thin vert partitions; but in incomplete contraction. 


The opening may be such that contraction will take place 
along one portion of its perimeter, or at the top of the open¬ 
ing a, Eig 17 ; while it is suppressed on another portion ; as 
at the bottom and two ends of the opeuiug a ; where suppres¬ 
sion is caused by the addition of short side and bottom pieces 
c, c, c. Or it may be caused by the bottom, or ends, or both, 
coinciding with the bottom aud sides of the reservoir. In 
such cases the disch will be greater than in those of complete 
contraction ; but less thau in those of full flow ; inasmuch as 
the opeuing now partakes somewhat of the character of the 
short tubes of Art 8; and the coeff will rise from .62, or that 
which usually pertains to openings in full contraction ; and 
will approach .8, or that of full flow, in proportion to the ex¬ 
tent of perimeter along which contraction is suppressed ; or 
even to .9 or .98 by the use of such openings as are shown by 
Figs 7, 8, 9. 



To find approximately a new coeff of disch; and the disch 
itself, in cases of incomplete contraction. 

Rule. First find by the foregoing rules, what would be the disch in the particular case that may 
be under consideration, supposing the contraction to be complete. Then div that portion of the 
perimeter of the opening on which contraction is suppressed, by the entire perimeter. Mult the quot 
by the dec .152 if the opeuing is rectangular, or by .128 if circular. To the prod add uuity, or 1. Call 
the sum, g. Then say, as unity, or 1, is to g, so is the coefft'or complete contraction in ordinary cases 
(usually .62) to the reqd new coeff. Finally, repeat the original calculation, only substituting this new 
coeff in the place of .62. 

According to this rule, we have the following coeff of discharge for rectangular openings within pro¬ 
bably 3 or 4 per cent when contraction is not suppressed on more than % of the perimeter. The theo¬ 
retical discharge multiplied by the corresponding coeff will give the actual discharge. Wheu the con¬ 
traction is carried farther, the coeff becomes extremely irregular, and is probably indeterminable. 


For complete contraction (ordinarily ).62 

When contraction is suppressed on perimeter .64 

•• “ “ “ •• % “ •• 67 

• • “ “ “ “ % “ *• 69 

“ “ “ “ entirely around the orifice .80 


Intermediate ones can be estimated nearly enough, mentally. 

Hem. 1. When, instead of a short spout, as in Fig 17. the 
opening- is provided with an indefinitely long hor trough, 

similarly attached, and open at top, there will be no practically appreciable diminution of disch below 
that through the simple opening as at a, Fig 11; provided the head measured above the cen of grav 
of the opening be at least as great as 2 or 2% times the height of the opening itself. Therefore, under 
such circumstances the disch may be calculated by the rules in Art 9. But with smaller heads the 
disch diminishes considerably ; so when the head above the center becomes but as great as the height 
of the opening, it will be but about -i of the calculated one. With still smaller heads, the flow 
becomes less much more rapidly; but has not been reduced to any rule. 

Rem. 2. If, instead of being hor, tlie trough is INCLINED 
































264 


HYDRAUJ -ICS, 


as milch as 1 in 10, the discli will be increased very slightly, (some 3 or 4 per 

c^nt) over that calculated by the rules in Art!), for the plain opening. These results were obtained 
by experiments on a very small scale; aud should be considered as mere approximations. 


Art. 12. In a case like Fis IS, where contraction is supposed 
to he suppressed at the bottom, and at both vert sides ol -the 

opening o, in consequence ot their coinciding 
with the bottom and sides of the reservoir; but where the 
front of the reservoir, instead of being vert, is sloped as at j ; 
aud when the water, after leaving the opening, flows away 
over a slightly sloping apron, g, then the disch in cub ft per 
sec may be approximately found by Rule 1, Case 1, Art 9. 
only substituting .8 in place of .62, when / slopes back 4> J , 
or 1 to 1; or .74 when / slopes back 63°, or with a base of 1 
to a rise of 2. In such cases of inclined fronts, the height of 
the opening must be measured vert, or rather at right ang.es 
to the floor of the reservoir; and not in a line with the 
sloping front. 



Rem. When the front, /, of the reservoir is vert, and a sloping 
apron or trough, < 7 , is used, having its upper edge level with the bottom 

ol the opening, the disch is not appreciably diminished below that which takes place freely into the 
air, provided the head above the cen of grav of the opening is not less than from 


18 to 24 ins, for an opening 6 to 9 ins high. 

12 to 16 “ “ •* “ 4 ins high. 

9 “ “ “ 2 ins or less, high. 


Art. 13. To find, approximately, the time reqd for the emp> 
tying of a pond, or any other reservoir, as l ig lb, which is 
not of a prismatic shape; through an opening, w, near the 
bottom. 


Rulk. First ascertain the exact shape and dimensions 
of the reservoir. If large, aud irregular, it must be care¬ 
fully surveyed; aud souudiugs taken, aud figured upou a 
correct plau aud cross-sections. Next, consider the entire 
body of water to be divided into a series of thiu hor strata. 
A, B, C, U ; the top liue of the lower one being at least a 
few ins above the top of the opening n. It is not necessary 
that these strata should be of equal thickness; although 
the thinner they are, the more correct will the result be. 
The depth of the lower one, D, will vary to some extent 
with the height of the opening; those next above it should 
not exceed about a foot in thickness, until a depth of 6 or 8 feet is reached; then they may conve¬ 
niently. and with sufficient accuracy, be increased to about 2 ft, for 6 or 8 ft more ; and so on ; be¬ 
coming thicker as they approach the surf. By aid of the drawings, calculate the content of each 
stratum in cub ft. Now, since the strata are thin, we may, without serious error, assume each of 
them to be prismatic, as shown by the dotted lines; and may assume that the head under which each 
stratum (except the lowest) empties itself through «, is equal to the vert height from the center of 
the opening to the center of the stratum. Thus, m n will be the head of A ; w n. the head of B ; xn, 
the head of C. Then, for the stratum A by Rule 1, Art 9, (only using inn as the head instead of on,', 
and instead of the coeff .62 of that rule (which can only be used if n is in complete contraction) using 
.61, or whatever other coeff near the end of Art 11 applies to the case, calculate the disch in cub ft 
per sec. Div the coutent of the stratum A by this disch, and the quot will be the number of sec reqd 
for discharging A. Using the head ten, proceed in precisely the same way with the stratum B; and 
using the head xn, do the same with C. Finally, for the lower stratum D. find by Rule 1, Art9. (with 
the same caution as before respecting the proper coeff,) in what time it would empty itself under a 
constant head equal to y n, measured from its surf to the center of the opening. Double this time will 
he that reqd to empty itself in the case before us. under its varying head. Finallv, add together all 
these separate times ; and their sum will be the entire time reqd to empty the pond, or reservoir, ap¬ 
proximately enough for practical purposes. 

Art. 14. On the discharge of water over weirs, or overfalls. 

Experiments and observations on a grand scale, were made on this subject at Lowell, Mass, bv 
31 r James It. 1'raucis, c E, one of the most accomplished hvdraulicians of the age. (See 
his “ Lowell Hydraulic Experiments.”) To apply the rule arrived at by Mr Francis, the following 
conditions must exist. 

The crest a. Fig 20, or top of the weir over which the water discharges, must, he a hor. sharp cor¬ 
ner, in thin plate, or thin partition. (See p 260; first footnote.) Its inner side must form a vertical 
straight line, a h, with the inner face of the dam, to a depth ah, not less than twice the depth, or 
head a m, measured vert from a to the level o m of the hor portion of the water surf: and not to c, the 
curved surf of the falling sheet of water. The head a m may vary from 6 to 24 ins in height. The 
ends ah, ah, of the weir, Figs 21 and 22, must be vert; and its length, a, a, not less th»n3 times the 
head a m. 

These conditions being observed, we may distinguish 2 cases; namely. Case 1st, that in which, as 
in Fig 21, the weir extends entirely across the reservoir: so that its ends ah. a h, coincide with, or 
form portions of, the sides s, s. of the reservoir: in which case contraction takes place only along the 
upper edge a. a, of the weir. Fig 21, as snown at a in Fig 20; but, is suppressed entirely at the ends, 
so that the water flows out as shown in plan by Fig 23. And Case 2d, in which, as in Fig 22, the 
vert ends a h, a h, as well as the crest a, a. are formed with a sharp corner iu thin plate; and'are, 
moreover, removed from the sides v. v, of the reservoir, a dist equal at least to the head am ; so that 
contraction takes place at the ends of the weir, as well as along its crest; and less water flows out, 
as shown in plan at a, a, Fig 24. 





















HYDRAULICS, 


265 



To find the discharge over a weir in thin plate. 

Case 1. When there is no contraction at either end of the 
weir. Figs 21 and 23 : 

Discharge_ 

3 33 X ' en 8 t ^ 1 a a °f x \./ cu ^ e head 
in cab ft per sec " * the overfall, in ft / V a min ft. 

Ex. How many cub ft per sec will flow over such a weir in thin plate, ‘200 ft long; having a head, 
a to, of 1.5 ft, measured to the level surf o to, of the reservoir ; and with no contraction at either end ? 

Here, the cube of 1.5 — 3.375. And the sq rt of 3.375 — 1.837. And 3.33 X 200 X 1-837 = 1223.4 
cub ft per sec, the reqd disch. 

This rule will also be very approximate even when there 
is contraction at both ends, provided the length of the weir 
is at least lO times as great as the head am; and provided the head 

is not less than 2 or 3 ins in depth. Indeed, it will be within about 6 per cent of the truth, for weirs 
with contraction at the ends, and whose lengths are but 4 times the head ; and for the many cases in 
which no closer approximation is reqd, the disch may be tuken at once from table 13. 

Case 2. When contraction takes place at both ends, as in Figs 
22 and 24, or at one end only, use the same formula as in Case 1; except that 
when there is contraction at both ends, one-fifth of the head a to is to be taken from the length a a 
before using it as a multiplier ; and when there is contraction at one end only, oue-tenth of a to must 
first be taken from the length a a. 

Ex. How many cub ft per sec will flow over such a weir in thin plate, 200 ft long; having a head 
a in, of 1.5 ft; with contraction at both ends 7 

Here, the cube of 1.5 = 3.375. And the sq rt of 3.375 = 1.837. Again, one-fifth of a to = .3 of a ft. 
And 200—.3 = 199.7. Therefore we have 3.33 X 199.7 X 1.837 = 1221.6 cub ft per sec, the reqd disch ; 
or (in this cas e's practically the same as when there is no contraction. 

Rem. If, instead of 3.33, we use 3.41. the two foregoing rules will apply to heads am, as small as 
% an inch ; and coeffs between 3.33 and 3.41 may be used for heads between about 5 inches, and 
an iuch, where more thau common accuracy is aimed at. We may also use 3.3 instead of 3.33, for 
heads greater than 2 feet. 

Ey telweinN formula for weirs, over a thin edge, and having no contrac¬ 

tion at either end, is identical with the above. His coefficient is 3.4. 

From the Eowell experiments, by Mr Francis.it appears that 

when the depth, a to, is 1 foot, and the entire sheet of water, after passing over the weir, strikes a 
bor solid floor placed only about 6 ins below the crest a of the weir, the disch is thereby diminished 
but about the one-thousandth part; and that when the head a to is about 10 ins, and falls into water 
of considerable depth, no ditf whatever is perceptible in the disch, whether the surf of that water be 
about 4, or about 13 ins below the crest a; and that a fall below the crest a, equal to one-half of the 
head a to. is quite sufficient. 

Art. 14 A. If the water in the reservoir, or in the feeding 
canal, instead of being stagnant, has a slight current toward 
the weir, the disch will be but very little increased thereby when the head am 
is several ins. Mr Francis observed that a current of 1 foot per sec, or nearly .7 of a mile per hour, 
increased the disch but about 2 per cent, when the head was 13 ins ; and one of 6 ins per sec, about 
1 per ct, when the head was 8 ins. Whenever the effect of the current., however, is so great as to re¬ 
quire uotice, proceed as follows: First find the approx disch as directed in Art 14, under Case 1 or 
Cane 2, according to circumstances. Then, ill C'asc 2. find the area in sq ft of 
cross section of channel of approach at o b 1 , Fig 20. measured between its two sides, and from bottom 
to surf of water. Divide this area by the length of the weir in ft, corrected for contraction as directed 
in Case 2 above. Call the quot D. In Fase 1, D is = the depth ob\ In either 

H 

case, divide the head am or H, in ft, by D. Find this last quot in the column j-j in table 13 A, p 266. 

Mult the approx disch by the corresponding coeff in column K. 

For this method of correction for vel of approach, and for table 13 A, we are indebted to Messrs A. 
W. Hunking and Frank S. Hart. Civ Engs, of Lowell, Mass. They are, in fact, simply a convenient 
method of applying Mr. Francis’ rule for similar cases, which is as follows: 
Find in Table 10. the theoretical head h corresponding to the observed vel of approach. Add 

this head to the head a to, or H, Fig 20. Then, in the formula, Art 14, instead of 


y cube of head a to, use ]/ cube of (H -f- h) — Y cube of A. 
















































266 


HYDRAULICS, 


TABLE 13. Of actual discharges in cub ft per sec, for each 
foot in length of weir in thin plate: and without contraction 
at either end; a h, Fig 20. being vert, and not less than twice 
the head am. Very approximate also, when there is con¬ 
traction at both ends, provided the length be at least 10 
times the head. And but about 6 per cent in excess ot the 
truth, if the length be but 4 times the head. (Original.) 

The decimal .01 of a foot, is precisely .V2 of an inch ; or scant % inch. 


Head, 

Cub. ft. 

Head, 

Cub. ft. 

Head, 

Cub. ft. 

Head, 

Cub. ft. 

Head, 

Cub. ft 

a in. in 

per 

am, in 

per 

am, in 

per 

avi, in 

per 

am, in 

per 

Feet. 

Second. 

Feet. 

Second. 

Feet. 

Second. 

Feet. 

Second. 

Feet. 

Second. 

.03 

.on 

.22 

.351 

.58 

1.47 

.94 

3.04 

2.4 

12.2 

.04 

.037 

.24 

.401 

.60 

1.54 

.96 

3.14 

2.5 

13.0 

.05 

.038 

.26 

.452 

.62 

1.62 

.98 

3.24 

2.6 

13.8 

.06 

.050 

.28 

.505 

.64 

1.70 

1 . 

3.33 

2.7 

14.6 

.07 

.063 

.30 

.560 

.66 

1.78 

1.1 

3.85 

2.8 

15.4 

.08 

.077 

.32 

.603 

.68 

1.86 

1.2 

4.3S 

2.9 

16.2 

.09 

.092 

.34 

.659 

.70 

1.95 

1.3 

4.94 

3. 

17.1 

.10 

.108 

.36 

.719 

.72 

2.03 

1.4 

5.51 

3.1 

18.0 

.11 

.124 

.38 

.789 

.74 

2.12 

1.5 

6.11 

3.2 

18.9 

.12 

.142 

.40 

.842 

.76 

2.21 

1.6 

6.73 

3.3 

19.8 

.13 

.160 

.42 

.907 

.78 

2.30 

1.7 

7.37 

3.4 

20.7 

.14 

.178 

.44 

.972 

.80 

2 38 

1.8 

8.04 

3.5 

21.6 

.15 

.198 

.46 

1.04 

.82 

2.47 

1.9 

8.72 

3.6 

22.5 

.16 

.218 

.48 

1.11 

.84 

2.56 

2 # 

9.42 

3.7 

235 

.17 

.239 

.50 

1.18 

.86 

2.65 

2.1 

10.0 

3.8 

24.4 

.18 

.260 

.52 

1.25 

.88 

2.74 

2.2 

10.8 

3.9 

25.4 

.19 

.282 

.54 

1,32 

.90 

2.84 

2.3 

11.5 

4. 

26.4 

.2 

.305 

.56 

1.40 

.92 

2.94 






In calculating this table, the coeff 3.41 was used for heads from .03 ft to .3 ft; then 3.33 to 2 ft; 
then 3.3 to the end. 


TABLE 13 A. Coefficients It for velocity of approach. For 

the use of this table, see Art 14 A, p265. 


Jff 

D 

K 

H 

D 

K 

H 

D 

K 

_H 

D 

K 

H 

D 

K 

.01 

1.0000 

.09 

1.0020 

.17 

1.0072 

.24 

1.0143 

.31 

1.0239 

.02 

1.0001 

.10 

1.0025 

.18 

1.0081 

.25 

1.0155 

.32 

1.0254 

.03 

1.0002 

.11 

1.0030 

.19 

1.0090 

.26 

1.0168 

.33 

1 0271 

.01 

1.0004 

.12 

1.0036 

.20 

1.0100 

.27 

1.0181 

.34 

1.0287 

.05 

1 0006 

.13 

1.0042 

.21 

1.0110 

.28 

1.0195 

.35 

1.0305 

.06 

1.0009 

.14 

1.0049 

.22 

1.0121 

.29 

1.0209 

.36 

1.0322 

.07 

1.0012 

.15 

1.0056 

.23 

1.0132 

.30 

1.0224 

.37 

1.0341 

.08 

1.0016 

.16 

1.0064 








The following rule for finding the discharge over an over- 
fall, or weir, approximately, has been prepared by the 
author from various data. 

Mult together the length, a, o, of the weir. Figs 21 and 22, In feet; the head, am. Fig 20, measured 
to the lentil surf of the reservoir, in ft; the theoretioal vel in ft per sec, corresponding to the head u m , 
(Table 10, p 258;) and that coeff from the following table, w hich agrees most uearly w ith the case. The 
prod will be the reqd disch in cub ft per sec, uear enough for ordinary purposes ; and probably quite 
as close as cau be arrived at without actual measurement in each case that presents itself. 

Kx. How much water will be dischd over a weir 60 ft long; the crest of which is level, smooth, and 
3 ft wide, or thick; and over which the head a m, Fig 20, is 8 ins, or ,6666 ft thick ? Here the theo¬ 
retical vel for a head of 8 ins, (Table 10. ) is 6.55 ft per sec. The coeff for a weir whose crest is 

level, and 3 ft wide, with a head of 8 ins, is by the following table .31. Consequently, 60X.6666 X 6.55 
X -31 —81.21 cub ft per sec ; the reqd disch approximately. 










































HYDRAULICS, 


267 


TABLE 14. Of coefficients of approximate discharge over 
weirs of different thicknesses, varying from a sharp edge to 
3 feet. —(Original.) ® 


Head 
a m 

in Feet. 

Head 
a in 

in Inches. 

Sharp 

Edge. 

2 Inches 
Thick. 

3 Ft Thick ; 

smooth; 
sloping out¬ 
ward; and 
downward, 
from 1 in 12 
to 1 in 18. 

3 Ft Thick ; 
smooth, 
and level. 

.0833 

1 

.41 

.37 

.32 

.27 

.1666 

2 

.40 

.38 

.34 

.30 

.25 

3 

.40 

.39 

.34 

.31 

.3333 

4 

.40 

.41 

.35 

.31 

.4166 

5 

.40 

.41 

.35 

.32 

.5 

6 

.39 

.41 

.35 

.33 

.5833 

7 

.39 

.41 

.35 

.32 

.6666 

8 

.39 

.41 

.34 

.31 

.8333 

10 

.38 

.40 

.34 

.31 

1 . 

12 

.38 

.40 

.33 

.31 

2. 

24 

.37 

.39 

.32 

.30 

3. 

36 

.37 

.39 

.32 

.30 


Art. 15. If the inner face of the weir and dam, instead of 
being vert, as a b, Fig 20. is sloped, as a or ft, Fig 25; the contraction 

on the crest will be diminished; and consequently the disch will be in¬ 
creased. This will also be the case if the inner corner or edge of the crest 
be rounded off, instead of being left sharp ; or if the sides of the reservoir 
converge more or less as they approach the weir; so as to form wings for 
guiding the water more directly to it; or if a b, Fig 20, be less than twice 
a m. Indeed, so many modifying circumstances exist to embarrass experi¬ 
ments on this, and similar subjects, that some of those which have been 
made with great care, are rendered inapplicable as other than tolerable 
approximations, in consequence of the neglect to take into consideration 
some local peculiarity, which was not at the time regarded as exerting an 
appreciable effect, llnless, therefore, circumstances admit of our com¬ 
bining all the conditions mentioned in the first part of Art 14, thereby securing very approxi¬ 

mate results, we must either resort to an actual measurement of the disch in a vessel of known 
capacity ; or else be contented with rules which may lead to errors of 5, 10, or more per cent, in pro¬ 
portion'as we deviate from these conditions. Frequently even 10 per ct of error may be of little real 
importance. 




Rem. 1. When the water, after pass¬ 
ing over a weir. Fig 2fi, instead o! 
failing freely into the air, is carried 
away by a slightly inclined apron 
or trough. T, the floor of which coincides 
with the crest, a. of the weir, then the disch is not appre¬ 
ciably diminished thereby when the head a m is 15 ins or 
more'. But if the head am is but 1 ft, then the calculated 
disch must be reduced about yly part; if 6 inches, -j- 2 ^-; if 2*-$ inches, ; and if 1 inch, or 
one-half, as approximations. 

Rem. 2. Professor Thomson, of Dublin, proposed the nse of 
Triangular notches, or Weirs, for measuring the disch; inasmuch as then 


fc\q ,26 


90° 


the periphery always bears the same ratio to the area of 
the stream flowing over it; which is not the case with any 
other form. F.xperimentlng with a right-angled triangu¬ 
lar notch in thin sheet iron, Fig ; with heads of from 
2 to 7 ins, measured vert from the bottom of the notch, to 
the level surf of the quiet water , he found the disch in cub 

ft per. sec to he as follows: ... ........ 

Find the fifth power of the head in inches , (Table 5, pages 251-2.) Take the sq rt of this 6th power, 
(Table 6, ) Mult this sq rt by .0051. Or by formula, 

ptsrh in __ }j ea< i i n i nc hee 2 X .0051. 
cub ft per sec 













































268 


HYDRAULICS. 


ON THE FLOW OF WATER IN OPEN CHANNELS. 

Art. lfi. Tile mean velocity of flow is an imaginary uniform one, 
which, if given to the water at every point in the cross section, would give the 
same discharge that the actual ununiform one does. Or 

volume of discharge 
mean velocity = area cross section 

In channels of uniform cross section, tlie maximum velocity is found 
about midway between the two banks, and generally at some dist below the sur¬ 
face. This dist varies in diff streams; but, as an average, it seems to be about 
one third of the total depth. Where the total depth is great in proportion to 
the width, (say 4 the width or more), the max vel has been found as deep as 
midway between surf and bottom; while in small shallow streams it appears to 
approacit the surf to within from .1 to .2 of the total depth. Many experiments 
upon shallow streams have indeed indicated that the max vel was at the surf. 

The ratio between tlie velocities in different parts of the 
cross section varies greatly in dirt'streams; so that but little dependence 
can be placed upon rules for obtaining one from the other. With the same surf 
vel wide and deep streams have greater mean and bottom vels than small shal¬ 
low ones. In order to approximate roughly to the mean vel when 
the greatest surf vel is given, it is frequently assumed that the former is = £ 
(or . 8 ) of the latter. But Mr. Francis found, in his experiments at Lowell, I hat 
surface floats of wax, 2 ins diam, floating down the center of a rectangular flume 
10 ft wide, and 8 ft deep, actually moved about 6 per cent slower than a tin tube 

2 ins diam, reaching from a few ins above the surf, down to within ins of tlie 
bottom of the flume; and loaded at bottom with lead, to insure its maintaining 
a nearly vert position. While the wax surf float moved at the rate of 8.73 ft per 
sec, the rate of the tube (which was evidently very nearly the same as that of 
the center vert thread of water) was 3.98 ft per sec. Also, that, in the same flume, 
with vels of the center tube varying from 1.55 to 4 ft per sec, the vel of the tube 
was less than that of the mean vel of the entire cross section of water in the 
flume, about as .96 to 1, for the lesser vel; and .93 to 1 for the greater vel. 
While, in another rectangular flume 20 ft wide and 8 ft deep, with vels varying 
from 1.16 to 1.84 ft per sec, that of the tubes was greater than that of the entire 
mass of water, about as 1.04 to 1 . In a flume 29 ft wide, by 8.1 ft deep, with vels 
of about 3 ft per sec, it was as 1 to .9; and in a flume 36£ ft wide, by 8.4 ft deep, 
with vels of about 3£ ft per sec, as 1 to .97. 

Charles El let. Jr. C E, found in the Mississippi “at diff points 

on the river, in depths varying from 54 to 100 ft; and in currents varying from 

3 to 7 miles an hour that the speed of a float supporting a line 50 ft long, is al¬ 
most always greater than that of the surf float alone.” The same results were 
obtained with lines 25 and 75 ft long; the excess of the speed of the line floats 
being about 2 per cent over that of the simple floats: and Mr. Ellet concludes, 
therefore, that the mean vel of the entire cross section of the Mississippi, instead 
of being less, is absolutely greater by about 2 per cent, than the mean surf vel. 
He, however,employed .8 of the greatest surf vel as representing approximately, 
in bis opinion, the mean vel of the entire cross section of water. In shallow 
st reams, he always found the surf float to travel more rapidly than a line float. 

European trials of the mean vel of separate single verticals , in tolerably deep 
ill vers, have resulted in from .85 to .96 of the surf vel at each vertical. The mean 
of all may be taken at .9. 

Bottom velocity. In streams of nearly uniform slope and cross section, 
there is a great reduction of vel near the bottom. As a very rough approxima¬ 
tion, the deepest measurable vel, in streams of uniform slope etc, appears to be 
from £ to £ of the mean vel. But see rents on “scour”, p 279/. 

Art. 17. To measure the surface velocity, select a place where 

the stream is for some dist (the longer the better) of tolerably uniform cross 
section; and free from counter-currents, slack water, eddies, rapids, etc. Ob¬ 
serve, by a seconds-watch,or pendulum, how long a time afloat (such as a small 
block of wood) placed in the swiftest part of the current, occupies in passing 
through some previously measured dist. From 50 feet for slow streams, to 150 ft 
for rapid ones, will answer very well. This dist in ft, or ins, div by the entire 
number of seconds reqd by the float to traverse it, will give the greatest surf vel 
in ft or ins per sec. 

The surf vel should he uieas<l in perfectly calm weather, 

so that the float may not be disturbed by wind; and, for the same reason, the 
float should not project much above the water. The measurement should be 



HYDRAULICS. 


269 


repeated several times to insure accuracy. In very small streams, the banks 
and bed may be trimmed tor a short, dist, so as to present a uniform channel- 
way. The float should be placed in the water a little dist above the point for 
commencing the observation ; so that it may acquire the full vel of the water, 
before reaching that point. 

Art. 18. To gauge a stream 
by means oi'its velocity. Select a 
place where the cross-section remains for a short 
dist, tolerably uniform, and free from counter-cur¬ 
rents, eddies, still water, or other irregularities. 
Prepare a careful cross-section, as Fig 27. By 
meaDS of poles, or buoys, n, n, divide the stream 
into sections, a. b, c, &c. Plant two range-poles, 
R, R, at the upper end; and two others at the 
lower end of the dist through which the floats are to pass; for observing the time, by a seconds 
watch, or a pendulum, which they occupy in the passage. Then measure the mean vel of each section 
a, 6, c, &c, separately, and directly, by means of long floats, as F L, reaching to near the bottom : and 
projecting a little above the surf. The floats may be tin tubes, or wooden rods; weighted in either 
case, at the lower end, until they will float nearly vert. They must be of diff lengths, to suit the 
depths of the diff sections. For this purpose the float may be made iu pieces, with screw-joints. The 
area of each separate section of the stream in sq ft, being mult by the observed mean vel of its water 
in ft per sec, will give the disch of that section in cub ft per sec. And the discharges of all the sepa¬ 
rate sections thus obtained, when added together, will give the total disch of the stream. And this 
total disch, div by the entire area of cross-section of the stream in sq ft, gives the mean vel of all the 
water of the stream, in ft per sec. 



Rem. If the channel is in common earth, especially if sandy, 
the loss bysoakage into the soil, and by evaporation, will frequently abstract so 
much water that the disch will gradually become less and less, the farther down 
stream it is measured. Long canal feeders thus generally deliver into the canal 
but a small proportion of the water that enters their upper ends. 

The double float is used for ascertaining vels at diff depths. It consists 
of a float resting upon the surface of the water, and of a heavier body, or “ lower 
float”, which is suspended from the upper float by means of a cord. The depth 
of tlie lower float of course depends upon the length of the suspending cord 
(which may be increased or diminished at pleasure until the lower float is be¬ 
lieved to be at that depth for which the vel is wanted), and upon its straight¬ 
ness, which is more or less affected by the current. Owing to this latter circum¬ 
stance, it is difficult to know whether the lower float is really at the proper 
depth. Moreover it is uncertain to what extent the two floats and the string 
interfere with one another’s motions. In deep water the string may oppose a 
greater area to the current than the lower float itself does. It thus becomes 
doubtful to what extent the vel of the upper float can be relied upon as indicat¬ 
ing that of the water at the depth of the lower one. 

Art. 19. eastern's quadrant, or hydrometric pendulum, 

consisted of a metallic ball suspended by a thread from the center of a graduated 
arc. The instrument was placed in the current, with the arc parallel to the 
direction of flow; and the vel was then calculated from the angle formed be¬ 
tween the thread and a vert line. 


Gauthev’s pressure plate was a sheet of metal suspended by one of its 
ends, about which it was left free to swing. The plate was immersed in the 
stream, with its face at right angles to the current. The vel was estimated by 
means of the weight required to make the plate hang vert in opposition to the 
force of the current. 

Pitot's tube is a tube bent at right angles like the letter L. One leg is 
held hor under the water, with its open end facing the current The vel ismeasd 
by the height to which the water rises in the vert leg above the general surf. 

These instruments could not well be used except for points near the surface; 
and they gave only the vel at the time of observation. 











270 


HYDRAULICS. 


Art. 20. Wlieel meters. For accurate current measurements, espe¬ 
cially in streams of irregular cross section, where long floats cannot well tie 
used; wheel meters, similar in principle to Woltmaun’s tachometer, are now 
largely employed. 

Such a meter consists of a wheel which is turned by the current, and which 
communicates its motion, by means of its axle and gearing, to indices which 
record the number of revolutions. The inst may be clamped to any part, of a 
iong pole reaching to the bottom of the stream, and thus may be used at. any 
depth. The observer, by means of a wire, rod or string, reaching down to the 
inst, throws the registering apparatus first into, and then out of, gear with the 
wheel (applying a brake to the former at the instant it is thrown out of giar), 
and carefully noting the times when he does so. The inst is then raised, the 
number of revolutions in the measd time is read off from the indices, and from 
it the vel is calculated. 

Various plans have been tried for so arrang-ing such wheel 
meters that their revolutions may be registered above the 
surface at the time. This is now generally accomplished by electricity ; 
the wheel, at each rev, automatically breaking and re-establishing a galvanic 
current generated by a battery. The wire carrying this current is thus made 
to operate Morse telegraphic registering apparatus placed in a boat or on shore. 

A number of meters, so arranged, can be attached at diff points on the same 
pole at the same time, and thus simultaneous observations of veloc¬ 
ities at ditrerent depths may be made and registered. 

Wheel meters are made by Messrs. Buflf A Berger, No 9 Province 
Court, Boston. The prices range approximately from $125 to $225 each. Most 
of their meters are so arranged that they can freely swing hor about the long 
vert pole to which they are clamped, and are provided each with a vane or tail 
similar to that of a wind-mill, for keeping the wheel in the proper position as 
regards the current. The wheels are generally made like those of a wind-mill; 
i e with blades set at such an angle as to present a sloping surface to the cur¬ 
rent; and with the axis of the wheel parallel to the direction of flow. The axis 
runs in agate bearings. When desired, the rim of the wheel is furnished with 
an air-chamber, which just counter-balances the weight of the wheel, and thus 
removes journal friction due to it. Meters are made both with and without 
electrical registering apparatus. In the latter case the gearing and indices etc 
are sometimes enclosed in a glass case, to prevent them from becoming clogged 
by weeds, sediment etc. 

In all of the above methods, except those with floats which move along with 
the current, it is necessary, in order to calculate the vel from an observation, to 
first rate the meter; i e, to ascertain what effects different vels produce 
upon the inst. This is best done by moving the inst at a known vel through still 
water, and noting the effect produced. In this way a coefficient is obtained for 
each meter, which, whenmultiplied by the number of revs etc recorded in any 
given case, gives the vel for that case.* 


HYDRAULICS. 271 


Art. 21. Kutter's lonimla for the mean vel of water flowing in open 
channels o! umionn cross section and slope throughout. 

Caution. The use of all such formulae is liable to error arising from the 
difficulty of ascertaining the exact condition of the stream as regards loneliness 
ol bed, surface slope,* etc. 

Rem. 1 . Caro must l»e taken that the bottom vel is not so 
great as to wear away the soil. If there is any such danger art ificial 
means must be applied to protect the channel-way; or'it may be advisable to 
] educe the rate ol lull, and increase the cross section of the channel: so as to 
secure the same disch, but with less vel. A liberal increase should also he made 
m the dimensions of such channels, to compensate for obstructions to the flow, 
arising trom the growth of aquatic plants, or deposits of mud from rain- 
washes, etc ; or even from very strong winds blowing against the current. See 
also Rem, p 2(19. 

Rem. 2. AVater running in a channel with a horizontal toed, 
or bottom, cannot have a uniform vel, or depth, through¬ 
out its course; because the action of gravity due to the inclined plane of a 
sloping bottom, is wanting in this case; and the water can flow only by forming 
its surface into an inclined plane; which evidently involves a dimiuutiou of 
depth at every successive dist from the reservoir. 



Fig. 28. Fig. 29. 



Fig. 30. 


Theory of flow. It is generally held that the resistances to the flow of 
water in a pipe or channel are directly proportional to the area of the bed sur¬ 
face with which the water comes in contact (i e, to the product of the “wetted 
perimeter” as abco Figs 28, 29, 30 mult by the length of the channel, or of the 
portion of it under consideration); and to the square of the vel of the flowing 
water; and, inasmuch as the resistance at any given point in the cross section 
appears to be inversely as the dist of that point from the bottom or sides, we 
conclude t hat the total resistances are inversely as the area of the cross section ; 
because the greater that area, the greater would be the mean dist of all the par¬ 
ticles from the bottom and sides. (The resistance is independent of the pressure. 
See p 374c.) 

In short, the resistances are assumed to be in proportion to 


vel 2 X wet perimeter X length v~pl 

- -t. -..- or -- 

area of cross section a 

and the head h” in feet or in metres etc, required to overcome those resist, 
ances, is 


resistance __ a coefficient vel 2 X wet. perimeter X length 
* ieat * C ^ area of wet cross section 


or 


h' f 


y-pt 


a 


from which we have 


V 2 


a h" 
Cpl 


and 


1 1 a h" 
\Cp l 


or 


vel = 



area of wet 
cross section 

wet perimeter 


X 


resistce 

head 

length 


* “ In measuring the slope of a large river, the ordinary errors of the most careful leveling are a 
large proportion of the whole fall; the variation of level in the cross section of the surface is often as 
great as the slope for ten miles or more; the exact point where the level should be taken is often 
uncertain : the rise and fall of the wafer makes it extremely difficult to decide when the levels should 
he taken at the upper and lower points : waves of translation may affect the inclination to a great 
and uncertain degree, and may even make the surface slopo the reverse way." Getil T. (i. Kills, 
Trans Am Soc Civ Engrs, Aug 1877. 




18 





















272 


HYDRAULICS. 


But 


area of wet cross section 


a 


or 


is the “hydraulic radius” or 


wet perimeter P . 

“ mean depth ” or “ mean radius,” II, oi the cross section ; 

resistance head or h _ j s t he i nc ]j n ation or slope, S, (fre- 

onentlv denomf'by “I”) of the hydraulic grade line, p 240, or the sine of the 
angle wso Fig H, p 240. In open channels, it is = the fall of the surface per 
unit of length. 


We therefore have 


velocity = X l/ meau riiilius X slope 


or, by using a coeff (c) = 


velocity = coefficient c X l/mean radius X slope 


or v — c \/R S 


The earlier hvdraulicians gave (each according to the results of his investiga¬ 
tions) fixed values for the coeir c, (generally about 95 to 100 for channels 
in earth or gravel, as in our early editions), making it, in other words, a con¬ 
stant -and independent of the siiape, size, slope and roughness of the channel. 
But according to Messrs. E. Ganguillet and W. R. Kutter, eminent Swiss 
engineers, the coeff is affected hy diffs in any of these particulars. 

According to their formula (generally called, for convenience, “ Kutter’s for¬ 
mula”) 


C 


For English measure. 

„ . .00281 , 1.811 

41.6 -)---f- - 

slope n 


/ „ .00281 \ 
(41.6 - )X 

\ slope / 


n 


1 + 


l/mean rad in feet 


For metric measure. 

23 + ^ + I 
slope n 


1 + 


/ .00155 \ w 

( 23 + -sh^e-) X 


n 


l/mean rad in metres 


Tables giving values of c for diff grades, mean radii and degrees of 
roughness, and for English and metric measures, are given on pp 275 etc. 

Here n is a “coefficient of roughness” of sides of channel as given 
below. These values of n were obtained from experience, by averaging a large 
number of experiments made under very different circumstances. They there¬ 
fore embrace all the disturbing effects arising from obstructions existing upon 
the bottom and sides of the channel in the cases experimented upon. In small 
artificial channels of uniform cross section and slope, these obstructions may be 
said to consist entirely of the comparatively minute roughnesses of the material 
of which the bed of the channel consists. But in rivers and earth canals, even 
where the general direction, slope and cross section are tolerably uniform, (as 
they were in the cases upon which our list is based), there are still many con¬ 
siderable irregularities in the sides and bottom ; and these exert a much greater 
retarding effect upon the mean vel than the mere roughness of the material of 
the banks. We therefore find larger values given for win such cases than for 
small regular art ificial channels, although the material of the sides etc was in 
many cases smooth mud; and we must not apply to such comparatively irregu¬ 
lar channels the small values of n obtained by experiments with small and care¬ 
fully made st raight flumes of uniform section and slope, even if we suppose the 
bottom and sides of the former to be made as smooth as those of the latter. 

No general formula is applicable to cases of decided bends in the course 
of a natural stream, or of marked irregularities in the cross sec¬ 
tion. Such cases would require still higher coefficients n than those here given 
for rivers and canals; but they would have to be ascertained by experiment for 
each case, and would be useless for other cases. For such streams we must there¬ 
fore depend upon actual measurements of the velocity,either direct or by means 
of thedisch. 


















HYDRAULICS. 


273 


There is much room for the exercise of judgment in the selection of the 
proper coefficient n for any given case, even where the condition of the 
channel is well known. It may frequently be necessary to usevaluesof n inter¬ 
mediate between those given ; for careless brickwork may be rougher than well 
finished rubble; side slopes in “very firm gravel” may have very diff degrees 
of roughness; etc etc. The engineer should make lists of values of n from his 
own experience, fully noting the peculiarities of each case, and calculating n 
from the tables, pp 275 etc, as directed. 

A given diff in the deg n of roughness exerts a much greater effect upon the 
coefficient c, and thus upon the velocity, in small channels than in larger ones. 
It is therefore especially necessary in small channels that care be exercised in 
finding (by experiment if necessary) the proper value of n; and, where a large 
disch is desired, the sides of small channels should be made particularly smooth. 

Table of n, or coefficient of roughness. 

In any given case the value of n is the same whether the mean 
radius is given in English, metric or any other measure. 


Artificial channels of uniform cross section. 


Sides and bottom of channel lined with n = 

well planed timber.009 

neat cement * (applies also to glazed pipes and very smooth iron pipes). .010 
plaster of 1 measure of sand to 3 of cement;* (or smooth iron pipes). .011 

unplaned timber (applies also to ordinary iron pipes).012 

ashlar or brickwork. .013 

rubble.017 

Channels subject to irregularity of cross section. 

Canals in very firm gravel.020 

Canals and rivers of tolerably uniform cross section, slope and direction, 
in moderately good order and regimen, and free from stones and 

weeds.025 

having stones and weeds occasionally.030 

in bad order and regimen, overgrown with vegetation, and strewn 

with stones and detritus.035 


Art. 22. The following tables give values of the coefficient 

C as obtained by Rutter’s formula for diff slopes (S) mean radii (R) and degrees 
of roughness (rc).f 

Caution. Different values of c must be used with English and with metric 
measures. We give tables for both measures. 

1st. Having the slope S, the mean rad R and the deg n of roughness; to 
find the coeff c. Turn to the division of the table corresponding to the 
given slope S. In the first column find the given mean rad, R. In the same 
line with this R, and under the given n, is the proper value of c.f 

2d. Having the slope S, the mean rad R and either cor the actual or reqd 
vel v; to find the actual, or the greatest permissible,deg n of 
roughness of channel. If the vel is given, and not c, first find 

( c _velocity-_ Turn to the division of the table corresponding to 

T/sIope X mean radius 

the given S, and in the first col find the given R. In the same line find the 
value given’, or just obtained, for c; over which will be found the reqd n .f 

3d. Having the slope S, the deg n of roughness, and the actual or required 
veil); to find the actual or necessary mean rad. R. Assume a 
mean rad ; and from the division of the table corresponding to the given S take 
out the value of c corresponding to the given n and the assumed R. Then say 

v' — c so found X l/assumed mean radius X slope 


# For experiments on abrasion of eements, see p 678. 

^ It is often necessary to interpolate values of S, R, h and c intermediate of those in the tables* 
jfhi* m:»y be done mentally by simple proportion. 

























274 


HYDRAULICS 


If this v' is the same as the given vel, or near enough to it, take the assumed R 
as the proper one. Otherwise, repeat the whole process, assuming a new K, 
greater than the former one if v' is less than the given vel, and vice versa * 

4th. Having the dimensions of the wetted portion ( abco Figs 28, 29, 30.) of 
the channel, the degn of roughness, and the actual or reqd vel; to finct the 
actual or necessary slope, S: 

area of wet cross section 

lindthemean rad,R— ~ a ft c0) G f we t perimeter 

Assume one of the four slopes of the tables to be the proper one. From the 
corresponding division of the table take out the value of c corresponding to the 

given It and n. „ , . , 

If R is 3.28 feet, or 1 metre, the value of c thus found is the proper one ( >e- 
cause tlmn e, for an}' given «, remains the same for all slopes); and the slope, S, 
may be found at once, thus: 

„ / given velocitv \2 

Slope, S = ( —- - ) 

\c X l/rnean radius/ 

Cut if R is greater or less than 3.28 feet, or 1 metre, say 

v' = c thus found X l/mean radius X assumed slope 

If this v' is near enough to the given vel, take the assumed S as the proper one. 
Otherwise, assume a new S, greater than the former one if v' is/mthan the given 
vel, and vice versa; and repeat the whole process.* 

*• It is often necessary to interpolate values of S, R, n and c intermediate of those in the tables. 
This may be done mentally by simple proportion. 


Table of coefllcient c, for mean radii in feet. 



Mean 




Coefficients n of roughness 




Mean 


rad IS 













rad 11 

rH 

|| 

feet 

.009 

.010 

.011 

.012 

.013 

.015 

.017 

O 

o 

.025 

.030 

.035 

.040 

feet 



c 

o 

c 

c 

c 

e 

c 

e 

e 

e 

c 

c 


bD 

.1 

65 

57 

50 

44 

40 

33 

28 

23 

17 

14 

12 

10 

.1 

? 0> 

.2 

87 

75 

67 

59 

53 

45 

38 

31 

24 

19 

16 

14 

.2 

^ ’S 

.4 

111 

97 

87 

78 

70 

59 

51 

42 

32 

26 

22 

19 

.4 

o a 

.6 

127 

112 

100 

90 

81 

69 

60 

49 

38 

31 

26 

22 

.6 


.8 

138 

122 

109 

99 

90 

77 

66 

55 

43 

35 

30 

25 

.8 

— w 

1 

148 

131 

118 

106 

97 

83 

72 

60 

47 

38 

32 

28 

1 

* 2 

1.5 

166 

148 

133 

121 

111 

95 

83 

69 

55 

45 

38 

33 

1.5 


2 

179 

160 

144 

131 

121 

104 

91 

77 

61 

50 

43 

37 

2 

1(5 co 

3 

197 

177 

160 

147 

135 

117 

103 

88 

70 

59 

50 

44 

3 


3.28 

201 

181 

164 

151 

139 

121 

106 

91 

72 

60 

52 

46 

3.28 

1 II 

4 

209 

188 

172 

158 

146 

127 

113 

96 

78 

65 

56 

49 

4 

C o' 

6 

226 

206 

188 

174 

161 

142 

126 

108 

88 

74 

64 

57 

6 

©I 

8 

238 

216 

199 

184 

171 

151 

135 

117 

96 

82 

71 

63 

8 

• O 

II ^ 

10 

246 

225 

207 

192 

179 

159 

142 

124 

102 

87 

76 

68 

10 

II 

12 

253 

231 

214 

198 

186 

165 

149 

129 

107 

92 

81 

72 

12 

x 

16 

263 

242 

223 

208 

195 

174 

157 

138 

115 

100 

88 

79 

16 


20 

271 

249 

231 

215 

202 

181 

164 

144 

121 

106 

94 

84 

20 


30 

283 

261 

243 

228 

215 

193 

176 

157 

133 

117 

104 

95 

30 


50 

297 

274 

257 

241 

228 

207 

190 

170 

147 

130 

117 

107 

50 

X 

75 

306 

284 

267 

251 

238 

217 

200 

180 

157 

140 

127 

117 

75 


100 

312 

290 

273 

257 

244 

223 

207 

187 

163 

147 

134 

124 

100 














































Slope S — .0002 per unit of length, Slope S = .0001 per unit of length, Slope S = .00005 per unit of length, 

I = 1 5000, = 1.056 feet per mile. = l in 10000, = .528 foot per mile. 20000, = .264 foot per mile. 


HYDRAULICS 


275 


Table of coefficient c, for mean radii in feet .— Continued. 


Mean 
rad It 

feet 

.009 

.010 

.011 

Coef 

.012 

[icier 

.013 

its 11 

.015 

of r 

.017 

Ollgl 

.020 

ness 

.025 

.030 

1.035 

.040 

Mean 
rad It 

feet 


c 

c 

c 

c 

c 

c 

e 

c 

c 

e 

c 

C 


.1 

78 

67 

59 

52 

47 

39 

33 

26 

20 

16 

13 

11 

.1 

.15 

91 

79 

69 

62 

56 

46 

39 

31 

23 

19 

16 

13 

.15 

.2 

100 

87 

77 

68 

62 

51 

44 

35 

26 

21 

18 

15 

.2 

.3 

114 

99 

88 

79 

71 

59 

50 

41 

31 

25 

21 

18 

.3 

.4 

124 

109 

97 

88 

79 

66 

57 

46 

35 

28 

24 

20 

.4 

.6 

139 

122 

109 

98 

90 

76 

65 

53 

41 

33 

28 

24 

.6 

.8 

150 

133 

119 

107 

98 

83 

71 

59 

46 

37 

31 

27 

.8 

1 

158 

140 

126 

114 

104 

89 

77 

64 

49 

40 

34 

29 

1 

1.5 

173 

154 

139 

126 

116 

99 

87 

72 

57 

47 

40 

34 

1.5 

2 

184 

164 

148 

135 

124 

107 

94 

79 

62 

51 

44 

38 

2 

3 

198 

178 

161 

148 

136 

118 

104 

88 

71 

59 

50 

44 

3 

3.28 

201 

181 

164 

151 

139 

121 

106 

91 

72 

60 

52 

46 

3.28 

4 

207 

187 

170 

156 

145 

126 

111 

95 

77 

64 

56 

49 

4 

6 

220 

199 

182 

168 

156 

137 

122 

105 

85 

72 

63 

56 

6 

8 

228 

206 

189 

175 

163 

144 

129 

111 

91 

78 

68 

61 

8 

10 

234 

212 

195 

181 

169 

149 

134 

116 

96 

82 

72 

64 

10 

12 

238 

217 

200 

185 

173 

153 

138 

120 

99 

86 

75 

68 

12 

16 

245 

223 

206 

191 

180 

160 

144 

126 

106 

91 

81 

73 

16 

20 

250 

228 

211 

196 

184 

165 

149 

131 

110 

96 

85 

77 

20 

30 

257 

236 

219 

204 

192 

172 

157 

139 

118 

103 

92 

84 

30 

50 

266 

245 

228 

213 

201 

181 

165 

148 

127 

112 

101 

93 

50 

75 

272 

250 

233 

218 

207 

187 

171 

153 

133 

119 

108 

99 

75 

100 

275 

254 

237 

222 

210 

190 

175 

158 

137 

123 

112 

104 

100 

.1 

90 

78 

68 

60 

54 

44 

37 

30 

22 

17 

14 

12 

.1 

.2 

112 

98 

86 

76 

69 

57 

48 

39 

29 

23 

19 

16 

.2 

.3 

125 

109 

97 

87 

78 

65 

56 

45 

34 

27 

22 

19 

.3 

.4 

136 

119 

106 

95 

86 

72 

62 

50 

38 

31 

25 

22 

.4 

.6 

149 

131 

118 

105 

96 

81 

70 

57 

44 

35 

30 

25 

.6 

.8 

158 

140 

126 

114 

103 

88 

76 

63 

48 

39 

33 

28 

.8 

1 

166 

147 

132 

120 

109 

93 

81 

67 

52 

42 

35 

31 

1 

1.5 

178 

159 

144 

130 

120 

103 

89 

75 

59 

48 

41 

35 

1.5 

2 

187 

168 

151 

138 

127 

109 

96 

81 

64 

53 

45 

39 

2 

3 

198 

178 

162 

149 

137 

119 

104 

89 

71 

59 

51 

45 

3 

3.28 

201 

181 

164 

151 

139 

121 

106 

91 

72 

60 

52 

46 

3.28 

4 

206 

186 

169 

155 

143 

125 

111 

94 

76 

64 

55 

49 

4 

6 

215 

195 

178 

164 

152 

134 

119 

102 

84 

71 

61 

54 

6 

8 

221 

201 

184 

170 

158 

139 

124 

107 

88 

/ ft 

66 

59 

8 

10 

226 

205 

188 

174 

162 

143 

128 

111 

92 

78 

69 

62 

10 

15 

233 

212 

195 

181 

169 

150 

135 

118 

98 

85 

75 

68 

15 

20 

237 

216 

200 

185 

173 

154 

139 

122 

102 

89 

79 

71 

20 

30 

243 

222 

206 

191 

179 

160 

145 

128 

108 

95 

84 

77 

30 

50 

249 

227 

211 

197 

185 

166 

151 

134 

114 

100 

91 

83 

50 

100 

255 

234 

218 

204 

191 

172 

158 

140 

121 

108 

98 

91 

100 

.1 

99 

85 

74 

65 

59 

48 

41 

32 

24 J 

18 

15 

12 

.1 

.2 

121 

105 

93 

83 

74 

61 

52 

42 

31 

25 

21 

17 

.2 

.3 

133 

116 

103 

92 

83 

69 

59 

48 

36 

29 

24 

20 

.3 

.4 

143 

125 

112 

100 

91 

76 

65 

53 

40 

32 

27 

23 

.4 

.6 

155 

138 

122 

111 

100 

85 

73 

60 

46 

37 

31 

26 

.6 

.8 

164 

145 

131 

118 

107 

91 

79 

65 

50 

41 

34 

29 

.8 

1 

170 

151 

136 

123 

113 

96 

83 

69 

54 

44 

37 

32 

1 

1.5 

181 

162 

146 

133 

122 

105 

91 

77 

60 

49 

42 

36 

1.5 

2 

188 

170 

154 

140 

129 

111 

97 

82 

64 

54 

45 

40 

2 

3 

200 

179 

163 

149 

137 

119 

105 

89 

72 

59 

51 

45 

3 

4 

205 

185 

168 

155 

143 

125 

111 

94 

76 

63 

55 

48 

4 

6 

213 

193 

176 

162 

150 

132 

117 

100 

82 

69 

60 

53 

6 

8 

218 

198 

181 

167 

155 

137 

122 

105 

87 

73 

64 

57 

8 

10 

222 

201 

185 

170 

158 

140 

125 

108 

89 

76 

67 

60 

10 

15 

228 

207 

190 

176 

164 

145 

131 

113 

95 

82 

72 

65 

15 

20 

231 

210 

194 

180 

168 

149 

134 

117 

98 

85 

76 

68 

20 

30 

235 

215 

198 

184 

172 

154 

139 

122 

103 

89 

80 

73 

30 

50 

240 

220 

203 

189 

177 

158 

143 

126 

108 

94 

85 

78 

50 

100 

245 

224 

208 

194 

182 1 

163 

148 

131 

113 

99 

90 

83 

100 

















































































276 


HYDRAULICS 


Table of coefficient c, for mean radii in fee/.— Continued. 



Mean 
ra<l It 




Coefficients n of rough 

ness. 




Mean 
ra<l R 

bO - 

feet 

.009 

.010 

.011 

.012 

.013 

.015 

.017 

.020 

.025 

.030 | 

.035 

.040 

leet 



e 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 


o 

.1 

104 

89 

78 

69 

62 

50 

43 

34 

25 

19 

16 

13 

.1 

P. 

.15 

116 

101 

90 

80 

71 

59 

50 

40 

29 

23 

19 

16 

.15 


.2 

126 

110 

97 

87 

78 

65 

54 

44 

32 

25 

21 

18 

.2 


.3 

138 

120 

107 

96 

87 

73 

62 

50 

37 

30 

24 

21 

.3 


.4 

148 

129 

115 

104 

94 

79 

68 

55 

42 

33 

27 

23 

.4 


.6 

157 

140 

126 

113 

103 

87 

75 

62 

47 

38 

31 

27 

.6 


.8 

166 

148 

133 

121 

110 

93 

81 

67 

51 

42 

35 

30 

.8 

© ;! 

1 

172 

154 

138 

125 

115 

98 

85 

70 

55 

45 

37 

32 

1 

c „ 

• o 

1.5 

183 

164 

148 

135 

124 

106 

93 

78 

61 

50 

42 

37 

1.5 

ii s 

2 

190 

170 

154 

141 

130 

112 

98 

83 

. 65 

54 

45 

40 

2 

3 

199 

179 

162 

149 

138 

119 

105 

89 

71 

59 

51 

45 

3 

4 

204 

184 

168 

154 

142 

124 

110 

94 

76 

63 

55 

48 

4 

w 

6 

211 

191 

175 

161 

149 

130 

116 

99 

81 

69 

60 

53 

6 

9 II 

10 

219 

199 

183 

168 

157 

138 

123 

107 

88 

75 

66 

59 

10 

7. 

20 

227 

207 

190 

176 

164 

146 

131 

115 

96 

83 

73 

66 

20 

50 

235 

215 

198 

184 

173 

154 

139 

123 

104 

91 

82 

75 

50 


[ ioo 

239 

219 

203 

189 

177 

158 

143 

127 

108 

96 

87 

80 

lfK) 

o a ' 

.1 

110 

94 

83 

73 

65 

54 

45 

36 

27 

21 

17 

14 

.1 


.2 

129 

113 

99 

89 

81 

66 

57 

45 

34 

27 

22 

18 

.2 

*5 ~ 

.3 

141 

124 

109 

98 

89 

74 

63 

51 

39 

30 

25 

21 

.3 

s d 

.4 

150 

131 

117 

105 

96 

80 

69 

56 

43 

34 

28 

24 

.4 

£ =5 

.6 

161 

142 

127 

115 

104 

88 

76 

63 

48 

39 

32 

27 

.6 

^to’ 

.8 

169 

150 

134 

122 

111 

94 

82 

68 

52 

42 

35 

30 

.8 


1 

175 

155 

139 

127 

116 

99 

86 

71 

56 

45 

38 

33 

1 

1.5 

184 

165 

149 

136 

124 

108 

93 

78 

62 

50 

43 

37 

1.5 

®o 

2 

191 

171 

155 

142 

130 

112 

98 

83 

66 

54 

46 

40 

2 

r- » 

3 

199 

179 

163 

149 

138 

119 

105 

89 

71 

59 

51 

45 

3 

ii a 

4 

204 

184 

168 

154 

142 

124 

110 

93 

75 

63 

54 

48 

4 

l-H 

6 

211 

190 

174 

160 

149 

130 

116 

99 

81 

68 

59 

52 

6 

& II 

10 

218 

197 

181 

167 

155 

136 

122 

105 

87 

74 

65 

58 

10 

ad 

20 

225 

205 

188 

175 

163 

144 

129 

113 

94 

81 

72 

65 

20 

* ii 

50 

232 

212 

196 

182 

170 

151 

137 

120 

101 

89 

79 

72 

50 

*1 

100 

236 

216 

200 

186 

174 

155 

141 

124 

105 

94 

85 

77 

100 

JS 

.1 

110 

95 

83 

74 

66 

54 

46 

36 

27 

21 

17 

iT 

.1 

bbji 

.15 

122 

105 

93 

83 

75 

62 

52 

42 

31 

24 

20 

17 

.15 

c ~ 

.2 

130 

114 

100 

90 

81 

67 

57 

46 

34 

27 

22 

19 

.2 

^ r 

.3 

143 

125 

111 

100 

90 

76 

64 

52 

39 

31 

25 

22 

.3 

c ^ 

.4 

151 

133 

119 

107 

98 

82 

70 

57 

44 

35 

29 

24 

.4 


.6 

162 

143 

129 

116 

106 

90 

77 

64 

49 

39 

33 

28 

.6 

2 1> 

.8 

170 

151 

135 

123 

112 

95 

82 

68 

53 

43 

35 

31 

.8 


1 

175 

156 

141 

128 

117 

99 

87 

72 

56 

45 

38 

33 

1 


1.5 

185 

165 

149 

136 

125 

107 

94 

79 

62 

51 

43 

37 

1.5 

2 

191 

171 

155 

142 

130 

112 

99 

83 

66 

55 

46 

40 

2 

C II 

3 

199 

179 

162 

149 

138 

119 

105 

89 

71 

59 

51 

45 

3 

• 

3.28 

201 

181 

164 

151 

139 

121 

106 

91 

72 

60 

52 

46 

3.28 

II 1 

4 

204 

184 

167 

154 

142 

123 

109 

93 

76 

63 

55 

48 

4 

* - 

6 

210 

190 

173 

160 

148 

129 

115 

99 

81 

68 

59 

52 

6 

v " H 

10 

217 

196 

180 

166 

154 

136 

121 

105 

86 

74 

65 

58 

10 

a- 

e II 

20 

225 

204 

187 

173 

161 

143 

128 

112 

93 

80 

71 

64 

20 

50 

231 

210 

194 

181 

168 

150 

135 

119 

100 

87 

78 

71 

50 

a 

l 100 

235 

214 

197 

184 

172 

153 

139 

122 

104 

91 

82 

1 75 

100 


For slopes steeper than .01 per unit of length, = 1 in 100 = 52.8 feet 
per mile, c rem ains practically the same as at that slope. But the velocity 

(being = c X 1/mean radius X slope) of course continues to increase as the 
slope becomes steeper. 











































































jSlope— .0002 unit Slope — .0001 per unit of Slope = .00005 per unit of Slope = .000025 per unit of length 

of length, = 1 in 5000. length, = 1 in 10000. length, = 1 in 20000. == 1 in 40000. 


HYDRAULICS 


277 


Table of eoeflicient c, for mean radii in metres. 


Mean 
rad It 



Coefficients 

11 ot 

roughness. 




Mean 
rad It 

metres 

.009 

.010 

.011 

.012 

.013 

.015 

.017 

.020 

.025 

.030 

.035 

.040 

metres 

.025 

c 

c 

c 

e 

c 

c 

e 

c 

c 

c 

c 

e 


34 

29 

25 

22 

20 

17 

14 

11 

9 

7 

6 

5 

.025 

.05 

44 

38 

33 

30 

27 

22 

19 

16 

12 

9 

8 

7 

.05 

.1 

58 

50 

44 

40 

36 

30 

26 

21 

16 

13 

11 

9 

.1 

.2 

72 

63 

56 

51 

46 

39 

34 

28 

21 

18 

15 

13 

.2 

.3 

82 

72 

64 

58 

53 

45 

39 

33 

25 

21 

17 

15 

.3 

.4 

89 

79 

71 

64 

59 

50 

44 

37 

29 

23 

20 

17 

.4 

.6 

99 

88 

80 

72 

67 

57 

50 

42 

33 

28 

23 

20 

.6 

1 . 

111 

100 

90 

83 

77 

67 

59 

50 

40 

33 

28 

25 

1 . 

1.50 

121 

109 

100 

92 

85 

74 

66 

57 

46 

38 

33 

29 

1.50 

2 

127 

115 

106 

98 

91 

80 

71 

61 

50 

42 

37 

32 

2 

3 

136 

124 

114 

106 

99 

87 

78 

68 

56 

48 

42 

37 

3 

4 

142 

130 

120 

111 

104 

93 

83 

73 

61 

52 

46 

41 

4 

6 

149 

137 

127 

119 

111 

100 

90 

80 

67 

58 

51 

46 

6 

10 

158 

145 

135 

127 

120 

108 

98 

88 

75 

66 

59 

53 

10 

15 

164 

151 

141 

133 

126 

114 

104 

94 

81 

72 

64 

59 

15 

20 

167 

155 

145 

137 

130 

118 

108 

98 

85 

75 

68 

62 

20 

30 

172 

160 

150 

142 

135 

123 

113 

103 

90 

81 

74 

68 

30 

.025 

40 

35 

30 

26 

24 

20 

17 

13 

10 

8 

7 

5 

.025 

.05 

52 

44 

39 

34 

31 

26 

22 

18 

13 

11 

9 

7 

.05 

.1 

65 

57 

50 

44 

40 

34 

29 

24 

18 

14 

12 

10 

.1 

o, 

79 

69 

62 

55 

51 

43 

37 

30 

23 

19 

16 

13 

.2 

.3 

87 

77 

69 

62 

57 

48 

42 

35 

27 

22 

18 

16 

.3 

.4 

93 

83 

74 

67 

62 

53 

46 

38 

30 

25 

21 

18 

.4 

.6 

102 

90 

82 

74 

69 

59 

52 

43 

34 

28 

24 

21 

.6 

1 . 

111 

100 

90 

83 

77 

67 

59 

50 

40 

33 

28 

25 

1 . 

1.5 

118 

107 

97 

90 

83 

73 

65 

55 

45 

38 

33 

28 

1.5 

2 

123 

111 

102 

94 

87 

77 

68 

59 

48 

41 

35 

31 

2 

3 

129 

117 

108 

100 

93 

83 

74 

64 

53 

45 

40 

35 

3 

4 

133 

121 

112 

104 

97 

86 

77 

68 

56 

49 

43 

38 

4 

6 

138 

126 

117 

109 

102 

91 

82 

72 

61 

53 

47 

42 

6 

10 

143 

131 

122 

114 

107 

96 

87 

78 

66 

58 

52 

47 

10 

15 

147 

135 

126 

118 

111 

100 

91 

82 

70 

62 

56 

51 

15 

20 

150 

137 

128 

120 

113 

103 

94 

84 

72 

64 

58 

53 

20 

30 

152 

140 

131 

123 

116 

105 

97 

87 

76 

68 

62 

57 

30 


.025 

.05 

.1 

.2 

.3 

.4 

.6 

1. 

1.5 

2 

4 

6 

10 

15 

30 

47 

59 

72 

84 

91 

97 

104 

111 

117 

120 

128 

131 

135 

137 

141 

40 

50 

62 

74 

81 

86 

92 

100 

105 

109 

116 

119 

123 

126 

129 

35 

44 

55 

66 

73 

77 

83 

90 

96 

100 

107 

110 

114 

116 

120 

31 

40 

50 

60 

66 

70 

76 

83 

88 

92 

99 

102 

106 

109 

112 

28 

35 

45 

54 

60 

64 

70 

77 

82 

85 

92 

96 

100 

102 

106 

22 

29 

37 

46 

51 

55 

60 

67 

72 

75 

82 

85 

89 

92 

95 

19 

25 

32 

39 

44 

48 

53 

59 

64 

67 

73 

77 

81 

83 

87 

15 

20 

26 

32 

37 

40 

45 

50 

54 

57 

64 

67 

71 

74 

78 

11 

15 

19 

25 

28 

31 

35 

40 

44 

47 

53 

56 

60 

63 

67 

9 

12 

16 

20 

23 

25 

29 

33 

37 

40 

46 

49 

53 

55 

59 

7 

10 

13 

17 

19 

21 

25 

28 

32 

34 

40 

43 

47 

50 

54 

6 

8 

11 

14 

17 

18 
21 
25 
28 
30 
36 
39 
43 
46 
50 

.025 

.05 

.1 

.2 

.3 

.4 

.6 

1. 

1.5 

2 

4 

6 

10 

15 

30 

.025 

52 

45 

40 

35 

31 

25 

21 

17 

12 

9 

8 

6 

.025 

.050 

63 

55 

48 

43 

39 

32 

27 

21 

16 

12 

10 

8 

.050 

.1 

75 

66 

59 

53 

48 

40 

34 

27 

21 

16 

13 

11 

.1 

.2 

87 

77 

69 

62 

57 

48 

41 

34 

26 

21 

17 

15 

.2 

.4 

99 

88 

80 

72 

66 

57 

49 

41 

32 

26 

22 

19 

.4 

.6 

104 

93 

84 

77 

71 

61 

53 

45 

36 

29 

25 

22 

.6 

1 

111 

100 

90 

83 

77 

67 

59 

50 

40 

33 

28 

25 

1 

2 

118 

107 

98 

90 

84 

74 

65 

56 

46 

39 

34 

30 

2 

4 

124 

113 

104 

97 

90 

79 

71 

62 

51 

44 

39 

35 

4 

10 

130 

119 

110 

102 

96 

85 

77 

67 

57 

50 

45 

40 

10 

30 

135 

124 

114 

107 

100 

90 

82 

73 

62 

55 

50 

46 

30 


















































































278 


HYDRAULICS 


Table of coefficient c, for mean radii in metres— Continued. 


xz 

-*-» 

mean 




Coefficients n of roughness 




Mean 


rad K 













rad K 

ri 

meters 

.ooy 

.010 

.011 

.012 

.013 

.015 

.017 

.020 

.025 

.030 

.035 

.040 

metres 

O 


c 

e 

e 

e 

c 

e 

c 

c 

c 

e 

e 

e 



.025 

55 

47 

41 

37 

33 

27 

22 

17 

13 

10 

8 

7 

.025 

= £ 

.050 

66 

58 

51 

45 

40 

33 

28 

23 

17 

13 

11 

9 

.050 

£ £ 

.1 

78 

68 

61 

55 

50 

42 

35 

28 

21 

17 

14 

12 

.1 

£ <N 
i— 

.2 

90 

80 

70 

64 

59 

49 

42 

35 

27 

22 

18 

15 

.2 


.3 

95 

85 

76 

70 

63 

54 

47 

39 

30 

24 

21 

17 

.3 

c- 

.4 

99 

89 

80 

73 

67 

57 

50 

42 

32 

27 

22 

20 

.4 

© 11 

.6 

105 

94 

85 

78 

72 

62 

54 

45 

36 

30 

25 

22 

.6 

• 

1 

111 

100 

90 

83 

77 

67 

59 

50 

40 

33 

28 

25 

1 

II 

2 

117 

106 

97 

89 

83 

73 

65 

56 

45 

38 

34 

30 

2 

0) 

4 

123 

111 

102 

95 

88 

78 

70 

61 

50 

48 

38 

34 

4 

& 

6 

125 

114 

105 

97 

91 

81 

72 

63 

53 

46 

40 

36 

6 

c 

10 

128 

117 

108 

100 

93 

83 

75 

66 

55 

48 

43 

39 

10 

X 

30 

132 

121 

112 

104 

98 

87 

79 

70 

60 

52 

48 

43 

30 

• rH 

.025 

57 

50 

43 

38 

34 

28 

23 

18 

13 

11 

9 

7 

.025 

So 

.050 

69 

59 

52 

47 

42 

34 

29 

23 

17 

13 

11 

9 

.050 

^ o 
° 

.1 

80 

70 

63 

56 

50 

42 

86 

30 

22 

17 

14 

12 

.1 

Q> 

.2 

90 

80 

72 

65 

60 

50 

43 

85 

27 

22 

18 

16 

.2 

^ G 

.3 

96 

86 

77 

70 

64 

54 

47 

39 

30 

25 

21 

18 

.3 

2- 

- .4 

100 

89 

81 

74 

67 

58 

50 

42 

33 

27 

23 

19 

.4 

© II t 

.6 

104 

94 

85 

78 

72 

62 

54 

46 

36 

30 

25 

22 

.6 

© - 

1 

111 

100 

90 

83 

77 

67 

59 

50 

40 

33 

28 

25 

1 

II m 

2 

116 

106 

97 

90 

83 

72 

64 

55 

45 

38 

33 

29 

2 

C ~ 

4 

121 

111 

102 

94 

87 

77 

69 

60 

50 

42 

37 

33 

4 

ft- 

6 

124 

113 

104 

97 

90 

80 

71 

62 

52 

45 

40 

36 

6 

*o 

10 

127 

115 

106 

99 

92 

82 

73 

64 

54 

47 

42 

38 

10 

* 

30 

130 

119 

110 

102 

96 

86 

77 

68 

58 

51 

46 

42 

30 


.025 

59 

50 

44 

39 

35 

28 

24 

19 

14 

10 

9 

7 

.025 


.05 

69 

60 

53 

48 

43 

35 

29 

24 

18 

14 

11 

9 

.05 

r- O 

.1 

81 

71 

63 

57 

51 

43 

36 

30 

22 

18 

15 

12 

.1 


.2 

91 

81 

72 

65 

60 

50 

44 

36 

27 

22 

18 

16 

.2 

D,— 

.3 

97 

86 

77 

71 

65 

55 

48 

40 

31 

25 

21 

18 

.3 


.4 

101 

90 

81 

74 

68 

58 

50 

42 

33 

27 

23 

20 

.4 

°. 1 

.6 

106 

95 

86 

78 

72 

62 

54 

46 

36 

30 

25 

22 

.6 


1. 

111 

100 

90 

83 

77 

67 

59 

50 

40 

33 

28 

25 

1. 

a- 2f> 

1.5 

115 

104 

94 

87 

80 

70 

62 

53 

43 

86 

31 

27 

1.5 

*1, 

2 

117 

105 

96 

89 

83 

72 

64 

55 

45 

38 

33 

29 

2 


4 

121 

110 

101 

93 

87 

76 

68 

59 

49 

42 

37 

33 

4 

** O 

10 

126 

114 

105 

98 

91 

81 

73 

04 

53 

46 

41 

37 

10 


L 30 

129 

118 

108 

101 

95 

84 

77 

67 

57 

50 

45 

41 

30 


For slopes steeper than .01 per unit of length, = 1 in 100, the co¬ 
efficient c remains practically the same as at that slope. The velocity, however, 

being = cX l/mean radiusX slope, continues to increase as the slope becomes 
steeper. 

To construct a ding-ram, fig 30 A, from which the values given 
by It utter’s formula may be taken by inspection. 

Draw xz hor, and say from 2 to 4 ft long; and cy vert at any point o within 
say the middle third of xz. On oy lay off, as shown on the left, the values of c 
for which the diagram will probably be used. If a scale of .05 inch, or .002 
metre, per unit of c be used, and be made to include c = 250 for English meas¬ 
ure, or 150 for metric measure, oy will be about 1 ft long. For the sake of 
clearness we show only the larger divisions in this and in what follows. 

On oz lay off, as shown on its upper side, the square roots of all the values of 
the mean rad R for which the diagram is to be used. One inch per ft, or .06 
metre per metre, of sq rt, is a convenient scale. Mark the dividing points with 
the respective values of the mean radii themselves. 

Having decided upon th e flattest slope to be embraced in the diagram, say 

.0028 


w = 41.6 + 


flattest slope per unit of length 


for English measure. 




































































HYDRAULICS 


279 



v — w = for English measure ; or y — w = - for metric measure. 

n n 

To each value of y — w, add w, thus obtaining values of y. We take .000025 
per unit of length as the flattest slope,* and .01, .02, .03 and .04 for n.f Hence 
1 ° 0028 

(using English measure) w = 41.6 + qqqq 2 5 ^ 41,6 + 112 = 153 - 6, 


1.811 1.811 1.811 1.811 . 
y — w —q 2 -, >03 » 04 > 


181.1, 90.5, 60.4, 45.3 respectively; 


* This is about as flat as is likely to occur in practice, 
t In most cases many intermediate ones would be used. 

























279 a 


HYDRAULICS. 


and 


y = 181.1 + 153.6, 90.5 + 153.6, 60.4 + 153.6 and 45.3 + 153.6; 


or 334 7 244 1 214.0 and 198.9 respectively. Layoff these values of y on oy in 
pencil,' as at y , y', y", and y"', using the scale already laid oti for c on oy. 

From each point, y, y’ etc, draw a hor pencil line yt . y’t' etc. and mark on it, 
in pencil, the value of n used in determining its height oy etc. 

Next say x = w X greatest value of n. Make ox = x bv the scale of sq rts> ol R 
on o? In our case = 153.6 X .04 = 6.144 by the scale of sq rts ol It, or = b.144* 

Divideox into as many equal spaces (4 in our case) as .01 is contained in 
greatest n. Mark the dividing points with the values of n as in our r 

From each dividing mark on oa: erect a perpendicular, (*<"' etc) in pencil, to 
cut that hor line {y"'t"’ etc) which corresponds to the same value oln. lhe 
intersections are points in a hyperbola. Join them by straight lines , > 

From r in oz (corresponding to a mean rad of 3.28 ft, or 1 metre) draw radial 
lines, rl, rt', rt " etc. Mark them “n = .01”, “» = .02” etc, the same as their 

corresponding lines yt, y't’ etc. . , 

For each slope (S) to be used in the diagram (except the flattest, for which 

this has already been done) say 


x\ x" etc 


= ^41j 


0Q98 \ 

6 + — ) X greatest n, 
slope/ 


for English measure. 


- ( 23 + X ° reatatn ' 


for metric measure. 


Thus, our slopes are = .000025, .00005, .0001 and .01 per unit of length. Hence, 

( 4 i - 6 +S) x - 04 " 3 - 904 ’ i ''“( 41 - 6 + S) x •° 4 - 2 - 784; 


x'" — ( 41.6 + -^p-) + .04 = 1.675. 

Lay off each value of x', x" etc from oy on a separate hor pencil line o'x' etc, 
using the scale of sq ris of R as on oz. 

Mark each line o'x' etc in pencil with the slope used in fixing its length. 

Divide each dist o'x' etc into the same number of equal parts as ox. From 
the dividing points (which, like those of ox, represent the values of n) erect perps 
to cut the radial lines rt'", rt" etc, each perp cutting that radial line which cor¬ 
responds to the value of n represented by the point at the foot of the perp. The 
intersections corresponding to each line "o' x' etc form a hyperbolic curve. Mark 
each curve with the slope of its corresponding line, ox, o'x' etc. 

The drawing is now in the shape proposed by Mess Ganguillet and Kutter, and 
is ready for use in finding either c, n, R or S when the other three are given. 
Thus: 

1st. Having R, S and n , to find c. For example let R = 20 ft. S = .00005, 
« = .03. From the intersection d of slope curve .00005 and radial line n = .03, 
draw*d-20 to the point (20) in oz corresponding to the given R. At e, where 
d-20 cuts oy, is the reqd c, = 96 in this case. 

2d. Having R, S and c, to find n. For example let R = 20 ft, S = .00005, 
c = 96. Through the points R = 20 in oz, and c = 96 in oy, draw* d-20 to cut 
curve .00005. n (= .03) is found by means of the radial lines nearest to the in¬ 
tersection, d. 

3d. Having S, v and c, to find R. For example let S = .00005, n = .03, 
c = 96. Find curve .00005 and radial line n = .03. From their intersection d 
draw d-20 through the point eshowingc = 96. Its intersection with oz shows 
the reqd R, 20 in this case. 


* Instead of drawing these lines, we mav use a fine black thread with a loop at one end. Drive a 
needle either into one of the points R or into one of the intersections, d etc. Slip the loop over the 
needle. The other end of the thread i« held between the fingers, and the thread is made to cut the 
other points as reqd. The diagram should lie perfectly flat, and the string he drawn tight at each ob¬ 
servation, in order that friction between string and paper may not prevent the string from forming a 
straight line. Or the free end of the string may rest on a pamphlet or other object about H inch thick, 
to keep the string clear of the diagram. Special care must then be taken to have the eye perp over 
the point observed. 






HYDRAULICS. 


279 b 


4th. Having R. c and n, to find S. For example let R = 20 ft c = 96 
”r = .' U3 ' Throu S 1 ' R - 20 and e = 96 draw d-20. S (.00005) is found bv means 
of the curves nearest to the point d of intersection of d-20 with radial line 
w ■— .03. 


Ihe following addition to Kutter’s diagram, proposed bv Mr Rudolph Hering 
Civil and Sanitary Engineer, Philadelphia,* enables us to read the veloc¬ 
ity from the diagram. 


Fin d the sq rt of the recipr ocal of each slope to be embraced in the diagram 

= A slope per unit of length' Lay ° ff these S< 1 rtS on the ri S ht of °V > usin S 
the scale of c already laid off on its left. In our fig we have so proportioned the 

c 1 o 

two scales that — - = Mark the dividing points with the slopes 

1/recip of S 1 
per vnil of length. 

On 02 lay off the vels to be embraced in the diagram, using the scale of sq rts 

vel 


V c 1 

of R already laid off on os, and making —= 


1/R \/recip of S 

1st. Having R, S and n; to find v. For example let R = 20 ft, S = .00005, 
n = .03. From R = 20 draw d-20 to the intersection d of curve .00005 with radial 
line n = .03. d-20 cuts oy at e, where c = 96. With a parallel ruler join R 
= 20 with S = .00005 on oy. Draw a parallel liue through c = 96. Itcuisozat 
ni, giving the reqd vel, 3.03 ft per sec. 


2d. Having R, S and t>; fo find n. For example let R = 20 ft, S = .00005, 
v = 3.03 ft per s^c. With a parallel ruler join R = 20 and slope .00005 on oy. 
Draw a parallel line through v = 3.03. It cuts oy at e, where c = 96. Through 
R = 20 and c = 96, draw d-20 to cut curve .00005. The point d of intersection, 
being on radial line n = .03, shows .03 to be the proper value of n. 

^ Any line drawn to the curves from R = 3.28 ft or 1 metre, is one of the radial 
’ lines used in makiug the diagram. It therefore necessarily cuts all the slope 
curves at points showing the same value of n. 

3d. Having S, n and v : to find R. For example, let S = .00005, n = .03, 
v = 3.03 ft per sec. Assume a value of R, say 10 ft. Find curve .00005 and radial 
linen = .03. Join their intersection d with R — 10 ft. The connecting line cuts 
oy at c = 82. with a parallel ruler join c = 82 with v = 3.03. Draw a parallel 
line through slope = .00005 on oy. It cuts os at R = 27.3. showing that a new 
trial is necessary, and with an assumed R greater than 10 ft. 

If R thus found is the same as the assumed one, the latter is correct. If they 
are nearly equal, their mean may be taken. 

4th. Having R, n and v ; to find S. For example, let R = 20 ft, n = .03, 
v = 3.03 ft per sec. Assume a slope (say .0001). Find its curve, and radial line 
n = .03. Join their intersection with R = 20, and note the value (89) of c where 
the connecting line cutso^. With a parallel ruler join c = 89 with v = 3.03. 
Draw a parallel line through R = 20. It cuts oy at slope .000058, showing that 
a new trial is necessary, and with an assumed S flatter than .0001. If R is 3.28 
ft, or 1 metre, the diagram gives the correct S at the first trial, no matter what 
S was assumed at starting. With any other R, if the diagram gives the same S 
as that assumed, the latter is correct. If the two differ but slightly, we may take 
their mean. 


» Transactions of the American Society of Civil Engineers, January 1879. 











279 c 


VELOCITIES IN SEWERS. 


Table of vels in Circular Ilrick Sewers when running full, by 

Rutter’s formula, p272, but taking n at .015 instead of his .013, in consideration 
of the rough character of sewer brickwork generally. 

When running' only half full the vel will be the same as when full, 
but this is not the case at any other depth whether greater or less. At greater 
ones it increases until the depth equals very nearly .9 of the diam, when it is 
about 10 per cent greater than when either full or half full. From depth of .9 of 
the diam the vel decreases whether the depth becomes greater or less. At depth 
of .25 diam the vel is about .78 of that when full; and then diminishes much 
more rapidly for less depths. All this applies also to pipes. 

The vel for any fall or diam intermediate of those in the table can be found by 
simple proportion. Original. 


Fall 

in ft 
per 
mile. 

2 

3 

4 

Dlamete 

6 

rs in fee 

8 

L 

12 

16 

20 

Fall 

in ft 
per 

100 ft. 




Velocities in feet per second. 




.1 

.19 

.27 

.35 

.50 

.64 

.89 

1.10 

1.34 

.0019 


.30 

.42 

.53 

.74 

.93 

1.26 

1.56 

1.84 

.0038 

.4 

.46 

.65 

.80 

1.08 

1.39 

1.81 

2.20 

2.60 

.0076 

.6 

.59 

.81 

1.00 

1.35 

1.70 

2.22 

2.70 

3.18 

.0114 

.8 

.69 

.95 

1.17 

1.57 

1.94 

2.56 

3.08 

3.60 

.0151 

1.0 

.79 

1.07 

1.32 

1.77 

2.16 

2.84 

3.43 

3.96 

.0189 

1.25 

.89 

1.21 

1.49 

1.98 

2.42 

3.17 

3.8 

4.5 

.0237 

1.50 

.98 

1.33 

1.64 

2.18 

2.64 

3.5 

4.2 

4.9 

.0284 

1.75 

1.06 

1.44 

1.78 

2.34 

2.85 

3.8 

4.5 

5.3 

.0331 

2.0 

1.15 

1.55 

1.91 

2.53 

3.1 

4.0 

4.8 

5.6 

.(-379 

2.5 

1.32 

1.78 

2.18 

2.85 

3.5 

4.5 

5.4 

6.3 

.0473 

3.0 

1.44 

1.94 

2.38 

3.2 

3.8 

5.0 

6.0 

69 

.0568 

3.5 

1.58 

2.10 

2.58 

3.4 

4.1 

5.3 

6.5 

74 

.0662 

4. 

1.68 

2.2 

2.7 

3.6 

4.4 

5.7 

6.9 

7.9 

.0758 

5. 

1.90 

2.5 

3.1 

4.1 

4.9 

6.3 

7.6 

8.7 

.0947 

6. 

2.06 

2.7 

3.3 

4.4 

5.4 

6.9 

8.3 

9.6 

.1136 

7. 

2.2 

3.0 

3.6 

4.8 

5.8 

7.5 

9.0 

10.4 

.1325 

8. 

2.4 

3.2 

3.8 

5.1 

6.2 

8.0 

9.7 

11.1 

.1514 

9. 

2.5 

3.4 

4.1 

5.4 

6.6 

8.5 

10.3 

11.8 

.17(3 

10. 

2.7 

3.5 

4.3 

5.7 

6.9 

9.0 

10.8 

12.5 

.1894 

12. 

2.9 

3.9 

4.8 

6.3 

7.6 

9.9 

11.9 

13.6 

.2273 

15. 

3.3 

4.4 

5.4 

7.1 

8.5 

11.0 

13.3 

15.3 

.2841 

18. 

3.6 

4.8 

5.9 

7.7 

9.3 

12.1 

14.5 

16.7 

.3409 1 

21. 

3.9 

5.1 

6.3 

8.4 

10.0 

13.0 

15.7 

17.9 

.8975 

24. 

4.2 

5.5 

6.8 

8.9 

10.8 

13.9 

16.8 

19.2 

.4546 

27. 

4.5 

5.9 

7.2 

9.5 

11.4 

14.8 

17.9 

20.4 

.51 < >9 

30. 

4.7 

6.2 

7.5 

9.9 

12.0 

15.6 

18.8 

21.5 

.5682 

35. 

5.0 

6.7 

8.2 

10.8 

13.0 

16.8 

20 4 

23.2 

.6629 

40. 

5.4 

7.1 

8.7 

11.5 

13.9 

18.0 

21.7 

24.8 

.7576 I 

45. 

5.6 

7.5 

9.2 

12.2 

14.8 

19.1 

23.0 

26.3 

.8523 

50. 

5.9 

8.0 

9.7 

12.8 

15.5 

20.1 

24.2 

27.7 

.9470 

60. 

6.5 

8.7 

10.7 

14.1 

17.0 

22.1 

26.5 

30.3 

1.136 

70. 

7.0 

9.4 

11.5 

15.2 

18.4 

23.9 

28.5 

32.8 

1.326 

80. 

7.4 

10.1 

12.3 

16.2 

19.7 

25.5 

31.0 

35.0 

1.515 

90. 

7.9 

10.7 

13.1 

17.2 

20.9 

27.0 

32.3 

37.1 

1.705 

100. 

8.4 

11.3 

13.8 

18.2 

22.0 

28.5 

34.1 

39.1 

1.894 


A vel of 10 ft per sec = 600 ft per minute = 36000 ft, or 6.818 miles per 
hour. About 5 ft per sec is as great as can be adopted in practice to prevent the 
lower parts of the sewers from wearing away too rapidly by the debris carried 
along by the water. 


Art. 251. The rate at wliieh rain water readies a sewer or 

culvert, etc, may, according to the admirable “ Report on European Sewerage 
Systems” by Mr. Rudolph Hering, Civ. and San. Eng. of Philada, be found 
approximately by the following formula by Mr Burkli-Ziegler. See Trans. Am. 
boc. c. E, Nov 1881. 



































HYDRAULICS. 


279 d 


Cub. ft. per 
second per 
acre, reach¬ 
ing sewer 


A coef Av. cub. ft. of rainfall 

according X per second per acre, 
to judgment during heaviest fall. 


^ [Av. slope of ground 
X in feet per 1000 ft 

\ No.ofacresdrained 


His coefficient for paved streets is .75; for ordinary cases .625; and for 
suburbs with gardens, lawns, and macadamized streets .31. His average 
heaviest fall is from If to 2f ins per hour. To this the writer will add 
that each inch of rainfall per hour, corresponds closely enough to 1 cub ft 
per sec per acre; so that, if we liberally allow for 3 or 4, etc, ins per hour of 
average heaviest rainfall, the third term of the above equation also becomes 
simply 3 or 4, etc. 

Example. If an area of 3100 acre3 (nearly 5 sq miles), with an average slope of 5 ft per 1000 ft, 
receives a rainfall averaging 3 ins per hour when heaviest, then, assuming a coefficient of .5, the rate 
at which the water would reach the mouth of a sewer at the lower end of the 3100 acres would be 


.5 X 3 X y/ 3TTT0 = X 3 X *203 = .305 cub ft per sec per acre; 
or .305 X 3100 — 945.5 cub ft per sec, total. 

Now suppose the fall of the intended sewer to be say 4 ft per mile; and that for fear of the too rapid 
wearing away of its brickwork by debris swept along by the water, we limit its vel to 6.3 ft per sec, 
which may be permitted on occasions as rare as rains of 3 ins per hour, although for tolerable constant 
flow, where liable to debris, it should not exceed about 5 ft per sec. To And the diameter, look 
in the Table of Vels in Sewers, p 279c. for a diarn corresponding as near as may he, to a vel of 6.3, 
and to a tall of 4 ft per mile. We find this diam to be 14 ft, the area of which is 154 sq ft. Hence, 
154 X 6.3 — 970 cub ft per sec = capacity of sewer. This is a trifle more than our 945.5 cub ft per 
sec of rainfall; nevertheless, to allow for deposits in the sewer, it would be advisable to increase tho 
diam say to 14.5 or 15 ft. 

Rem. Mr Wicksiml, ail ex peri on cod English hydrauli- 
cian, gives the following - table of the least vels and grades 
or falls, to be given to drain-pipes and sewers in cities, in order 

that they may under ordinary circumstances keep themselves clean, or free from deposits. He re¬ 
commends that no drain pipe, even for a single common dwelling, shall be less than 6 ins diam. 


Diam. 
in Inches. 

Vel. In ft. 
per Min. 

Grade, 

1 in 

Grade. 
Feet per 
Mile. 

Diam. 
in Inches. 

Vel. in ft. 
per Min. 

Grade, 

1 iu 

Grade. 

Feet per 

Mile. 

4 

240 

36 

146.7 

18 

180 

294 

18 0 

6 

220 

65 

81.2 

21 

ISO 

343 

15.4 

7 

220 

76 

60 5 

24 

180 

392 

13 5 

8 

220 

87 

60.7 

30 

180 

490 

10 8 

9 

220 

98 

53.9 

3G 

180 

588 

9.0 

10 

210 

119 

44.4 

42 

180 

686 

7.7 

11 

200 

145 

36.7 

48 

180 

784 

6.8 

12 

190 

175 

30.2 

54 

180 

882 

6.0 

15 

180 

244 

21.6 

60 

180 

980 

54 


Weight per foot run, and price per foot run, of glazed 
terra cotta pipes for drains etc; made by Moorhead Clay Works, Spring 
Mill (office No 11 South 7th St) Philadelphia. 


Drain pipe, with socket joint 


Bore 

Wt 

Price 

Bore 

Wt 

Price 

ins 

lbs 

S 

ins 

lbs 

$ 

2 

4 

0.13 

6 

18 

0.30 

3 

7 

0.16 

8 

22 

0.45 

4 

10 

0.20 

10 

30 

0.70 

5 

12 

0.25 

12 

33 

0.80 


Sewer pipe, with sleeve joint 


Bore 

Wt 

Price 

Bore 

Wt 

Price 

ins 

lbs 

$ 

ins 

lbs 

$ 

15 

45 

1.25 

30 

150 

4.50 

18 

65 

1.60 

36 

195 

Special 

con- 

21 

89 

2.30 

42 

203 

24 

100 

3.00 

48 

230 

tract. 


The joints are filled with cement mortar; or, when used for drainage only, 
with clay. Drain pipes (3 to 12 ins bore) are about § inch thick. A bend or 
branch costs about as much as from 3 to 5 feet of pipe. The 48-inch pipes are 
about 2 ins thick. 

Art. 24. When the area of cross section of channel Is re¬ 
duced at any point, as by a dam (Fig 33, p 279 e), or by narrowing it, either 
at its sides (Fig 32) or by placing in it a pier etc, Fig 34; a portion at least of 
the force of grav (which would otherwise be giving vel to the water up-stream 
from the point where the obstruction takes place), causes pressure against the 
dam etc. This pres maintains the up-stream water at a higher level than it 

















































279 e 


HYDRAULICS. 


would otherwise have. Said water is then practically in a reservoir', i e, it has 
less vel and greater pres than before. If the reservoir has no outlet, there is no 
vel; and all of the head, or force of grav, acting on the water is expended in pres 

But if there is an outlet, as over the dam, or between the piers etc, a portion 
co, Figs 31, 33, 34, of this pres or head, is expended in giving vel (or an accelera¬ 
tion of vel) to the water escapiug by that outlet: after which only so much 
head (in the shape of surface slope) is needed as will overcome the resistances 
of the channel down-stream from the obstruction, and so maintain uniform the 
vel given to the water by the head co. 

Where a large canal, such as those intended for navigation, is fed from a reser¬ 
voir, the fall co in feet is approximately 

= mean velocity* in canal, in feet per second, X .017; 

and in smaller canals, such as mill courses, 

= mean velocity 2 in canal, in feet per second, X -02. 

The abruptness of the fall may be diminished by rounding off or sloping the 
edges of the piers, or the corners at the sides of the channel (Fig 32) or the 
approach to the dam (Figs 1 to 4, pp 283, 284). 

Fig 33 is a cross section of C'le$s - g' , s (lam. across Cape Fear River, N. C. It 
is from measurements made by Ell wood Morris, C E; by whom they were com¬ 
municated to the writer. The dam is of wooden cribwork ; and its level crest, 
8 ft 5 ins wide, is covered with plank ; along which the water glides in a smoot h 
sheet, 6 ins deep, (at the time of measurement). At the upper end of this 
sheet, and in a dist of about 2 ft, a head co of 9 ins forms itself, as In the fig. 







































HYDRAULICS, 


279/ 


Art. 2«>. Scour. In a channel of uniform and constant slope and cross 
section, the vel of the particles of water immediately adjoining the bottom and 
sides is very slight; and but little scouring takes place. But when irregular¬ 
ities in the slope or cross section occur, as in the last article, the scour is greatly 
increased in their immediate neighborhood. 

The erection of one or two piers in a quite large stream, will frequently pro¬ 
duce an almost incredible amount of scour, if the bottom is at all of a yielding 
nature. The greatest scour of course takes place during freshets: and near the 
obstruction. 


Scouring action is supposed to be as square of vel. 

■ n A 9® OI l^' n 8 to Smeaton, a vel of 8 miles an hour will not derange quarry rubble stones, not exceed¬ 
ing nan a cub ft. deposited around piers. Ac; except by washing the soil from uuder them, 
i inch per sec. rr 5 ft per min, .05(1818 of a mile, or 300 ft per hour. 

1 foot per sec, = 60 ft per min, = .681816 of a mile, or 3600 ft per hour. 


To reduce inches per sec, to feet per minute, multiply by 5. 

“ “ “ “ “ “ “ hour, “ “ so 

„ “. ‘‘ “ “ to miles per hour, divide by 17.6. 

One mile per hour =r 88 ft per min = 1.4667 ft, or 17.6 ins per sec. 


300. 


The two following tables are (with many corrections) from Nicholson’s 
Architecture ; and must be looked upon merely as probable approximations. They suppose the piers, 
&c, to be properly rounded or pointed at their upstream ends, so as to give as free a passage as pos¬ 
sible to the water. He says that if they are square-ended, the head will be increased about 50 per ct. 
The subject is an extremely intricate one, and admits of no precise solution. If the increased vel 
scours away the bottom uutil the area of water-way becomes as great as it originally was, the head 
disappears; and the vel also becomes reduced to its original rate. This is common iu soft bottoms. 

TABLE Of heads produced by obstructions to streams. 


Kind of Bottom 
which begins to 
Original Vel. wear away under 

of Stream.* Bottom Vel. equal 

to those in the 
first three cols. 


Proportion of Area of original Water-way, 
occupied by the Obstructions. 


1 1 

1 

1 

1 

1 

1 

1 

5 

3 

1 1 

1 0 

8 

6 


3 


8 

4 


Head of Water produced at the Obstructions} in 


ns. 

3 

Ft. 

A 

Miles. 

.170 

Ooze, and Mud... 

.0003 

.0004 

.0004 

.0006 

Fe< 

.001 

it. 

.0014 

.0033 

.0067 

.0162 

6 

A 

.341 

Clay. 

.0011 

.0014 

.0017 

.0023 

.004 

.0058 

.0133 

.0267 

.0646 

12 

l 

681 

Saud. 

.0045 

.0056 

.0069 

.0091 

.015 

.0231 

.0532 

.1069 

.2584 

24 

2 

1.26 

Gravel. 

.0182 

.0225 

.0276 

.0364 

.060 

.0924 

.2128 

.4276 

1.036 

36 

3 

2.04 

Small Shingle.... 
Large “ .... 

.0409 

.0507 

.0621 

.0819 

.135 

.2079 

.4788 

.9621 

2.326 

48 

4 

2.72 

.0728 

.0902 

.1104 

.1456 

.240 

.3696 

.8412 

1.710 

4.144 


0 

3.41 

Soft Shistus. 

.1137 

.1410 

.1725 

.2275 

.375 

.5775 

1.320 

2.672 

6 4 75 


6 

4.09 

Stratified Rocks.. 

.1638 

.2030 

.2484 

.3276 

.540 

.8316 

1.915 

3.848 

9.304 

20 

10 

6.81 

Hard Rocks. 

.4550 

.5640 

.6901 

.9100 

1.50 

2.310 

5.280 

10.69 

25.9 


TABLE Increased velocities produced at and by 

minded, or pointed obstructions. If square, these vels must, accord- 
ng to Nicholson, be increased % part. 


Original Vel. 
of Stream.* 

Proportion of Area of Water-way, occupied by the Obstructions. 

iiiii ii ii l i 1 i l ft I -a 


Per 

T2 


8 1 

6 1 

¥ 1 

3 1 

2 1 

8 1 

¥ 

Ins. 

Ft. 

Hour. 

Miles. 

Velocity produced at the Obstruction in Feet per Second. 

3 

A 

.170 

.28 

.29 

.30 

.32 

.35 

.394 

.52 

.7 

1.05 

6 

A 

.341 

.56 

.58 

.60 

.64 

.70 

.788 

1.05 

1.4 

2 1 

> 

1 

.681 

1.13 

1.16 

1.20 

1.26 

1.40 

1.58 

2.1 

2.8 

4.2 

24 

2 

1.36 

2.27 

2.33 

2.40 

2.52 

2.80 

3.16 

4.2 

5.6 

8.4 

36 

3 

2.04 

3.39 

3.48 

3.60 

3.78 

4.20 

4.74 

6.3 

8.4 

12.6 

48 

4 

2.72 

4.54 

4.66 

4.80 

5.04 

5.60 

6.32 

8.4 

11.2 

16.8 

60 

5 

3.41 

5.60 

5.80 

6.00 

6 40 

7.00 

7.88 

10.5 

14.0 

21.0 

72 

6 

4.09 

6.78 

6.96 

7.20 

7.56 

8.40 

9.48 

12.6 

16 8 

25.2 

20 

10 

6.81 

11.3 

11.6 

12.0 

12.6 

14.0 

15.8 

21.0 

28.0 

42.0 


* A very vague expression. Does it refer to the greatest surface vel at mid-chaunel; or to the mean 
i of the entire cross-section ? 



















































280 


HYDRAULICS, 




Art. 26. The resistance of water against a flat surface mov¬ 
ing through it at right angles, is nearly as the squares of the yel; ami, I 
according to Hutton, its amount in lbs per sq ft approx = Square of vel m It per 
see. Or like the ores of a running stream against a perp faxed fiat i 

surface, it is = wt of a col of water whose base = pressed surf, and Whose ht — head due to the vel us 
per table p 258. 

The resist of a sphere is to that of its great circle about as 1 to 2.9. _ . 

When the moving surf, instead of being at right angles to the direction in which it moves, forms 
another angle with it. the resistance becomes less in about the follow ing proportions. 1 hereto! e, | 
when the surf is inclined, first calculate the resistance as if at right angles: and then mult by the 
following decimals opposite the angle of inclination: , J 


90°.. 

..1.00 

603.... 

.88 

40°.... 

.58 

20°.... 

.16 

80 .. 

.. .98 

55 .... 

.83 

35 .• • . 

.46 

15 .... 

.10 

70 

.. .95 

50 .... 

.76 

30 .... 

.34 

10 .... 

.06 

65 .. 

.. .92 

45 .... 

.68 

25 .... 

.24 

5 .... 

.02 


The scour, or abrading power of moving water is considered to 

be as the square of its vel. 

Art. 27. To calculate the horse-power of falling water, on 

the ordinary assumption that a horse-power is equal to 38000 tt>s lifted 1 foot vert per min. That of I 
average horses is really but about % as much, or 22000 lbs, 1 foot high per min. Mult together the 
numberofoub ft, of water which fall per min ; the vert height or bead in feet, through which it falls , 
and the number 62.3, (the wt of a cub ft of water in lbs ;) and div the prod by 33000. Or, by formula, 


Ctlb ft y. 
The number of __ per min A 
horse-powers 


vert 

height in ft 
83000! 


X 


lbs 

62.3 


Ex. Over a fall 16 ft in vert height, 800 cub ft of water are dischd per min. 
powers does the fall afford ? 

cub ft ft lt>9 

„ 800 X 16 X 62.3 __ 797440 

Here, -“ „„ nnrL - — 24.17 h-powr. 


How many horse- 


33000 


33000 


AYatCr-wheels do not realize all the power inherent in the water, as found by on 
rule. Thus, undershots realize but from y A to % ; breast-wheels, overshots, from % to % ; tur¬ 
bines, % to .85 of it; according to the skill of design, and the perfection of workmanship. Even when 
the wheel revolves in a close-fitting casing, or breast, elbow buckets give considerably more power 
than plain radial or center-buckets. Of the power actually received by a wheel, part is expended in 
friction, Ac; while the remainder does the useful or paying net work of raising water, grindiug 
grain, sawing, &c. 


Observations by Genl If aupt, in 1866, gave the following results for a 
small hydraulic ram. Head of water to ram = 8.812 ft; diam of drive-pipe = 
iy± ins; length 15 ft. Diam of delivery-pipe — % inch; length 200 ft. Vert height to which the 
water was raised by the delivery-pipe, 63.4 feet. Strokes of ram per min, 170. Quantity of water 
which worked the ram — 768 cub ins. =3.31 galls, = 27.73 lbs per min. Quantity raised 63.4 ft, high 

tbs water ft ft-lbs 

per min, = 48 cub ins, = 1.736 lbs. Hence the power expended per min, wa* 27.73 X 8.812 = 244.35. 
lbs water ft ft-lbs 

And the useful effect, was 1.736 X 63.4 = 110.06. Hence the ratio which the useful effect bears to the 
110.06 

power in this instance, is ——, or .45. The actual power of the ram is, however, greater thau tins, 
244.35 

inasmuch as it has to overcome the friction of the water along the delivery-pipe.* 


To find the horse-power of a running stream. Water-wheels 

with simple float-boards,! instead of buckets, are sometimes driven by the mere force of the ordinary 
natural current of a stream, without any appreciable fall like that iu the foregoing case. In such 
cases, we must substitute the virtual or theoretic head ; which is that which would impart to it the 
same vel which it actually has. This virtual head may be taken at once from Table, p 258. Tiius, a 
stream has a vel of 2.386 miles per hour: or 210 ft per niin ; or 3J^ ft per sec ; and in the column of 
heads iu Table 10, opposite to 3.5 vel per sec, we find the reqd head .190 of a ft. Having thus found 
the head, we must now find the quantity of water which passes any given area of the stream iu a 
min. Thus, suppose that the immersed part of a float when vert is 5 ft long, and 1 ft wide or deep; 
then the area of this part which receives the force of the current, is 5 X 1 — 5 square feet. Hence, 
area vel 

5 sq ft X 210 = 1050 cub ft per min. Having now the cub ft per min, and the vert height or head, 
the number of horse powers of the stream of the given area, is found by the foregoing rule, or formula. 


* A committee of the Franklin Institute, in 1850, gave .71 as th 

coefficient for a ram at the Girard College, in which the diam of drive-pipe was 2>^ ins: its lengt* , 
160 ft; fall. 14 ft. Delivery-pipe, 1 inch diam ; 2260 ft long: vert rise, or height to which the watfc. 1 
was raised. 93 ft. No details of the experiment are giveu. Some large rams in France give a use/iu 
effect of from .6 to .65 of the whole power expended. It is an excellent machine for many purposes- : 
and is sometimes used for filling railway tanks at water stations. 

T Sueli wheels, for floating mills, in Europe, rarely exceed 15 ft 

diam. Whatever the diam, they may have about 18 to 20 floats. The floats are from 8 to 16 ft long; 
and about ^ to as deep as the diam of the wheel. They should not dip their entire depth into 
the water, but nearly so. They should not be in the same straight line with the radii • but should 
incline from them 30° up stream, to produce their full effect.. All these remarks apply to wheels 
moving freely in a wide or indefinite channel; as in the case or a floating mill built on a sew and 
anchored out in a stream : but. not to wheels for which the water is dammed up'and acts with a prac¬ 
tical fall. No (treat exactness is to be expected in rules on this subject. The best vel for the wheel 
is about .4 that of the stream. 





















HYDRAULICS, 


281 


cub ft per min v vert lit in ft v Ihs 
Thns. No -°f — 1060 * - 190 * 62.3 124*29 

’ H. Pow. 33000 ~ 33000 = * 377 °f tt 11 ■ Pow - 

But iu practice the wheels actually realise but about ^ of this power of the stream, when working 
in an open channel; and still less when the writer flows with the same vet through a narrow artificial 
chaunel, but little wider thau the wheel. Therefore, the actual power of our wheel will be but .377 X 
.4=: .1608; or about ^ of ahorse power; or 33000 X .1508 = 4076 ft-Ibs per min. Making a rough 
allowance for the friction of the machine at its journals, &c. we should have say about 4400 ft-tbs of 
useful power ; that is, the wheel w-uld actually raise about 440 tbs 10 ft high : or 44 tbs 100 ft high. &c, 
l»er min. The vel of the stream must not be measured at the surface ; hut at about of the depth to 
which the floats are to dip, or be immersed. This, however, is ehiefly necessary in shallow streams, 
iu which the depth of the float bears a considerable ratio to that of the water. 

This power of a running* stream, (for any given area of 
transverse section,) increases as the cubes of the vels; for, as 

we have seen, the power in ft-fl)s per miu is found by mult together the weight of water which passes 
through the section iu a miu, and the virtual head in ft; and since this weight increases as the vel, 
and this head as the square of the vel, the prod of the two (or the power) must be as the cube of the 
vel. Therefore, if the vel in the foregoing case had been 10.5 ft per sec, or 3 times 3.5 ft, the power 
of the wheel would have been 27 times as great, or .1508 X 27 := 4.07 horse-powers. 




DAMS. 


We can devote bnt little space to this subject, in addition to what is said oij 
earthen dams for reservoirs, p 287 ; and on stone ones, p 229, &c. Those we shal 
now describe will also answer for such reservoirs, when the perishable nature of 
timber is not an objection. 

Primary requisites, in the erection of dains, are, a foundation suffi-j 
cieutly firm to prevent them from settling, and thus leaking; the prevention 
of leaks through their backs, or under their bases; and the prevention of wear j 
of the bottom of the stream in front of the dam, by the action of the falling 
water. For the first purpose, hard level rock bottom is of course the best; and 
should be chosen, if possible. In that case, thick planks, tt , Fig 6, (single or 
double, as the case may be,) closely jointed, aud reaching from the crest, c, iO ( 
the back lower edge tv, (where they should be scribed down to the rock ;) with a 
good backing, b, of gravel, will suffice to prevent leaks. Gravel, or rather very 
gravelly soil, is far better than earth for this purpose; for if the water should . 
chance to form a void in it, the gravel falls and stops it. To prevent this back¬ 
ing from being disturbed near the crest of the dam, by floating bodies swept! 
along by freshets, a rough pavement of stones, about 15 to 18 inches d^ep, asi 
shown in Fig 7, should be added for a width of about 10 to 20 feet: or until its! 
top becomes 3 to 5 feet below the crest c of the dam, according to circumstances. 



In Fig 1, (a dam on the Schuylkill navigation,) the upper timbers, e, are all 
close jointed, and laid touching, so as not to require planking in addition. 

But if the bottom of the stream is gravel or earth, there must in addit'on to 
these be used two thicknesses of sheet piles, p, Fig 2, Ac, close driven, breaking 
joint, to a depth of several feet, to prevent leaking through the soil beneath 
the base of the dam. Frequently but one thickness is used. If the bottom is 
soft oi open ior a depth of only a few leet, it is at times better to remove t, and i 
base the dam on the firmer stratum below ; still, however, using the sheet piles I 
Old decayed timber and other rubbish should be removed from the base. In I 
very bad soils of greater depth, it may he necessary to support the dam entirely 
upon a platform resting on bearing piles. Here great precautions are neces¬ 
sary against leaks; but the case occurs so rarely, that we shall not stop to con- 
siuer it. 

As to the wearing away of the bottom of the stream by the water falling over 
the front of the dam, precautions should be used in all cases except that of very 



hard rock, or of medium rock protected by a considerable depth of water. The 
dam, Iig 1, was built upon a tolerably firm micaceous gneiss in nearly vertical 
strata, covered by .about 2 feet of water in ordinary stages. In 39 years the rock was 




























DAMS, 


283 





wnm away in front of the dam, as shown in the fig, to the average depth of 3 feet; or very nearly 1 
jnch per year. The depth of water on the crest c, was usually from 6 to 18 ins; rarely 5 or 6 ft dur¬ 
ing freshets ; aud but a few times during the whole period, 8 or 9 ft. 

At Jones’s dam, on Cape Fear River; height of dam, 16 ft; front vert; 

. fall, usually 10 ft, into 6 ft depth of water; the soft shale rock, iu vert strata, was, in the course of 
a few years, worn away 16 ft; and the dam was uudermiued to such an extent as to fall into the cavity. 
In another case, dam 36 ft high ; trout vert; the water Tailing upon nearly vert strata of hard shalti 
111 rock, usually covered by but about 2 ft of water : in about 20 years wore it to an irregular depth of 
ill from 10 to 20 ft; and extending from the very face of the dam, to 70 or 80 ft in front of it. 
if Iu Fig 2, upon a stream subject to very violeut freshets, the gravel was washed awav for a consid¬ 
erable width aud depth beyond the apron, as at A. To prevent a repetition, the cavity was tilled 
with cribwork full of stone, clear across the river. 

!■ ' C A deposit of blocks of loose stone, of even 

a ton weight or more, will uot serve as a pro¬ 
tection iu front of a dam exposed to nigh 
freshets; but will soou be swept away. A 
common precaution against this wear, in low 
dams, is an apron, a a , Fig 2 ; or d d, Fig 3; 
of either rough round tree trunks, or of hewn 
timber, laid close together; extending under 
the eutire base of the dam, aud from 15 to 30 
ft in front of its face. These are sometimes 
bolted to pieces, ss, Fig 2; or yy, Fig 3; laid 
under them across the stream. Iu Fig 3, 
with very soft bottom, these pieces yy are 
supposed to be bolted to short piles £ t, driven 
for that purpose. 

1 At times a distinct wide low timber crib, filled with stone, and covered on top with stout plank. 

' has been placed in front of the dam, to receive the fall of the water; and is effective in protecting 

!. the bottom. Also, in some cases, a dam of less height, and of cheap character, has been built at a 
short distance down stream from the main one, in order to secure at all times a deep pool in front 
of the latter for breaking the force. 

Another precaution is to 
substitute a sloping front like 
cl, Fig 4, or such as Figs 1 
and 2 would form if reversed, 
for the nearly vert one of the 
other figs; thus to some extent 
reducing the force of the wa¬ 
ter. This, however, is but a 
partial remedy, especially 
for soft bottoms in shallow 
water; for the sliding sheet 
still descends with great force. 
The best form of dam, per¬ 
haps, in such cases, is that 
shown in Fig 5, in w'hich the 
front consists of a series of steps of 
about 1 vert, to 3 or 4 hor. These ef¬ 
fectually break the force of the water; 
aud, with the addition of an apron oo, 
secure a satisfactory result. It is ob¬ 
jected against this form, as also 
against Figs 4 and 6, that their fronts 
are liable to be torn by descending 
trees, ice, and other bodies swept 
along during freshets ; but experience shows that this objection has but little weight; for when such 
bodies pass, the sheet of water is thicker than usual; and protects the front timbers. On the Sch 
Nav, the timbers cl, Fig 6, scarcely wear thin at the rate of an inch in 10 to 15 years. 



ROCK 


Tlie forms of wooden dams are many; (see the figs, which show 

those most used :) varying with the circumstances of the case, and with the fancy of the designer. 
In the United States they are usually of cribwork. of either rough round logs with the bark on, or of 
hewn timber; in either case about a foot through. These timbers are merely laid on top of each 































































284 


DAMS, 


other, forming in plan a series of rectangles with sides of about 7 to 12 ft. They » re °J **°teh d 
together, but simply bolted by 1 inch square bolts (often ragged or jagged) about i to 2H feet long, 
through two timbers at every intersection. These are not found to rust or wear seriously, even when 
exposed to a current. Square bolts hold best. Round logs are flattened where they lie upon each 
other. Experience shows that firmer but more expensive connections are entirely unnecessary 


The 


cribs are usually, but uot always, tilled with 
rough stoue. In triangular dams, disposed 
as in Figs 1, 2, and 7, this stoue filling is 
uot so essential as iu other forms ; because 
the weight of the water, aud ot the gravel 
baching, teuds to hold the daui down on its 
base. Still, even iu these, when the lower 
timbers are uot bolted to a rock bottom, or 
otherwise secured iu place, some stone may 
be necessary to prevent the timbers from 
floating away w hile the work is unfinished, 
aud the gravel uot yet deposited behind it. 
On rock, the lowest'timbers are often bolted 
to it, to prevent them from floating away 
during construction; and when the water 
is some feet deep, this requires coffer-dams. 



__ r- _^ Or. the cribs may be built at first only a few feet high ; 

then floated iuto place, and sunk by londiug them with stone; for the reception of which a rough 
platform or flooring will be reqd in the cribs, a little above their lowest timbers. The bolting to the 
rock may then be dispensed with. The water may flow through the open cribwork as the building 
higher goes on ; attention being paid to adding stone enough to prevent it floating away if a freshet' 
should happen. Or, cribs shown in plan at cc, Fig 8, loaded with 
stone, may be sunk, leaving one or more intervals, like that at oooo, 
between them, for the free escape of the water. These openings to 
be finally closed by floating into them closing-cribs shaped like n. 

The workmanship of a dam in deep water can of course be much 
better executed in coffer-dams, than by merely sinking cribs. The 
joints can be made tighter: the stone filling better packed ; the sheet 
piling more closely fitted, &c. 

When a very uneven rock bottom in deep water, or the introduc¬ 
tion of sluices in the dam, or any other considerations, make it ex¬ 
pedient to build dams within coffer-dams, both should he carried on 
in sections ; so as to leave part of the channel-way open for the es¬ 
cape of the water. Commencing at one or both shores, the first section of the coffer dam may reach 
say quarter way or more across the stream. In the section of the dam itself built w ithin this enclos-p 



ing coffer-dam. ample sluices should be left for the water to flow through when we come to build the 
closing section of the coffer-dam. When the dam has been finished, these sluices may be closed 
by drop-timbers*. Before removing one section of coffer-dam, the outer end of tile enclosed) 
section of dam itself must be firmly finished in such a manner as to constitute a part of the innej 
end of the next section of coffer-dam. It is impossible to give details for every contingency ; the en 
gineer must rely upon his own ingenuity to meet the peculiarities of the case before him. In some 
cases of shallow water, mere mounds of earth may answer for coffer-dams; or rough stone motiuds, 
backed with earth or gravel. 

After the water has passed beyond the crest, c in the figs, there is no necessity for preventing its 
leaking down among the crib timbers: on the contrary, the thick sheeting planks, (or squared tim¬ 
bers, as occasion may require.) cl. Figs 4 and 6, which form the slopes along which the water tlienl 
flows in some dams, are usually not laid close together, but with open joints of about inch wide be- J 
tween them, for the express purpose of allowing part of the water to fall through them, so as tcj 
keep the timbers beneath them partially wet: which, to some extent, renders them more durable. In j 
Figs 1, 4, 6, and 7. the water of the lower pool flows freely back among the crib timbers, and rough; 
quarry stones with which the cribs are filled either partly or entirely. In Figs 4 and 6, these stones, 
are not shown. In the dam, Fig I, none were used. In Fig 2. they were as shown. 

A substantial, and not very expensive dam of the form of Fig 7. may be built of rough stone ini 
cement. Some hewn timbers should be firmly built horizontally into the masonry of the sloping] 
hack en to. at, a few feet apart, with their tops level with the surf of the masonry. To those must be 1 
well spiked close-jointed sheeting-plank cnto, for protecting the masonry from the action of tie 
water, and of floating bodies. The gravel backing b, may be omitted ; but the sheet piles p, aud ani 
apron in front of the dam, will be as indispensable in yielding soils, as if the dam were of timber. 

Figs 1, 2, 4, 6. and 7. are sections drawn to a scale, of existing dams in Pennsylvania, that havt] 
stood successfully the force of heavy freshets for a long series of years.t These freshets at times carr; j 
along large bodies of ice, trees, houses, bridges, &c. ; and have risen to 11 ft above the crests. Fig 1. 
on the Sch Nav. was built in 1819, and served perfectly for 39 years, until in 1858 the decay of much 
of its timber, especially of the close-laid top ones, e, rendered it necessary to build a new one just in 
front of it.. It was of extremely simple construction ; and was never filled with stone. The bottom tim¬ 
bers. oo, 10 ft apart, were bolted to the rock : and immediately over each of them, was such a series of 
inclined timbers as is shown in the fig. The top ones, e, however, were close-jointed, and laid touching, 
so as to form the top sheeting, instead of thinner planks. The short pieces at t were laid in the same 
way. No coffer dam was used ; but the bottom pieces were first bolted to the rock ; 10 ft apart; then 
the stringers and the sloping pieces were added. The close covering (e) was carried forward from 
each end of the dam. until at last a space of only about 60 ft was left in the center, for the water to 
pass. T he close covering for this space being then all got ready, a strong force of men was set to 
work, and the space was covered so rapidly that the river had not time to rise sufficiently high to 
impede the operation. 


* Timbers ready prepared for closing an opening through which water is flowing; and suddenly 
dropped into place by means of grooves or guides of some kind for retaining them in position Sev¬ 
eral such timbers may at times be firmly framed together, and then be all dropped at once • closing 
the opening or sluice at one operation; especially when it is of small size. In some cases a crib 
may he sunk on the up-stream side of such an opening, for closing it. 

t Those on the Schuylkill Navigation were obligingly furnished by .Tames F Smith Fso chief 
engineer and superintendent of that work. Other valuable information from the same source will 
be found in different parts of this volume. 























DAMS 


285 


Fig 2 is a canal feeder dam on the Juniata. Here s s are timbers stretching clear across the stream, 
(about 300 ft,) and sustaining the apron an, of stout hewn timbers laid touching. This dam was tilled 
with stone, for the retention of which the front sheeting planks were added. 

Fig 6 is on the Soh Nav ; was built iu 1855. it is a form much approved of on that work, for such 
situations; namely, firm rock foundation, with a considerable depth of water in front. The highest 
dam (32 ft) on the Sch Nav. is very similar to it; built iu 1851. All the dams on this work are of 
hewn timber, chiefly white and yellow pine. The water occasionally runs from 8 to 12 feet deep over 
their crests; and then overflows and surrounds mauy of the abuts. The vertical back allows the 
overflowing water to leak down among all the lower timbers of the dam, and thus tend to their 
preservation. 

Fig 4 shows the dams on the Monongahela slackwater navigation ; W. Milnor Roberts, eng. They 
are of round logs, w-ith the bark on; flattened at crossings. The longest ones iu the fig are 10 feet 
apart along the length of the dam. Experience shows that such dams possess all the strength neces¬ 
sary for violent streams. On rock, the lowest timbers are bolted to it. 

Fig 7 has been successfully used to heights of 40 ft.* 

Fig 3 is intended merely as a hint for a very low dam on yielding bottom. Its main supports are 
piles it, from 4 to 8 ft apart, according to the height of the dam; and other circumstances ; and tt 
are short piles for sustaining the apron dd. It may be extended to greater heights by addiug braces 
in front; which may be covered by stout planks, to form an inclined slide for the overfalling water. 
Many effective arrangements of piles, and sloping timbers for dams on soft ground, will suggest them¬ 
selves to the engineer. Thus, at intervals of several feet, rows of 3 or more piles may be driven trans¬ 
versely of the dam ; the top of the outer pile of each row being left at the intended height of the crest, 
while those behind are successively driven lower and lower; so that when all are afterward con¬ 
nected by transverse and longitudinal timbers, and covered by stout planking, and gravel, they will 
form a dam somewhat of the triangular form of Fig 7. It would be well to drive the piles with an 
inclination of their tops up stream. 

There is much scope for ingenuity both in designing, and in constructing dams under various cir¬ 
cumstances ; and in turning the course of the water from one channel to another, by means of ditches, 
pipes, or troughs, &c., at diff heights; aided at times by low temporary dams or mounds of earth ; or 
of sheet piles, &c; or by coffer-dams; so as to keep it away from the part being built. Each locality 
will have its peculiar features ; and the engineer must depend on his judgment to make the most of 
them. 

Abutments of flams as a general rule should not contract the natural 

width of the stream ; or, if they must do so. as little as possible; for contractions increase the height, 
aud violence of the overflowing water in time of freshets ; during which a great length of overfall is 
especially desirable. They should be very firmly connected with the ends of the dams; and should, 
if the section of the valley admits of it. be so high, and carri d so far inland, that the high water 
of freshets will not sweep either over them, or around their extremities; aud thus endanger under¬ 
mining, and destruction. In wide, flat valleys they cannot be so extended without too much ex¬ 
pense; and the only alternative is to found them so deeply and securely as to withstand such 
action ; making their height such that they will, at least, be overflowed but seldom. Their euds 
adjacent to the dam, should be rounded off, so as to facilitate the flow of the water over the crest. 

They are best built of large stone in cement; for although sufficient strength may bs secured by 
timber, that material decays rapidly in such exposures. If of earth only, they are very apt to be 
carried away if a freshet should overtop them. 

Sluices should be placed In every important dam. in order that 

all the water may be drawn off. if necessary, for the purpose of repairs; or of removing mud deposits; 
or finding lost articles of importance, &c. They may be merely strong boxings, with floor, sides, and 
top of squared timbers; and passing through the breadth of the dam, just above the bottom. To pre¬ 
vent trees, &c, from entering and sticking fast in them, some kind of strong screen is expedient. In 
common cases a sluice should not exceed about ft by 5 ft in cross-section ; otherwise it becomes 
hard to work. Two or more such openings may be used when much water is to be voided. They 
should be near the abutments. The gates or valves for opening and shutting them, should be at the 
up-stream end; for if at the lower one, accumulations of mud, &c, will till the sluices, and prevent 
them from working. They are usually of timber; and slide vertically in rebates; being raised aud 
lowered by rack and pinion ; but in very important dams they may be of cast iron. Two sets of sluice! 
are desirable; that one may be always ready for use if the other is stopped for repairs. 

The part of the apron in front of the sluice should be particularly firm, so as not to be deranged by 
the water rushing out under a high head. 

Dams are sometimes, but rarely, built in the form of an 
arch ; convex up stream. This form is strong; and when the shores are of rock 
it may be expedient to use it; but if the banks are soft, they will be exposed to wear by the current 
thrown agaiust them at the abuts of the arch. 

At times (lams are built obliquely across the stream, with 

the object of increasing the length, and consequently reducing the depth of water over the crest in 
times of freshets. The argument, however, appears to the writer to be of but little weight, inasmuch 
as the reduction of depth would extend but a trifling distance up stream from the dam; and would 
therefore scarcely have an appreciable effect in diminishing the injury to the overflowed district above. 
M oreover, the increased expense is probably always more than commensurate with any advantage 
gained. 

Nome dams are subject to “trembling’s,” which have not been 

satisfactorily accounted for. They exhibit themselves chieflv as undulations of the air, produced by 
the falling water; aud which occasionally cause a rattling of windows within a distance of % a mile 
or more. We have known this to be stopped unintentionally in one case, by building a well-covered 


*t’ost of crib dams. With common labor at $1.50 per day; $20 per 1000 ft 

board measure, delivered ; stone for filling. $1 per cub yard ; gravel 50 cts per cub yd; iron for bolts, 
&c, 4 cts per lb; such dams in shallow water usually cost complete, from 9 to 12 cts per cub ft; or 
$2.43 to $3.24 per cub yd of crib. 







286 


DAMS. 


wide crib apron, a few feet high, against the front of the dam foi• prevet.ting tt.e ** 

> n other cases a series of obliuue timbers placed against the front ot the dam, ami pari waj up 
U at alpeofaboutlAtol, and covered with plank, has been perfectly effective »u stopping it. 

Tlio proper time (or building dams is of course at the longest period 

of low stage of water. 

To ascertain in advance approximately, the height to 
which the water will rise above tlie crest of a «lam: 01 rather, 

a little back from it • the crest being above the level of the original v ater. This will vary with the shape 
of the crest, asniav be seen by refe.ei.ee to Fig 26*, page 268. which, however, is a very peculiar 

.- Still, until we have more experiments, appreciable deviations from the results of such ruies 

i • .. .. o_ _ • dionh tho ttlranm in Cllh ft HPT SBC. C.lll this nQUSUB. 8 . 


C C A1 II 1 lim-ii v", «. I' I - ■ ' - ~ .. 

must be expected in practice. Square the disch of the stream in cub ft per sec. squares. 

Square the length of the overfall in ft. Mult this square by 7. Call the piod p. I>'."'ie s by p. lake 
the cube rt of the quot.* This cube rt will be the reqd approximate height of rise in ft. 

When in times of freshets, the water rises above the crest to a height equal to that of the dam 
itself, there is no perceptible fall at the dam; aud boats may pass in salety over the crest. 

For measuring the disci: over dams, see pp 264, 267. See also Art 1 of Hydrostatics, 
shape of a formula, the foregoing rule will be, 


In 


Rise _ , r , f (di scharge in cub ft per_sec*\ 

*» ft \7 (length of overfall in ft 2). / 


When the dam is originally a 
submerged, or drowned one. as I), 

Fig 9 ; the following is a rough approximation, probably 
somewhat in excess; og being the natural level of the 
water previous to building the dam; and oc the natural 
depth in ft, of the water above the intended crest. Then 
the required depth oc, of the up-stream water, above the 
crest when built, will be, approximately. 


ac-oc + cube root of ( 2 \ 

\ 7 X length 



I 


Having a c, deduct o c; and the rem will be ao, or the required rise produced by the dam.* 

The rnles given for the varying rise of surface for consid¬ 
erable distances np stream from dams ; as well as for some allied sub¬ 
jects in hydraulics; are extremely complicated : and require much greater knowledge of mathematics 
than is usually found among civil engineers; and so far as regards their application to the actualities 
of common occurrence, they are probably no less useless than complicated. 


Table of thickness of white pine plank required not to bend 
more than part of its clear horizontal stretch, under 
different heads of water. (Original ) 




Heads in feet. 


Stretch 






in Ft. 

40 

30 

20 

10 

5 



Thick 

ness in Inches. 


3 

3X4 

3 

2 ; X 

2 H 

1 H 

4 

6 

4 4 

4 

6 

5t| 

2% 

4 /4 

m 

8 

9 

8 

4 

5/6 


10 

u% 

10 


7 

5*4 

12 


12 K 

10$ 


6 % 

15 


15 

13 

io U 


20 

2 M 

20 

17^ 

14 

11 


* These two rules are from Raukine, who says they apply to “ crests either flat or slightly rounded.' 

But that, in itself, is very vague. 

































WATER SUPPLY. 


287 


WATER SUPPLY. 

Tilt- quantity of water required in cities, lias been found bv ex¬ 
perience to increase faster than the population. About 60 gallons, or 8cubic feet, 
per day, to each inhabitant, is usually considered a fair ample allowance. Manv 
European cities have not half as much ; while New York, and some others, use 
and waste half as much more. With efficient means for preventing waste, 60 
gallons would probably suffice for any commercial city; but inasmuch as clean¬ 
liness and health are promoted by its free use, as few restrictions as possible 
should be introduced. 

Water for city use should not he drawn from the very bot¬ 
tom of the reservoir, because it will then be apt to carry along the sedi¬ 
ment ; which not only injures the water, but creates deposits within the pipes; 
thus obstructing the flow. In fixing upon the necessary capacity of a reservoir, 
this must be taken into consideration ; inasmuch as all the water below the level 
for drawing off, must be regarded as lost. When circumstances justify the ex¬ 
pense, it is well to curve up the reservoir end of the service main, so as to pro¬ 
vide it with valves at different heights; for drawing off only the purest stratum 
that may be in the reservoir. With this view, the valve-tower (page 289) gen¬ 
erally has such valves communicating with the water in the reservoir; and by 
this means only the purest is admitted into the tower: and from it, into the 
city pipes. This refinement, however, is rarely practicable. Such valves must 
of course be worked by watchmen. For rainfall, see p 220. 

Art. 1. Reservoirs. In important reservoirs of earth, for storing water 
to moderate depths for cities, experience appears not to sanction dimensions 
bolder than 10 feet thick at top ; inney slope 2 to 1; outer slope to 1.* A top 
width of 15 feet to 20 feet, and inside slopes of 3 to 1, are adopted in some im¬ 
portant. cases; with outer slopes of 2 to 1. Both slopes, however, are at times 
made only 1% to 1. The level water surface should be kept at least 3 or 4 feet 
below the top of the embankment; or more, if liable to waves. In a large 
i reservoir, a quite moderate breeze will raise waves that will run 3 feet (measured 
vertically) up the inner slope. Alow wall, or close fence, w, Fig. 37, is some¬ 
times used as a defence against them. The top and the outer slopes should be 

{ .protected at least by sod or by grass. To assist in keeping the top dry, it 
should be either a little rounding, or else sloped toward the outside.f The soft 
soil and vegetable matter should be carefully removed from under the entire 
base o p the embankments ; which should be carried down to soil itself imper¬ 
vious to water, in order that, leakage may not take place under them. To aid in 
this, a double row of sheet piles, or a sunk wall of cement masonry, carried to 
a suitable depth below the bottom, may be placed along the inner toe in bad 
cases. If there are springs beneath the base, they must either be stopped, or 
led away by pipes. The embankment should be carried up in layers, slightly 
hollowing toward the center, and not exceeding a foot in thickness; and all 
stones, stumps, and other foreign material, such as clean gravel, sand, and de¬ 
composed mica schists, &c, that may produce leakage, carefully excluded. These 
layers should be well consolidated by the carts; and the easier the siopes are, 
the more effectively can this be done. The layers, however, should not be dis¬ 
tinct, and separated by actual plane surfaces; but each succeeding one shoul 1 
be well incorporated with the one below. This has sometimes been done by 
driving a drove of oxen, or even sheep, repeatedly over each layer; in addition 
to the carting. Rollers are not to be recommended, as they tend to produce 
i seams between the layers. This might possibly be obviated by projections on 
the circumference of the roller. 

Gravelly earth is an excellent material, perhaps the best. The choicest 
material should be placed in the slope next to the water; and should be de¬ 
posited and compacted with special care in that portion, so as to prevent the 
water from leaking into the main body of the dam, and thus weakening it. It 
is not amiss to introduce a bench, fe, Fig 37, in the outer slope, to diminish 
danger from rainwash by breaking the rapidity of its descent. 

If the bottom of the reservoir itself is on a leaky soil, or on fissured rock, 
through the seams of which water may escape, it must be carefully covered 
with from 1% to 3 feet of good puddle; which, in turn, should be protected from 
abrasion and disturbance, by a layer of gravel; or of concrete, either paved or 
not. according to circumstances. 


* The writer suggests that a top width equal to 2 feet + twice the square root 
of the height in feet, will be safe for any height whatever of reservoir properly 
constructed in other respects. 
fSome engineers slope the top toward the inside. 








288 


RESERVOIRS. 


Reservoirs constructed with the foregoing dimensions and with care way 
TP J®f n safe for an indefinite period; but where serious damage would result 
?rom failure the following additional precautions should be taken 
Tlie inner slopes should be carefully faced up to the very top, with at least a 
close drv rubble-stone pitching, not less than 15 to 18 inches thick ; as a pro - 
tion against wash and against muskrats. These animals, we believe, alway. 
commence to burrow under water. If the slopes are much steeper than 2 to 
this dry pitching will be apt to be overthrown by the sliding down ot the so 
ened earth behind it, if the water in the reservoir should for any cause be 
draw n down rather suddenly. It will be much more effective, but of course 
more costly, if laid in hydraulic cement; and still more so if la d upon a layer 
•i few inches thick of cement and- rav< 1 concrete; especially it this last he 
underlaid bv a layer about lV£to 3 feet thick of good puddle, spread over the j 
f a ce of the slope; the great object being to protect the inner slope iiom actual 
contact with the water. If this can be effectually accomplished, slopes as steep 
as ll< to 1 will he perfectly secure ; for the danger does not arise from any want 
of weight of the earth for resisting overthrow. Special care should he 
bestowed upon the inner toe of the slope, to prevent water from 
finding its wav beneath it, and softening the earth so as to undermine the stone 
pitching. Near the top, reference should he had to danger ol derangement by 
ice frost, rain, and waves. Flat inner slopes tend not only to prevent the dis¬ 
placement of the pitching; but increase the stability of the embankment, by 
causing the pressure of the water (which is always at right angles to the slope) 
to become more nearly vertical; and thus to hold the embankment more firmly 
to its base than if there were no water behind it. Sometimes the toes of both 
the inner and outer slopes abut against low retaining-walls in cement. 11ns 
gives a t eat finish, and tends to preservation from injury. 

Many engineers, ill order to prevent leaking, either through or beneath the 
embankment, construct a pmldle-wall, p, .Fig. 37, of well-rammed imper¬ 
vious soil,'grave ly clay is the’! 
- - 



Fig. 37. 


best,) reaching from the top 
to several feet below the base. 
This wall should not he less j 
than 6 or 8 feet thick on top,, 
for a deep reservoir; aid' 
should increase downward by' 
offsets (and not by slopes or 
hatters) at the rate of about 


1 in tntnl t ir*lr n « to 


in depth. Other engineers object to these puddle-walls; and contend that leak¬ 
age should be prevented by making both the inner slopes, and the bottom of the 
reservoir, water-tight, by means of puddle, concrete, and stone facing in cement, 
as just alluded to. They argue that if the embankment is well constructed, it 
is itself a puddle-wall throughout. 

Near San Francisco. Cal, are two cartlien reservoir dams 

built about 1864. one 95 feet high, 26 on top, inner slope 2.75 to 1, outer 2.5 to 1. 
The other 93 high, 25 on top, inner slope 3.5 to 1, outer 3 to 1. In each the pud¬ 
dle-wall is carried 47 feet deeper than the base. No stone facing. 

It is difficult to prevent water under high pressure from 
finding- its way through considerable distances along seams 
where earth is in contact with smooth rock, wood, or metal; as, for instance, 
a'ong the surfaces of iron pipes laid under reservoir embankments; or along 
the tie-rods sometimes used through the puddle of cotter-dams; and the same 
is apt to occur under the bases of embankments which rest on smooth rock. 
Special care should he taken that the earth used in such positions is not of a 
porous nature; and that it is thoroughly compacted all along the seam; and the 
straight continuity of the seam should be interrupted or broken as frequently 
as possible by projections. Faucets or flanges do this to a limited extent in the 
case of iron pipes; and something similar, but on a larger scale, should at short 
intervals be constructed in the shape of collars or yokes of cement stonework, 
in the case of rock or masonry. See also Dams, p 282, also p 229, &c. 

Tt is usually advisable to divide reservoirs into two parts, so that 
whi'e the water in one part is being drawn off for use, that in the other may 
purify itseli by settling its sediment. Also, one part may remain in use. while 
the other is being cleaned or repaired. Many days, or even two or three weeks, 
sometimes, are required for the complete settlement of the very fine clayey par¬ 
ticles in muddy water; depending on the depth of the reservoir. One or more 
flights of steps to the bottom of the reservoir should be provided. 

NInd in Reservoirs. The reservoirs of the New River Water To, Lon¬ 
don, England, were uncleaned for 100 years, during wh ch mud 8 feet deep was 



















RESERVOIRS. 


289 


deposited, or about an inch annually. At Philadelphia it is about .25 inch per 
annum from the Schuylkill, and 1 inch from the Delaware Kiver. At St Louis, 
Missouri, about 3 to 4 feet per year! Vegetation is apt to take place in shadow 
reservoirs and near the edges of deep ones, especially in very warm weather; 
and the plants, on decaying, injure the water. 

Water flowing' til* rough niarsli lands is sometimes unfit for drink¬ 
ing purposes. That, for instance, in some sections of the Concord River, Massa¬ 
chusetts, was reported by the eminent hydraulic engineer, Loammi Baldwin, of 
Boston, to be absolutely poisonous from this cause. 

The construction of a large deep reservoir is not only a very costly, but a 
very hazardous undertaking. With every watchfulness and cafe, it is almost 
impossible entirely to prevent leaking; although this may not manifest itself 
for months, or even years. Should a break occur, especially near a city, it 
would probably be attended by great loss of life and property. If the water 
once finds its way in a stream, either across the unpaved top, or through the 
body of the embankment, the rapid destruction of the whole becomes almost 
certain. 

Art. la. Storing Reservoirs. The entire annual yield of a stream 
may be much more than sufficient for supplying a certain population with 
water; and yet in its natural condition the stream may not be available for this 
purpose, because it becomes nearly dry in summer, when water is most needed; 
while, at other seasons, the rains and melted snows produce floods which supply 
vastly more than is required; and which must be allowed to run to waste. A 
storing reservoir is intended to collect aud store up this excess of water, so that 
it may be drawn off as required during the droughts of summer, and thus 
equalize the supply throughout the entire year. This, when the locality per¬ 
mits, is effected by building a dam across the stream, to form one side of the 
reservoir; while the hill-slopes of the valley of the stream form the other sides. 
The stream itself flows into this reservoir at its up-stream end. When the 
stream is liable to become nearly dry during long summer droughts experience 
shows that the capacity of the reservoir should be equal to from 4 to 6 
months’ supply, according to circumstances. During the construction of the 
dam, a free channel must be provided, to pass the stream without allowing it 
to do injury to the work. If the dam were built precisely like Fig 37, entirely 
of earth, it would plainly be liable to destruction by being washed away in case 
the reservoir should become so full that the water would begin to flow over its 
top. To provide against this we may, by means of masonry, or of cribs filled 
with broken stone, or otherwise, construct either the whole, or part of the dam, 
to serve as an overfall, or a waste-weir. Or a side channel (an open cut, 
pipes, or a culvert, &c) may be provided at one or both ends of the dam, and in 
the natural soil, at such a level as to carry away the surplus flood water before 
it can rise high enough to overtop the earthen dam. Besides these, and the 
pipes for carrying the water to the town, there should be an outlet, with a valve 
or gate, at the level of the bottom of the reservoir; in order that, if necessary 
for repairs, or for cleaning by scouring, all the water may be drawn off. The 
entrances to the city pipes should be protected by gratings, to exclude fish, &e. 

To facilitate repairs or renewals of all valves, Ac, whirl* 
are under water, the reservoir ends of the pipes or culverts to which they 
are attached, may be surrounded by a water-tight, box or chamber, which will 
usually be left open to the reservoir ; but may be closed when repairs are re¬ 
quired. Access may then be had to them by entering at the outer end, after 
the water has flowed away from inside. In case the outlet is through a long 
line of pipes which cannot thus be entered, a special entry for this purpose may 
be cast in the pipe itself, near the outer toe of the embankment; to be kept 
closed except in case of repairs. Sometimes a better, but more expensive means 
of access to such valves, is secured by enclosing them in a valve-tower of 
masonry. This is a hollow vertical water-tight chamber, like a well; but near 
the toe of the inner slope; having its foundation at the bottom of the reservoir; 
whence the tower rises through the water to above its surface. This chamber 
is provided with valves or gates usually left open to the reservoir; but which 
may be closed when repairs are needed ; and the water in the tower allowed to 
escape from it through the open valves of the outlets. This done, workmen can 
descend through the tower by ladders from the aperture at its top. 

At times the outlets for the discharge of surplus flood water are, like those for 
scouring, placed at, or just above, the level of the bottom of the reservoir. In 
order that these may work in case of a sudden flood at night, &c, they must be 
furnished with self-acting valves, which will open of their own accord when the 
flood is about to rise too high. This may be effected by attaching them to floats, 
the rising of which, when the water is high, will pull them open. All such out¬ 
lets should be large enough to let men enter them for repairs. They should by 








290 


WATER-PIPES. 


no means be laid through the artificial earthen body of the dam itself, without 
being supported upon masonry reaching down to a firm natural foundation; 
otherwise i hey are very apt to be broken by the subsidence of the embankment. 

It is usually safer to carrv them through the firm natural soil near one end of 
the dam. Their valves, if only single, should be at their inner or reservoir end, 
so as to leave the outlets themselves usually empty, for inspection; but it is 
better to have two valves, so that one may be used when the other needs repair: 
and in this case one may be placed at each end. Reservoirs which are supplied 
by pumps, need no precautions against overflow'; because the pumping is 
stopped when they are filled to the proper height. Large storing reservoirs 
necessarily submerge more or less land, which has therefore to be purchased. 

By intercepting the descending water, they frequently prevent spring floods 
from injuring low lands farther down stream. If there are mills down stream , 
from the reservoir, they would evidently be deprived of water for driving them, 
unless a portion of that stored in the reservoir be devoted to that purpose. 
Water thus applied to compensate for the loss of the natural stream, is called 
compensation water; and the reservoir, a compensating one. 

Art. lb. Distributing: reservoirs. Frequently a valley fit for a storing 

reservoir cau be found only at a long dist (sometimes many miles) from the town; and it tben be¬ 
comes expedient to construct also an additional one of smaller size than the storing one, near the 
town ; and at as great an elevation above it as circumstances will permit; but lower than the storing 
one. This is called, by way of distinction, a distributing reservoir, because from it the water, after 
having flowed into it from the storing reservoir, through the long supply pipe which connects them, is 
distributed in various directions through the town, bv means of the street mains, or pi(>es. This 
small reservoir should hold a supply sufficient at least for a few days; a few weeks would he better; 
and the end of the supply pipe which terminates in it, should be provided with a valve for shuttiug 
off the supply from the storing reservoir. These precautions permit repairs to he made along the line 
of supply pipe without depriving the town of water in the mean time. With a view to such repairs ; as 
well as to scouring out sediment from the supply pipe, this last should be provided with Outlet 
waives at various low points along the entire interval between the two reservoirs ; especially at 
those at which the valves may disch into natural watercourses. On opening these valves, the out- 
rush of the water carries away sediment; and leaves the pipe empty for inspection. ) 

In fixing upon the diams of pipes for supplying cities, it is necessary 

to bear in mind, that by far the greater portion of the 24 hour?’ yield is actually drawn from them 
during only 8 to 12 hours of daylight; and therefore the capacity of the pipes must be sufficient to 
furnish the daily supply in much less than 24 hours. Again, duriug the hot summer mouths, much 
more water is used thau during the winter ones; and this consideration necessitates a still larger diaiu. ‘ 

Art. 2. Systems of street pipes for supplying cities. The 

writer know3 of no practical rules for proportioning the diams for such systems. The various com¬ 
plications involved, render a purely scientific investigation of little or no service. With much hesi¬ 
tation, he ventures the following purely empirical rules of his own ; based on such limited observa¬ 
tions as have casually fallen under his notice. 

Rule 1. When, at no point in a system of city pipes, is the head, or vert dist below the surface 
of the reservoir, compared with the hor dist from the reservoir , less than at the rate of M ft per mile, 
then the population in the last column of the following Table A, may be abundantly supplied, for all 
city purposes, by either one pipe of the inner diam or bore in the 1st col; or by 2, 3, Ac, pipes of the 
diams in the other cols. These diams are given to the nearest safe % inch. The supply is assumed 
to he about 60 gallons per day to each inhabitant. 


TABLE A. (Original.) 


1 

2 

NUMBER 

3 | 4 

OF PIPE 

6 

:s. 

8 

12 

24 

Population. 

Diam. 

Diam. 

Diam. 

Diam. 

Diam. 

Diam. 

Diam. 

Diam. 

Ins. 

Ins. 

Ius. 

Ius. 

Ins. 

Ins. 

Ins. 

Ins. 


6 

4% 

3% 

3% 

3 

2% 

2% 

1% 

1647 

8 

6% 

5% 

4% 

4 

3% 

3% 

2% 

3165 

JO 

7% 

6% 

5% 

5 

4% 

3% 

3 

5908 

12 

9% 

7% 

7 


5% 

4% 

3% 

9324 

14 

10% 

9% 

8% 

7 

6% 

5% 

4% 

13706 

16 

12% 

10% 

9% 

7% 

6% 

6 

4% 

19141 

18 

13% 

11% 

10% 

8% 

7% 

6% 

5% 

25677 

20 

i5 % 

13 

11% 

9% 

8% 

7% 

5% 

33426 

22 

16% 

14% 

12% 

10% 

9% 

8% 

6% 

42433 

24 

18 % 

15% 

13% 

11% 

10% 

9 

6% 

52671 

26 

19% 

16% 

15 

12% 

11% 

9% 

7% 

64447 

28 

21% 

18% 

16% 

13% 

12% 

10% 

8 

77565 

30 

22 % 

19% 

17% 

14% 

13% 

11% 

8% 

91580 

32 

24 Vs 

25 % 

20% 

18% 

15% 

14 

U% 

9 

108160 

34 

22 

19% 

16% 

15 

12% 

9% 

125840 

36 

27% 

23% 

20% 

17% 

15% 

13% 

10% 

144480 

40 

30% 

25% 

23% 

19% 

17% 

15 

11% 

188320 

44 

33% 

28% 

25% 

21% 

19% 

16% 

12% 

239600 

48 

36% 

31 

27% 

23% 

21% 

18 

13% 

297600 

54 

41 

34% 

31% 

26% 

23% 

20% 

15% 

391200 

60 

45% 

38% 

34% 

29% 

26% 

22% 

17% 

511200 

66 

50% 

42% 

38% 

32% 

29% 

24% 

18% 

650*00 

72 

54% 

46% 

41% 

35% 

31% 

26% 

20% 

800000 

80 

60% 

51% 

46% 

39% 

35% 

29% 

22% 

1064000 



































WATER-PIPES. 


291 


It is well to allow in addition from % inch to 1 inch, or more, (depending on 
the character of the water,) to each diameter; for deposits and concretions. 

The water, after reaching the city through one or more large main pipes from 
the reservoir, must be distributed through the- streets by means of smaller 
mains branching from the larger ones. The diameters of these smaller ones 
also may be found by Table A. Thus, if a street, with its alleys, Ac, contains 
about 6000 persons, (the rate of head being, as before, not less than 50 feet to a 
mile at any point of the system,) then we see by the table that a 10-inch pipe 
will answer. It would be well to lay no city street pipes of less than 6 inches 
diameter. 

Mains which cross eacli other should he connected at some 
of their intersections, to allow the water a more free circulation through¬ 
out the entire system ; so that if the supply at any point is temporarily cut off 
from one direction by closing the valves for repairs, or is diminished by exces¬ 
sive demand, it may be maintained by the flow from other directions. 

Avoid dead ends when possible, as the water in them becomes foul and 
un wholesome. 

Rulk 2. With the same diameters , different rates of head will supply the propor¬ 
tionate populations in column 3 of Table B. Or, to find the diameters which at different 
rates of head will supply the same populations given in the last column of Table A, 
multiply the diameter given in Table A, by the corresponding number in col¬ 
umn 4 of Table B; or (approximately) do as directed in column 5. 

TABLE B. (Original.) 


Col. 1. 

Col. 2. 

Col. 3. 

Col. 4. 

Col. 5. 

Bate of Head, 

Rate of Head, 
compared with 
that in Table A. 

Proportionate 

Proportionate 
Diam. to supply 

Remarks. 

iu Feet per Mile. 

Populations. 

the Populations 
in Table A. 

5 

.1 

.32 

1.58 


10 

.2 

.45 

1.37 


12X 

.25 

.50 

1.32 

Add one-third. 

15 

.3 

.55 

1.27 

Add full one-fourth. 

20 

.4 

.64 

1.20 

Add one-tifth. 

25 

.5 

.71 

1.14 

Add one-seventh. 

30 

.6 

.78 

1.11 

Add one ninth. 

35 

.7 

.84 

1.07 

Add one fourteenth. 

37% 

.75 

.87 

1.06 

Add one-sixteenth. 

40 

.8 

.90 

1.05 

Add oDe-twentieth. 

45 

.9 

.95 

1.02 

Add one-fiftieth. 

50 

1.0 

1.00 

1.00 


75 

1.5 

1.23 

.92 

Deduct one-thirteenth. 

100 

2.0 

1.41 

.88 

Deduct one eighth. 

125 

2.5 

1.59 

.83 

Deduct full one-sixth. 

150 

3.0 

1.73 

.80 

Deduct one-fifth. 

200 

4.0 

2.00 

.76 

Deduct nearly one-fourth. 

250 

5.0 

2 25 

.73 

Deduct nearly two-sevenths. 

300 

6 0 

2 46 

.69 

Deduct three-tenths. 

400 

8.0 

2.83 

.66 

Deduct full one-third. 

500 

10.0 

3.18 

.63 



Example. By Table A we see that with the rate of head of 50 feet per 
mile, a 30-inch pipe will supply a population of 91580; hut with three times that 
rate of head, or 150 feet per mile, we see by column 3, Table B, that the same 
pipe will supply 1.73 times as many persons, or 91580 X 1.73= 158433 persons. 
But if, at this greater rate of head, we still wish to supply only 91580 persons, 
then we find in column 4, Table B, that we may diminish the diameter of the pipe 
from 30, down to 30 X-80 = 24 inches; or, by column 5, we have 30 — 6 = 24 
inches. 

Again, after the water has reached the city by the 30-inch pipe of Table A, 
if we wish to distribute it through the city by say eight branches or smaller 
mains, we see by column 6, Table A, that each of them must have at least 13% 
inches diameter. From these eight, other smaller ones may branch off into the 
cross streets, alleys, &c; and in estimating the supply required for any partic¬ 
ular street main, we must evidently add what is required also for such cross 
streets, &e, &c, as are to be fed from said main. 

If certain limited parts of a city pipe system have considerably less rates of 
head than most of the remainder, it may become expedient to supply the former 
by a special separate main of larger diameter; which may start either directly 
























292 


WATER-PIPES. 


from the reservoir; or as a branch from the grand leading main which feeds the 
lower parts, according to circumstances. 

It must be remembered, that although by increasing the diameters, an abun¬ 
dant supply may be obtained under a small rate of head, as well as under a great 
one, yet the water will not rise to as great a height in the service pipes for sup¬ 
plying the different stories of dwellings. &c. Even with the diameters in Table 
A, the water, under ordinary use, will not rise in these pipes to the full height 
of the surface of the reservoir; and if an unusual drawing-off is going on at 
the same time at many parts of the system, as in case of an extensive fire, or 
frequently during the hot summer mouths, it may not rise to eveu one-half of 
that height. 

Art. 3. The following has been found very effective for 
preventing concretions in water pipes. Formerly in Boston, cast- 

iron city pipes, 4 inches diameter, became closed up in 7 years; and those of 
larger diameter became seriously reduced in the same time. But later, during 
8 years, in which this varnish was used, no concretions formed. 


Coal-pitch varnish to be applied to pipes and castings, 

made for the Water Department of Philadelphia, under 

the following conditions: 

First. Every pipe must be thoroughly dressed and made clean, free from the 
earth or sand which clings to the iron in the moulds : hard brushes to be used 
in finishing the process to remove the loose dust. 

Second. Every pipe must be entirely free from rust when the varnish is ap¬ 
plied. If the pipe cannot be dipped immediately after being cleansed, the sur¬ 
face must be oiled with linseed oil to preserve it until it is ready to be dipped: 
no pipe to be dipped after rust has set in. 

Third. The coal-tar pitch is made from coal tar, distilled until the naphtha 
is entirely removed, and the material deodorized. It should be distilled until it 
has about the consistency of wax. The mixture of five or six per cent of linseed 
oil is recommended. Pitch which becomes hard and brittle when cold, will not 
answer for this use. 

Fourth. Pitch of the proper quality having been obtained, it must be care¬ 
fully heated in a suitable vessel to a temperature of 300 degrees Fahrenheit, and 
must be maintained at not less than this temperature during the time of dip¬ 
ping. The material will thicken and deteriorate after a number of pipes have 
been dipped ; fresh pitch must therefore be frequently added ; and occasionally 
the vessel must be entirely emptied of its old contents, and refilled with fresh 
pitch; the refuse will be hard and brittle like common pitch. 

Fifth. Every pipe must attain a temperature of 300 degrees Fahrenheit before 
it is removed from the vessel of hot pitch. It may then be slowly removed and 
laid upon skids to drip. 

All pipes of 20 inches diameter and upward, will require to remain at least 
thirty minutes in the hot fluid, to attain this temperature; probably more in 
cold weather. 

Sixth. The application must be made to the satisfaction of the Chief Engineer 
of the Water Department; and the material be subject at all times to his ex¬ 
amination, inspection, and rejection. 

Seventh. Payment for coating the pipes will only be made on such pipes as 
are sound and sufficient according to the specifications, and are acceptable inde¬ 
pendent of the coating. 

Eighth. No pipe to be dipped until the authorized inspector has examined it 
as to cleaning and rust; and subjected it thoroughly to the hammer proof It 
may then be dipped, after which, it will be passed to the hydraulic press to meet 
the required water proof. 

Ninth. I he proper coating will be tough and tenacious when cold on the 
pipes, and not brittle or with any tendency to scale off. When the coating of 
any pipe has not been properly applied, and does not give satisfaction, whether 
from defect in material tools, or manipulations, it shall not.be paid for- if it 
scales off or shows a tendency that way, the pipe shall be cleansed inside before 
it can be recoated or be receivable as an ordinary pipe. 






WATER-PIPES. 


293 


Art. 4. The pipes are laid to conform to the vert undulations of the street sur¬ 
faces. The tops of the pipes are laid not less than 3 y 2 feet below the surface of the 
street; but in 3-inch pipes the water has at times been frozen at that depth. 

In Philada, in 1885, there are about 784 miles of street 
pipes ; or about 1 mile to every 1100 inhabitants. The population is about 
86o,000; residing in about 150,000 dwellings. 

No K ill van ie action has been observed where lead pipes or brass unite with 
cast-iron ones. No pipe less than 6 inches diam should be laid in cities; and 
even they only for lengths of a few hundred feet. Their insufficiency is chiefly felt in 
case of fire. 8 ins would be a better minimum. No more leakage occurs in winter 
than in summer; except from the bursting of private service-pipes by freezing. 

To compact the earth thoroughly against the pipes excludes air, and greatly im¬ 
pedes rust. Pipes may be corroded by the leakage of gas through the body as well as 
through the joints of adjacent gas-pipes. 

For thickness of metal pipes to resist safely the pressure of various 
heads, see p 233 of Hydrostatics. 

WEIGHT OF CAST-IRON WATER-PIPES, 

As used in Phila, and tested by hydraulic press before delivery to an internal 
pres of 300 lbs per sq inch. This table includes spigots, and faucets or bells. The 
pipes are required to be made of remelted strong tough gray pig iron, such as may 
be readily drilled and chipped ; and all of more than 3 ins diam to be cast vertically, 
with the bell end down. Deviations of 5 per cent above or below the theoreti¬ 
cal weights, are allowed for irregularities in casting, which it seems impossible to 
avoid. 

The pipes are in lengths from 3 to 3^ ins longer than 12 ft; so that when laid they 
measure 12 ft from the mouth,/, Fig 38, of one bell to that of the next. 


Diam. 

Thick¬ 

ness. 

Wt per 
length. 

Diam. 

Thick¬ 

ness. 

Wt per 
length. 

Diam. 

Thick¬ 

ness. 

Wt per 
length. 

Ins. 

Ins. 

Lbs. 

Ins. 

Ins. 

Lbs. 

Ins. 

Ins. 

Lbs. 

3 

JS 

% 

158 

16 

% 

1322 

36 

T§ 

4334 

4 

211 

20 

It 

1654 

36 


4862 

6 

7 

IS 

385 

20 

T& 

1798 

36 

ik 

5366 

8 

»4 

460 

30 

if 

3313 

48 

7282 

10 

K 

667 

30 

.9 

3610 

48 


8667 

12 

% 

899 

30 

1 

3964 

48 

i k 

9378 


Price, Phila, 1886, about $35 to $45 per ton; or say to 2 cents per lb; de¬ 
pending on size and quantity ordered. Elbows, connections, &c, about 75 to 100 per 
cent more. In ordering anything by the ton, be careful to specify the number of 
lbs (2240). This prevents misunderstandings. 

The following sizes of lap-welded wrong-tit-iron water-pipe are 

made by the National Tube Works Co, McKeesport, Pa, and fitted with their 
Converse lock-joint.” One end of each length of pipe has the lock-joint 
permanently attached (leaded) to it at the works before shipping. The “ weights 
per foot” include these joints. The weight of “lead per joint” given is that re¬ 
quired to be poured in laying the pipe, or that for one side only of the joint. 


Outer diam, ins. 2 3 4 5 6 8 10 12 16 

Weight per ft, lbs. 1.86 3.48 5.26 7.33 8.76 13.20 17.08 25.12 47.70 

Read per joint, lbs. % l x 9 g 2% 3% 3 T 9 g 6 8 % 16 

Average car load: 

Number of lengths. 800 380 275 145 126 128 80 56 40 

“ “ feet. 11500 5600 4500 2600 2000 2000 1200 800 630 


The pipes are tested for a bursting pressure of 500 lbs per square inch, or higher 
if desired. They are furnished either coated with asphaltum, or “kala- 
meined ; ” or, if desired, first kalameined and then coated with asphaltum. 
Kalameining consists in “incorporating upon and into the body of the iron a non- 
corrosive metal alloy, largely composed of tin.” The surface thus formed is not 
cracked by blows, or by bendiug the pipe, either hot or cold. 


































294 


WATER-PIPES. 


The joint, or coupling, is of cast-iron, and has internal recesses which receive and 
hold lugs on the outside of each length of pipe, near each of its ends. The joint is 
then poured with lead in the usual way (see next page), either with clay collars, oi 
with a special pouring clamp furnished by the Co. This clamp resembles the 
“jointer,” Figs 39 &c, except that it is in two rigid semi-circular pieces, connected 
together by a hinge-joint, and furnished with handles like those ot a lemon-squeezer, 
and has a hole in one side for pouring. The coupling forms a flush inner surface 
with the pipe at the joint, thus avoiding much of the resistance ot cast-iron pipes 
to flow. For cases where it may be necessary to make frequent changes, the coup¬ 
lings are made in two pieces, which are bolted together by flanges. 

Wrouslit-iron, for pipes, has the great advantages over cast-iron 
of lightness, toughness, and pliability. The lightness ot wrought-iron pipes len¬ 
ders them easier to handle, and cheaper per foot notwithstanding that their cost per 
ton is about 25 per cent greater. They are not liable to breakage in transportation 
or from rough handling, and they may be bent through angles up to about 259. 
They therefore require no special bend castings for such angles. The National Co 
supply bending machines, to be worked by two men. One machine can, by changing 
the dies, be used in beuding all sizes of pipe. The pipes are in lengths of from lo to 
18 feet, instead of 12 feet, as in the case of cast-iron, so that fewer joints are 


required per mile. 

The Co furnish special “service clamps” and tapping machines for attacliing- 
service pipes to mains. This may he done (as in the case of the Fayne 
machine, page 299) while the main is under pressure. The service clamp is a cast- 
iron saddle, which, before the main is tapped, is attached to it by means ot a U 
bolt, aud which remains permanently so attached after the tapping. A sheet-lead 
gasket is placed between clamp and main. The clamp has a tapped cylindrical 
opening through it, into which the corporation stop (see page 299) is screwed before 
the pipe is tapped. The drill of the tapping machine passes through the stop, and 
through the cylindrical opening in the clamp, aud drills through the lead gasket 
and through the side of the main. , ■ 

The Co furnish also pipe-cutting machines, and special castings (reducers, 
crosses, &c, &c) fitted with the Converse joint. 


Art. 5. Wrou^lit-iron pipes corrode much more rapidly than cast. 
A g'litta-perclia pipe, % inch thick, and % inch bore, has sustained safely 

an internal pres of more than 250 lbs per sq inch; equal to nearly *>00 feet head. It merely swelled 
slightly at 337 tbs. Iu 1851 a tube of that material, 2y$ ins bore, about tfc inch thick, and 1350 ft loug. 
was sunk in the East River, New York, to carry the Croton water to Blackwell's Island. It was held 
down hy weights. It proven unsatisfactory owing to abrasion caused by tidal currents, aud injury 
from the anchors of dragging vessels. A wrapping of canvas, confined hy spun yarn, was useful in 
preventing the former, but not the latter. This pipe was replaced in 1870 by wrought-irou pipes. 

lSall'N patent iron and cement pipe, is made by The Patent Water 

and Gas Pipe Co, of Jersey City, N. J. It is formed of riveted sheet-iron, and each length is dipped 
into, and coated with, a hot mixture of coal tar aud asphalt. The lining of hydraulic cement is then 
applied. This ranges, in thickness, from % inch for 12-inch pipes to 1 inch for 20-inch pipe. This 
pipe is made up to diams of 36 ins. It is laid in a bed of cement mortar, aud completely covered with 
the same. Suitable means are provided for makiug all the attachments, &c, required iu city pipes 
for water and gas. More than 1300 miles of it are in use in various towns, some of it for 35 years ; 
and it appears to give general satisfaction. Tubercles do not form in these pipes, as they are apt to 
do in cast-iron ones. There is every reason to suppose that they are durable. The trenches being 
dug, the Jersey City Co furnish pipes and lay them (including the cement). 

The Wyckoff Pipe Co, Williamsport, Pa, make woodon water pipes. For 

pressures of 15 to 20 lbs per sq inch, they furnish either plain pipes, 3>^ to 7 ins square externally, 
and from 114 to 4 ins internal diam ; or round pipes, 1 inch to 16 ins bore, coated externally with 
asphaltuni cement. At their ends, both the square and the round pipes are banded with iron. For 
pressures from 40 to 160 lbs per sq iuch, the round wooden pipes, before being coated with cement, 
are spirally wrapped, by steam power, with hoop iron, which is first passed through a preparation 
of coal-tar. The iron is wound so tightly as to be imbedded in the pipe, leaving its outer surface 
flush with that of the wood. The ends of each length of pipe receive extra banding. The asphaltum 
cement coating is then applied. These pipes have been extensively and successfully used for both 
water and gas. Suitable arrangements are provided for joints and connections. 

Water pipes of t>ore<l oak aud pine lo^s, laid in Philndit 50 to 00 

years ago, are frequently quite sound, and still Ht for use, except where outer sap wood is decayed. 
When this is removed, many of these old pipes have been relaid in factories, Ac. Clay well parked 
around wooden pipes, excludes the coutact of air, and thus contributes greatly to their durability. 
Loose porous soils, such as gravel, &c, on the contrary, are unfavorable. 

IMp*** inadf 1 ot biiuminized |iapor, prepared under great pressure, 

have been used for both water and gas. They are much less liable to break than cast-iron, and do 
not weigh or cost more than about half as much. Pipes of 5 ins bore and ^ iuch thick, have resisted 
test strains of 220 lbs per sq iuch; equal to a water head of 507 ft. 





WATER-PIPES. 


295 


Art. 6. Fig 38 is a standard form of pipe-joint, Phila, 1886. Tlie 
clear distance, d, between the spigot and the faucet, is nearly uniform for all sizes 
of pipe, varying only from x 5 g inch for 4-inch 
pipe, to T 7 g inch for 30-inch pipe. The depth, 
vi n, of the faucet varies from 3 ins in 4-inch 
pipe, to 4 ins in 30-inch pipe. 

The small beads at »• and m, s' and m' on the 
spigot end of the pipe, project about ^ * lu "b ; 
and are to prevent the calking material from 
entering the pipe. The calking consists of about 
1 to 2 ins in depth of well-rammed, untarred 
gasket, or rope yarn; above which is poured 
melted lead, confined from spreading by means 
of clay plastered around the joint. The lead is 
afterwards compacted by a calking hammer. 

The lead is poured through a hole left in the 
clay on the upper side of the pipe. In large 
pipes two additional holes are left in the clay, 
one at each side of the pipe, and lead is first 
poured into the side holes by two men at once, 
one man pouring into each side hole until the 
joint is half full. The side holes are then 
stopped, and, after the lead already poured has 
hardened, the two men finish the pouring by 
means of the top hole. This course is necessary, 
because the great weight of melted lead in the 
entire large joint would press away the clay at the lower side of the joint, and 
thus escape. 

The moisture in the clay is liable to freeze in cold weather, and to render it too 
hard to be used. It is also liable, at all times, as is also any dampness in the pipe, 
to be converted into steam by the heat of the melted lead. The steam sometimes 
breaks out, or “blows” through the clay, allowing the lead to escape. 

Art. 7. The Watkins patent Pipe Jointer ” avoids these difficulties by- 
dispensing with the ring of clay. It consists of a ring R, Figs 39 and 40, of square 




^ig. 33 


cross-section, and made of packing composed of alternate layers of hemp cloth and 
India rubber. This ring is encircled by one or more thin strips of spring steel, 
which are riveted to it at intervals, as shown. E E are iron-elbows riveted outside 
of the steel bands. After the gasket has been rammed into its place, the ring is 
placed around the spigot near the faucet, in the position shown in Fig 40, and is 
held loosely by the clamp, Fig 41, one point of which enters a small pit in each 
of the elbows, E E. The ring is then, by means of a hammer, driven close up against 
the end,/, of the faucet. Fig 88; the screw of the clamp is tightened somewhat, so 
as to bring the ring close to the spigot; a small dam of clay is placed in front of the 
aperture between the two elbows, E E; and the joint is ready for pouring. After 
the lead has hardened, the “jointer” is removed, and is ready for use at another 
joint. One can be used for several hundred joints. They of course dispense with 
the services of the men who prepare the clay collars, and supply them to the pool ¬ 
ers. Upon the removal of the “joiuter” the lead is found smooth, requiring no 
chipping, as it is apt to do when poured in the ordinary way. One of these jointers, 
for 4-inch pipe, costs, 1886, $4; for 8-inch pipe, $8, and so on, at $1 for each inch of 
diam. of pipe. Thos. Watkins, Patentee and Sole Manufacturer, Johnstown, Pa. 

































296 


WATER-PIPES. 



D> 


Art. 8 . As a further preventive against the escape of any of the gaS' 
ket into the pipe, Mr Chas. G. Darrach, Hydraulic Engineer, Piiila, places r 
ring of lead pipe in the joint before the gasket is inserted. This lead pipe is of 
such diameter that it can just be pushed through the space, d, Fig 38, between the 
spigot and the faucet; and of such length as just to encircle the water-pipe. It if 
driven as closely as possible into the narrow annular space at o o, Fig 38. The gas¬ 
ket is then rammed in. and the lead poured, as usual. 

Art. 9. John F. Ward's flexible joint for pipes laid across the irreg¬ 
ular bottoms of streams, is shown at Fig 4‘2. A portion, a o, of the inside of the 

faucet F, is bored out truly to form the middle zonej 
of a sphere; and the spigot end, e o, of the other 
pipe is cast with two raised collars, o and e. The 
inner collar, o, is of such a height as barely to al¬ 
low it to pass into the faucet. The outer one, e, is 
a little lower, so as to allow melted lead (shown 
black) to be poured in at a. The outer edge or 
diam of the spigot end at o is carefully turned so 
as to fit the turned spherical zone; so that the 
joint will admit of considerable play without dan¬ 
ger of leaking. In laying the pipes under water, 
the joints are filled with melted lead, as usual, on 
board of suitable vessels or floats. As fast as they 
are thus filled, the floats are moved forward, and 
the pipes, if small, and the water shallow, are 
passed into the water without further care. But for large pipes in deep water, suit¬ 
able. apparatus is used for lowering them without undue strain on the joints. Mr 
Ward has been perfectiy successful in laying this pipe under water, in one case 40 
feet deep. One of the mains of the Philada water-works was thus laid across the 
Schuylkill River. 

In some cases preliminary dredging may be expedient, to diminish abrupt irregu¬ 
larities of the bottom. 

Art. 10. In Figs 43, A is a double brancli ; which is a pipe having, in 
addition to the faucet, c, at one end, two others, s and i, to which pipes leading in 
opposite directions (as at cross-streets) may be attached. If either s or i be omitted, 
the pipe is a single branch. The pipe is stronger when these extra faucets! 
are near its end, than if they were at its middle. In a long line of pipes, for the sake 
of expedition, different gangs of men are frequently laying detached portions some 
distance apart; and when two ends of different portions are brought near enough 
together to be united, as h and r, Fig C, their junction cannot be effected by the 
usual spigot-and-faucet joint. In this case a cast-iron sleeve, tt, is used, which isi 
first slid upon one of the pieces of pipe; and^(after the other piece also is laid) is 
slid back into the position in the fig, so as to cover the joint. Sleeves are usually 

about a foot long; as thick as 
the pipe; and their diam is suf¬ 
ficient to allow the usual joint 
of gasket and lead. There is .* 
of course such a joint at each 
end of the sleeve. 

Art. 11. Wlicn a, 
crack occurs in a pipe, 
a a. Fig B, already in use, it is 
repaired by mea.ns of a cast- 
iron sleeve, g g, made in two 
parts, bolted together by means 
of flanges as at n n. In other 
respects it is like the preceding 
sleeve. The intermediate white 
ring is the lead joint. If the 
crack is too long, or otherwise 


B 



Fig-s. 43. 


too bad to be remedied by a sleeve, the pipe is broken to pieces; and the lead joints 
at its ends melted out, so as to allow of its removal. Then, since an entire new pipe 
cannot now be inserted, owing to the overlapping of the spigot-and-faucet ends, two 
short pieces must be substituted for it. One end of each of these is lead-jointed to 
the pipes already laid ; while the other two ends, which will probably be a few 
inches apart, are covered by a sleeve, t t, Fig C. 

































WATER-PIPES. 


297 


Cracks may at times be temporarily repaired in an emergency, by a wrapping of 
folds of canvas thoroughly saturated with white-lead paint; and tightly confined to 
the pipe by a spiral banding of thin hoop-iron or wire. 

Or. by an iron band, made in two parts, B B, Fig 44, 
and clamped together by screw-bolts, S S. Such bands 
v are useful, also, for strengthening pipes that are con¬ 
sidered to be in danger of bursting. 

To attach a pipe, e, Fig 43, to one, /, 
already ill use, but in which no provision has 
been made for such attachment, a piece may be cut 
out of f, as at v v, and a casting, e, furnished with 
' flanges, m m, bolted over the opening, by screw-bolts 

passing through female screws tapped in the thickness of the pipe. If the new 
pipe is so Large that the opening, v v, if circular, would be inconveniently wide, 
it may be made oval, with the longest diameter in the direction of the length of 
the pipe,y. In that case the casting e will be oval at its flanges; and circular 
at c c. 

} Art. 12. The following table, arranged from data kindly furnished by the late 
Isaac Newton, C E, Ch Eng, Dept of Pub Wks, New York, and his Asst, Mr. James 
Duane, gives the average prices, «fcc, of pipes and laying-, in that 
city, for three years prior to 1885. The pipes are in lengths of 12 ft, and cost $35 per 
2000 pounds. Unpaving, digging trenches, re-filling and re-paving not included. 


]>iam of pipe in ins . 


12 

20 

36 

48 

Weight in lbs of pipe alone, per length. 

. 430 

1000 

2000 

4860 

8250 

44 44 44 44 44 44 44 

.. 36 

83 

167 

405 

688 

Cost in $ of pipe alone, per length. 

. 7.52 

17.50 35.00 

85.05 

144.38 

44 44 44 44 44 44 44 j'f- 

. .63 

1.46 

2.92 

7.09 

12.03 

Leiul* per joint, lbs, at 5 cts per lb. 


22 

41 

80 

146 

“ “ “ $ . 


1.10 

2.05 

4.00 

7.30 

Yarn per joint, lbs, at 9 cts per lb . 

• Vs 


1 

2^ 

5 

44 44 44 £ 

. . 

. .01 

.02 

.09 

.23 

.45 

Hemlock lumber per length, $ . 


... 

.40 

.75 

1.04 

* Coke, clay, &c, per joint, $ . 

. .02 

.04 

.08 

.15 

.25 

>18 jin ling per length, $ . 

. .40 

.80 

2.00 

3.40 

5.00 

Labor per length, $ . 

.60 

.86 

1.27 

1.95 

3.25 

Cost, in $, of laying, exclusive of pipe, per length.. 

1.68 

2.82 

5.89 

10.48 

17.65 

44 44 44 44 44 44 44 44 44 

.14 

.24 

.49 

.87 

1.47 

Total cost, in $, of pipe and laying, per length. 

9.20 

20.32 40.89 

95.53 

162.03 

4C 44 44 44 44 44 44 44 44 ff. 

.77 

1.69 

3.41 

7.96 

13.50 



Art. 13» Air valves. , Air is apt to collect gradually at the high points of 
vert curves along the supply pipes; and, uuless removed, produces more or less obstruction to the 


flow. This may be prevented by air valves, 
f seeFig 44A,which is hi of the full size of those 
once used in Philada. This simple device 
consists of a cast-iron box, ccd d. confined 
to the main pipe m m, by screw-bolts passing 
through its flange dd. It has a cover yng, 
confined to it by screws tt\ and at the top 
, of which is an opening n, for the escape of 
air from within. In this box is a float /, 
which may be a close tin or copper vessel, 
or of layers of cork, as supposed in the fig; 
or A'C. This float has a spindle or stem s s, 
fast to it; which passes through openings in 
the bridge-bars a a. and o; thereby allowing 
the float to rise and fall freely, but prevent- 
king it from moving sideways. Yhen the 
pipe mm is empty, the float is down; its 
base y resting on the cross-bar aa. The 
Btem ss has fixed to it a valve v, which rises 
and falls with it and the float. Suppose the 
pipe mm to be empty, and consequently the 
float, and the valve v, down. Then, if water 
• be admitted into the pipe, it will rise and 
fill also the box as far up as e; and in 
doing so will lift, the float/, and the valve v, 
to the position in the fig; thus preventing 


lb 



v j^j r Thomas Wicksteed, engineer of the East London water-works, England, says that more than 
50 vears’ experience proves that slightly tapering wedges of pine, about 4 ins long, 2 ins wide, and % 
ins thick at the butt, carefully shaped to suit the curve of the pipe, and well dnveu, answer all the 
purposes of lead joints, at considerably less cost. 

20 















































































298 


WATEK-PIPES. 


egress to the outer air, by closing the opening at v. Now, air carried along by the 
water, will, on account of its lightness, ascend to the highest points it meets with. 

Hence, when such air arrives under the opening aa, it will rise through it, 
and ascend to e; the closed valve preventing it from going farther. Thus 
successive portions of air ascend, and in time accumulate to such an extent 
as gradually to force much of the water downward out of the box. When j 
this takes place, the float, which is held up only by the water, of course de- j 
scends also; and in doing so, pulls down with it the valve v. The accumulated 
air then instantly escapes through the openings at v and n, into the atmosphere; j 
and the water in the pipe mm, immediately ascends again into the box, carry¬ 
ing with it the float; and thus again closing the valve v. The valve, and the ; 
valve-seat e, are faced with brass, to avoid rust, and consequent bad fit. The 
whole .is protected by an iron or wooden cover, reaching to the level of the street. 

Air valves are no longer used in city pipes; their place being 

supplied by the fireplugs at average distances of about 150 yards apart. These, 
being placed as much as possible at the summits of undulations in the lines of 
pipes, for convenience of washing the streets, and being frequently opened 
for that purpose, permit also the escape of accumulated air. 

The escape of compressed air tlirougii an air valve, or 
other opening', has been known to produce bursting' of the 
main pipes: for the escape is instantaneous, and permits the columns of 
water in the pipes on both sides of the valve, to rush together with great 
forces, which arrest each other, aud react against the pipes. 

Air-Vessels. Motion is imparted to the water in a line of pipes, by the 
forward stroke of the piston of a single-acting pump; but during the backward 
stroke, this motion is stopped ; and the water in the pipes comes to rest. There- ; 
fore, at the next forward stroke, all the water has to be again set in motion; 
and the force that must be exerted by tlie pump to do this is much greatei than i 
would he required if the motion previously imparted had been maintained ■ 
during the time of the backstroke. The addition of an air-vessel secures this I 
maintenance of motion, and thus effects a great saving of power; besides dimin¬ 
ishing the danger of bursting the pipes at each forward stroke. It is merely a 
tall and strong air-tight iron box, usually cylindrical, strongly bolted on top* 
of the pipes just beyond the pump, and communicating fnely with them 
through an opening iu its base. It is full of air. The forward stroke of the 
piston then forces water not only along the pipes, but also into the lower part 
of the air-vessel, through the opening in its base; thus compressing its con¬ 
tained air. But during the backstroke, this compressed air, being relieved from 
the pressure of the pump, expands; and in so doing presses upon the water in 
the pipes, and thus keeps it in motion until the next forward stroke ; and so on. 
An air-vessel also acts as an air-cushion; permittingthe piston to apply its force 
to the water in the pipes gradually: thus preserving both the pipes and tlie 
pump from violent shocks. The air in the vessel, however, becomes by degrees 
absorbed and taken away by the water; and its action as a regulator then 
ceases. To prevent this, fresh air must be forced into the vessel from time to 
time by a condenser, or forcing air-pump. A double-acting pump does not so 
much need an air-vessel. There is no particular rule for the size or capacity of 
air-vessels. In practice it appears to vary from about5 to 50 times that of the 
pump; with a height equal to two or more times the diameter. A stand-pipe 
(see below) is sometimes used instead of an air-vessel. , 

A staml-pipp is sometimes used for the same purpose as an air-vessel (see 
above). It is a tall pipe, open to the air at top; and communicating freely at t 
its foot, with the water-pipe, in the same manner as in an air-vessel. Its top t 
must he somewhat higher than that to which the pump has to force the water 
through the system of pipes; otherwise the water would be wasted bv flowing 
ovpr its top. The area of its transverse section should be at least equal to that , 
°i the pipe or pipes which conduct the water from it; but it is at times better 
to have it much larger, as a stand-pipe may then answer, especiallv in a small , 
town, as a reservoir, if the pumping should cease for a few hours. A stand-pipe - 
should be cylindrical, not conical; for if thick ice should form on top of the 
water in a conical one, a sudden forcing of it upward by the pump might strain 
the stand-pipe seriously. The stand-pipes connected' with the Philadelphia 
Water-V\ orks are from 125 to 170 feet high ; 5 feet diameter; and made of riveted 
boiler-iron about % inch thick near tlie base, and about inch near the top 
They have no protection from the weather; nor are thev braced in any manner- 
but retain their positions by their own inherent strength, although exposed at 
times to violent winds. 














WATER-PIPES. 


299 


mains, 



Fig. 45. 


The service-pipes for supplying single dwellings, 

are of lead; and of to % inch bore. They are connected with the street 
n n, lng 4o, by a brass ferrule,/, here shown at 
yg real size. The dotted lines show its % inch bore. 

The tapering ferrule is merely hard driven into a cyl¬ 
indrical hole reamed out of the main, as at s. The lead 
pipe, o, is attached to the other end of the ferrule; 
overlapping i t about 1% ins ; and the joint soldered, l. 

The extra thickness near/', is for giving proper shape 
and strength for hammering the ferrule into the main. 

The pipe and solder are shown in section. Besides the 
stopcocks attached to each service-pipe, and to its 
branches through the house, there is an underground one by which the city authori¬ 
ties cau stop off the water in case of delinquency in payment of dues ; and another 
by winch the plumber can stop it off when so required during indoor repairs. {Gal¬ 
vanized iron tubes are being much used for service-pipes, especially for hot 
water; being less subject to contraction and expansion, which produce leaks. See 
near bottom of page 218, for such water pipes. Brass service-pipes are now much 
.used in Boston. See bottom of page 218; also page 417. 

Art. 15. The so-called “corporation stops” or “ corporation cocks” 
are inserted into the pipe by a special machine, Fig 4G. Their great advantage over 
the ferrule, Fig 45, is that they can be inserted into a pipe when the 
latter is full of water under pressure. Besides, 
inasmuch as they are screwed into the pipe, they are in no danger 
of being forced out of it by any pressure within it. As their 
; jfi name implies, they are furnished with a stop-valve, which is kept 

closed while the valve is being inserted into the pipe, and is then 
opened, and generally remains open permanently. 

JPig>c-tapping' machines, for drilling and tapping the 
pipe, and for attaching these stops, are made in a variety of forms. 

Fig 46 shows one made by 
Walter S. Payne A 
Co.. Fostoria, Ohio. Each 
of these machines is furnished 
with a number of malleable 




M 


cast-iron saddles, which fit the various dianis of pipe 
with which it is to be used. The saddle is not shown 
in the fig. It is fastened to the pipe by a chain slung 
around the latter The chain is tightened by a bolt 
and nut at each end. 

The brass cylinder, C C (into which a tap-and-drill, 
ubitt anc * ^ ,e sto Pi S, have first been inserted), is then 

6'iflllBllHillf^^H Bill - screwed into the saddle by means of the thread at A. 

The stop is temporarily screwed on to a mandrel, M. 
This mandrel, and the drill-shank, K, pass through 
stuffing-boxes cast in one with the head of the cyl. 

| si By means of a handle, not shown in the fig, this head 

is now revolved (while the body of the cyl remains 
stationary) so as to bring the drill,T, and stop, S, into 

_ the respective positions shown in the fig. When the 

cyl head comes to the proper position, it is stopped by 
a lug inside of the cyl. The drill is then immediately over the center of a large 
circular opening in the base of the cyl, C C, and over a similar opening, through the 
saddle, to the surface of the pipe to be tapped. It is then pushed down until it 
touches said pipe. The ratchet-wrench, W W, is then set on the square head of the 
drill-shank, K; the feeder-yoke, Y, with feed screw, F, is put in position as shown; 
and the pipe is drilled and tapped by working the wrench; whereupon the water in 
the pipe, if under pressure, rushes out through the hole thus made, and fills the 
cylinder. 

By reversing the position of the switch on the ratchet in the wrench, W, and by 
working the latter, the tap is now withdrawn from the hole, but remains in the cyl. 
The cyl head is now revolved so as to reverse the positions of S and T ; the lug in¬ 
side of the cyl stopping the head when the stop is immediately over the hole. By 
means of the ratchet-wrench, applied to the square head of the mandrel, M, the stop, 
S (the. valve of which must be closed), is now screwed into the hole, but only far enough 
to hold securely, and thus prevent the further escape of the water from the pipe 
when the machine is now removed. The stop is now screwed firmly into place by 
means of a wrench applied to a square on the stop itself. When the pres in the pipe 
exceeds about 200 lbs per sq inch, the feeding apparatus, as used with the drill (see 
fig), may be also used in aiding the insertion of the stop. 



























300 


WATER-PIPES. 


The mandrel, M, is made in two lengths (one of which screws into the other) in 
order that the upper part may be out of the way of the wrench-handle while drill¬ 
ing. It has three or more diff threads at its foot, to suit did' sizes of stop. Stops, 
made to suit the machine, are furnished as wanted. 

The machine can work in any direction radial to the pipe, and can therefore be 
used tor tapping a pipe in any part of its circumference. 

After the stop is inserted, the service-pipe is attached to its outer end by a coup¬ 
ling nut passing over the thread there shown. 

The machines are guaranteed to tap under a pressure of 600 lbs per square in eh 
iThey are made in four sizes. No 0 taps holes from ]A to V 2 inch diatn, and weighs 15 
tbs; No 1 from to t inch, 22 tbs; No 2, from '/( to 1 Vo ins, 30 tbs; No 3, from X A to 2 
ins, 35 tbs. Card prices, 1886, No 0, $75; No 1, $100; No 2, $125; No 3, $150. These 
prices are for the machines complete, including 3 taps and drills and 3 saddles, but 
exclusive of stops. 

Other forms of pipe-tapping machines are the Boston, made by Whittier Machine 
Co., office, Granite and First Sts, Boston, Mass; the Lennox, by Lennox Foundry and 
Machine Works, Marshalltown, Iowa; Young’s by the Easton (Pa) Brass Works; 
Letzkus’, James H. Harlow, Eng’r, agent, Pittsburgh, Pa; Sperring’s, Mueller’s, and 
Iladesty’s. Any of these can be had through the larger manufacturers and dealers 
in plumbers’ supplies; as Haines, Jones & Cadbury, 1136 Ridge Ave, Phila; McCani- 
bridge & Co., 527 Cherry St, Phila; Chas Perkes, 627 Arch St, Phila. 






STOP-V ALVES. 


301 


Art. 16. Stop-valves, or gates, opening vertically in grooves, are placed 
across the street pipes at intervals of from 100 to 300 yds. Their use is to shut off 
the water from any section during repairs; the water of such sections being allowed 
to run to waste, and to soak into the ground. 

The details are much varied by diff makers. 



Figs 47 and 48 show such a gate made by Chapman Valve Mfg Co, Indian 
Orchard, Mass. The valve, v, is cast in one piece. When down, as in the figs, it 
closes the pipe. As in other styles, it opens vert by means of a screw, D, the valve 
rising into the cast-iron case or box, B B, and leaving, when all the way up, an 
opening of the full diam of the pipe. The screw is turned by a wrench fitting on 
its square head, h. The screw, D, itself, is prevented from moving vert by the col¬ 
lar, C. 

The two principal castings which compose the box or cover are bolted together by 
means of flanges, g. The joint faces of the castings are carefully smoothed; and a 
thin strip of lead is inserted between them, as a precaution against leaks. The recess, 
R. admits small particles of foreign matter which might otherwise prevent the gate 
from closing perfectly. The valve seats are faced with Babbitt metal. At the top 
of the cover, the screw stem passes through a stuffing-box. which prevents leaking 
at that point. Very careful workmanship is required throughout. 

The following ate the weights and approx prices (188(3) of these valves. 
They give a tolerable average of the wts and prices of similar gates by other first- 
class makers, among whom are Isaac S. Cassin, 2d St and Germantown Ave, Phila; 
Ludlow Valve Mfg Co. 938 River St, Troy, N Y; and Whittier Mach Co, office, Granite 
and First Sts, Boston. The latter make Coffin’s patent double-disk valve, besides 
other patterns, fire hydrants, locomotive and stationary engines, dredging machines, 
iron bridges, and heavy tools and machinery in general. 
















































































302 


STOP-VALVES. 


Chapman Bell-eml Water-gates. 


Bure. 

ins. 

Wt. 

lbs. 

List 

price.* 

$ 

i Bore. 

ins. 

Wt. 

lbs. 

List 

price.* 

$ 

Bore. 

ins. 

Wt. 

lbs. 

List 

price.* 

$ 

Bore. 

ins 

Wt. 

lbs. 

List 

price.* 

$ 

0 

32 

10 

1 fi 

195 

30>< 

12 

600 

82 

20 

1700 

230 

3 

55 

15 

I 7 

245 

36 

14 

843 

112 

24 

2750 

333 

4 

116 

19 

1 8 

290 

45 

16 

1080 

150 

30f 

6400 

700 

5 

135 

25 

1 10 

439 

62 

18 

1475 

194 

36f 

8300 

1200 


Art. 17. Fig 49 shows an arrangement witli outside screw for raising 
ami lowering the valve. Here the screw, D. does not revolve, but is attached to the 

valve, and rises and falls with it, being raised or low¬ 
ered by turning the wheel, W, at the center of which is 
a nut through which the screw passes. The nut is fixed 
in the wheel, and is so confined that it, and the wheel, 
cannot move vertically. 

Art. 18. A four-way stop, or four-way 
valve. Figs 50 and 51, is placed at the intersection of 
two mains ; the four ends of which areattached, respec¬ 
tively, to the four openings, M M M M. At the bottom 
is an additional opening, connecting, by means of an 
elbow, II. with pipes running to a fire-hydrant at the 
street curb. See Arts 20 and 21. Two or more of such 
bottom openings may be made, if desired, for the sup¬ 
ply of as many fire-hydrants. All of the openings are 
opened or closed at one time by raising or lowering 
the valve or plug, P, by means of a wrench or key ap¬ 
plied to the square head, S, of the screw stem. As in 
Figs 47 and 48, the screw turns, but is prevented from 
rising and falling, and the plug moves up and down on 
the screw. 

Inasmuch as all sediment escapes into the bottom 
opening which leads to the fire-hydrant, the valve is not 
liable to clogging through this cause. The fire-hy- 




Fijr. 40 


Fig. SO 


Fig. SI 


draut, being fed from both of the mains, obtains a fuller supply than would be possi¬ 
ble it it were fed, as usual, through only one main. 

Figs 50 and 51 represent Yiney’s four-way valve, made by Keystone Valve Co, office 
1510 Brown St Philadelphia. The plug, P, is a hollow iron casting, in the shape of a 
truncated four-sided pyramid. Each of its sloping sides is laced with brass; and the 
seats are of white metal. For prices and discounts, address as above. The weights 
are as follows: for 4 inch pipes, 170 lbs ; 0 inch, 395 lbs ; 8 inch, 495 lbs; 10 inch 700 
Itis; 12 inch, 1000 lbs. The Co. also makes three -way valves. 

* hiscount, 1886, about35 percent. 

t Furnished with gearing to aid in opening and closing the valve. The 20 inch and 24 inch cates 
can also be furnished with such geariug when desired. This increases the weight about one eichth 
and the costahoutone fourth. 61 










































































STOP-VALVES. 


303 * 


Art. 19. Whatever the sty 16of the gate may be. it is, when attached to the pipe, 
protected by a surrounding- box, generally of plank or cast-iron, with 
four sides, which taper so that the box is of smaller hor section at top than at bottom. 
It is open at bottom, but has a movable iron top, level with the street. This top is 
taken off when the valve is to be opened or closed, or inspected. Two of the oppo¬ 
site sides of the box of course have openings for the passage of the pipes to or from 
the valve. 

The gates, especially of large mains, must be closed very slowly. Otherwise, the 
too sudden arresting of the momentum of the flowing water would be apt to break 
either them or the covers; or burst the pipes. As a precaution against this, the 
covers for very large valves are cast with outside strengthening ribs. 

No self-acting air-valves (Fig 44 A) are now placed at street summits, to allow 
confined air to escape. The fire-plugs answer instead. The rad for hor bends in 
mains is if possible not less than about 12 times their diams; they are made as large 
as the widths of the streets will admit; usually about 50 ft. Fire-plujrs, Figs 
52, &c, are placed as much as possible at summits, so as to serve also for washing 
the streets; and for the escape of accumulated air. They average about 8 in num-, 
ber to each mile of pipe; or 1 to each block of buildings. 


( 


. I 


il 





304 


FIRE-HYDRANTS. 


Art. SO. Fig 52 represents a common street fire-plug, or fire-hydrant, 

& * as made by Gloucester Iron Works, 

office, No 6 N 7th St, Phila, and used 
in that city. 

The valve, v, is of layers of well- 
hammered sole-leather; and,when 
closed, shuts against a brass ring i 
seat, o, which is confined to its place 
by a lead joint. The valve is opened 
by working down the screw, s, 
which, by means of a swivel joint 
at a, can revolve without turning 
the valve-rod, y. When the valve, 
v, is closed, after the plug has been 
in use, the chamber, c, is full of 
water, which, if allowed to remain, 
would be in danger of freezing, 
and of bursting the plug. But, in 
closing the valve, we raise the 
flange, J,on the rod, and thus allow 
the water to escape through the 
opening at i, whence it runs to 
waste into the ground, through the 
open lower end of the frost- 
jacket, jj, which is a hollow 
cast-iron cylinder surrounding the 
working parts of the hydrant. 
Being free to slide vert it rises and 
falls when the level of the ground 
is disturbed by frost, and the hy¬ 
drant is thus protected against in¬ 
jury from this cause. 

The top, £,of the hydrant case, is 
cast in one piece with the cham¬ 
ber. c. 

The stopper, e, screws on over 
the nozzle, n. 

Art, 21. In the Chapman 
fire.hydrant. Figs 53, 54, and 
55, made by the Chapman Valve Mfg 
Co, Indian Orchard. Mass, the 
valve, v v, is a sliding one. 
The stem, y, Fig 53, to which the 
screw, s, is attached, is, like that in 
Figs 47 and 48, prevented from mov¬ 
ing vert by a collar, fast to it near 
its top, and confined in a circular 
groove. When the rod and screw 
are made to revolve, by means of a 
wrench applied to the square head 
of the former, the valve slides up 
or down on the screw, admitting 
the waterto,orshutting it off from, 
the hydrant. The valve slides on 
the two guides, g g, which are cast in one piece, respectively, with the two vert sides 
of the lower part of the hydrant. Its circular face, where it comes into contact 
with the hydrant case, h, is faced with a gun-metal ring, e e, which bears against a 
similar ring, made of Babbitt metal, let into the hydrant case. 

The water, left in the hydrant case after closing the valve, escapes through 
a cylindrical hole, d, about % inch diam, bored through the guide, g, and case, h. 
This hole is at such a ht as to be just above the top of the loose plate, p , when the 
valve is closed, as in Fig 53. This plate lies in avert groove in the side of the valve- 
casting, and is pressed against the side of the case, h, by two spiral springs confined 
in cavities, cc, in the valve casting. When the valve begins to rise, the hole, d, is 
closed by the plate, p, and remains so until the valve is again entirely closed. 























































FORCE IN RIGID BODIES. 


305 



-- » .»■- 

MECHANICS. FORCE IN RIGID BODIES. 


Art. 1. NIeeliaiiios fthe very foundation of civil engineering) is that«branch 
of science which treats of the effects of force upon matter. This broad application 
of the term necessarily includes hydrostatics, pneumatics, &c; but ordinarily it is 
restricted to the effects produced upon matter by the application of outward, extrane¬ 
ous, or mechanical force, from whatever source it may be derived; whether from 
steam, water, air, wind, gravity, animals, &c. Some of these effects consist in either 
deranging, or separating the particles of matter which compose a body, by pushing 
them close together: or by pulling them farther apart; and in all such cases the 
extraneous force is to he considered only as acting against the natural inherent 
forces, or strength of the materials of which the bodies consist. This class of effects 




















































































306 


FORCE IN RIGID BODIES. 


is comprised under the distinct head of St rength of Materials. Another and very 
important class refers only to the action of force upon entire bodies; either to move \ 
them ; or to keep them stationary. Force, when it moves bodies, is called Dyna¬ 
mic force; and when it keeps them stationary, Static. Hence, we have the 
sciences of Statics and Dynamics; which imply merely the effects ol lorce 
as to giving motion to bodies ; or to keeping them at rest. 

In examining the abstract static and dynamic effects of force thus applied to entire \ 
bodies, we of course have to assume that it does not break, bend, penetrate, or alter 
in any way, the shape of the body. The body is to be regarded merely as something 
that receives the force ; but is in no way affected thereby, further than that it mores, or j 
stands (or rests) when the force does so. That is, the forces are assumed to act upon 
each other only, when there are more than one; the body being only the field o*f their 
action. This assumption is of course not strictly true in any case, because no body i 
is so perfectly rigid as not to have its shape somewhat changed by the application of 1 
force ; still, it will be seen, as we proceed, that it is not liable to produce error. Thus, , 
it will be evident that the illustrations of force in the following pages would not 
apply to walls,'arches, beams, Ac, of snow, cotton, loose sand, or soft clay; and in || 
the same manner, although to a less extent, would they be influenced by the assump¬ 
tion of any yielding whatever of the particles composing bodies of metal, stone, or V 
wood. Such yielding has of course to be taken into consideration in nearly all cases in i 
practice ; but it must be done by a totally distinct process, under the head of Strength 
of Materials. The two processes do not interfere with each other. 

It is so absolutely essential in the study of Statics and Dynamics, and in reading the following articles, l 
to keep this assumption constantly in mind, that we repeat it; namely, that the force is to be con¬ 
sidered only as acting upou entire bodies; or upon bodies as a whole, incapable of being broken, 
beut, or affected by it in any manner whatever, other than in merely being moved; or kept station¬ 
ary by it. For instance, if we wish to ascertain the effects of a given force f h to overturn or upset a 
stone. S, p 337, around one of its edges, n, we suppose the stone to remain whole. But if we wish to 
know whether the edge n will be in dauger of being fractured when the whole weight of the stone 
comes upon it during the process of upsetting, we resort to the crushing strength of stone. It is 
plain, that a body when pushed, or pulled in several directions at once, may not as a whole, have 
the slightest tendency to move in one direction rather than another; yet some of its particles must 
tend to move iu one direction; and others in others. Statics and Dyuamics regard the body only 
in the first point of view. 

Rem 1. When not otherwise stated, or apparent, the force in many of the following 
examples, is supposed to be imparted to the body in a direction passing through its 
cen of grav: so as to move, pull, or press it, without tending to make it revolve, or 
upset. By examining first the action of forces imparted in that manner, certain 
elementary principles become much more easy of comprehension. 

Rein. 2. If the direction of the imparted force does not, 
pass through the cen of grav of a body free to move, the body will still 

move forward in that direction just as far as before; but while so doing, it will also revolve around 
its cen of grav to the same extent as if it did not move forward at all, and as it would around a fixed 
axis, and under the full action of the force. 

Rem 3. Lastly, when not otherwise stated, the force is supposed to he applied to 
the rigid body in a direction at right angles to its surf at the point where it is ap¬ 
plied; otherwise (except as per Art 19) only a part of the force will be imparted to 
the body ; that is, will be put into it, or enter it; and produce an effect unon it. 

Art. 2. Matter is any substance whatever, as metal, stone, wood, water, air, 
steam, gas, &c. A body is any piece of matter, as a stone, a house, a mountain, Ac. 
The weight of a body is the amount of vert pres, or pull, which the force of grav¬ 
ity in that body exerts upon it. In ordinary practical mechanics, the quantitv 
of matter in a body is measured by its wt; for a method which is the only theoret¬ 
ically correct one, but which is not adapted to popular use, see note to p. 317. Mo¬ 
tion is change of place, or of position. Since all bodies are constantly in motion 
in consequence of the revolution of the earth, the ordinary use of the word here 
refers to the motion produced by the extraneous force under consideration at the 
time. 

The hose of a body, as ordinarily understood, is that part of it which is 

underneath, and upon which it stands when acted upou by its own wt, or by the wt of other hodies also, 
which may be placed upon it. But in fact, a base may be at any part whatever of a body ; at its top, 
on one side, &c: as when we press a block of wood upward against the ceiling of a room ; or horizon¬ 
tally against a wall; the part which touches the ceiling, or wall, is the base. 

Strain is the effect produced by the action of equal forces or of equal parts of 
unequal forces, against each other; or in opposite, directions. It may consequently 
he applied to the effect which these equal parts or wholes produce among the par¬ 
ticles of the body upon which they act; as tending either to push them closer to¬ 
gether; or to pull them farther apart; in other words, to crush,or to rend the body. 
But as before stated, this view of strain belongs, not to pure Statics and Dynamics, 
hut to Strength of Materials. Again, it may refer to the action of the forces upon 
each other; or, as usually stated, upon the body as a whole. 






FORCE IN RIGID BODIES. 


307 


Thus, if two men pull or push, with equal forces, a body that is between them, their forces merely 
react against, counterbalance, or destroy each other; but they produce no effect 

whatever upou the body us a whole ; it remains at rest; and has no more tendency to move in any 
direction, than if both forces were absent. The forces then merely strain (pull or push against, 
without moving; each other, and keep each other at rest; but no practical error will arise from 
adopting the common phraseology, and saying they act upon the body to keep it at rest. 

Strain is measured by weight: as by pounds, tons, &c. Its amount or 
quantity is equal to that ot only one of the two equal opposing forces. Thus, if two 
men pull against each other at two ends of a rope, each with a force of 30 lbs, the 
strain on the rope is but 30 lbs; and it is equal throughout the length of the rope. 
Ihe two 30 lb forces strain each other - , 30 lbs ; as is made manifest if a spring-balance 
is inserted at any part of the rope. If a rope passes over a pulley, and equal wts be 
suspended at each end of it, then the two equal forces of gravity of the two wts strain 
against each other; and also strain the rope ; to an amount equal to one of them. 

bor more on Strain, see Art 13. 

Two equal opposing forces produce strain and a tendency to motion, among the 
particles which compose a body to which they are imparted; but exert no tendency 
to motion upon the body as a whole.; because the particles are held together by 
their internal cohesive force; and this cohesive force reacts against our extraneous 
imparted forces. If it is not sufficient to do so completely, the body is broken. 


Art. 3. Force is that principle of which, considered simply as a mechanical 
! agent, we know but little more than that when it is imparted , that is, put into , a body, 
I it produces either motion alone ; or strain, with or without motion. This is all that 
j force, mechanically considered, can do under any circumstances whatever. When it 
i produces motion alone, it is called moving force. When it produces strain alone, or 
i without motion, straining force ; or simply strain, pull, push, tension, or pres,as the 
case may be. As. for instance, when a body rests quietly on a table, or on a post, &c ; 
or is suspended by a chain, rope, &c; its effect is simply strain; a push or pres on 
the post or table, and a pull or tension, on the rope. There can be no strain except 
where there are at least two forces to strain against each other. When it produces 
strain and motion at the same time; or in other words, when a part of it produces 
strain; and a part of it, motion ; it is called working force; and its effect is work. 
Thus, when a man is lifting a wt, a part of his force is straining against the equal 
force of gravity of the wt ; while the other part of bis force is giving motion to the 
wt; so that all his force combined, is at work; or is working force; or performs 
work. See Art 11. 

These are not different kinds of force; but diff effects of the same force ; for there are no diff kinds of 
mechanical force; it is all the same: uo matter from what source it may be derived. Indeed, we 
might generally, and without fear of being misunderstood, call it simply motion, work, or strain, 
according to which of the three effects it is producing at the time; and we shall frequently do so 
in the following pages. The only means we have of measuring it, is by measuring the quantity of 
these its three effects. 

Nothing - but force can resist force. When all the force imparted, or 

/ put into, a body, is unresisted by opposing force; it produces motion only ; when it is all resisted, it 
produces strain only; when a part of it is unresisted, and a part resisted, the first part produces 
motion ; and the last part strain ; which two combined constitute work. A single force can produce 
motion ; but at least two are required to produce strain. See Art. 13. 

Force is sometimes defined to be that principle which either does produce, or tends to produce 
motion : or which either does prevent, or tends to prevent it. 

Rkm 1. We infer from both reason and observation, that force, when once imparted, 
or put into a body, would, if not resisted by other force, remain in it, and move it 
forever, in a straight line; in the direction in which the force was imparted; and at 
a uniform velocity, or rate of speed. 

For we observe in practice, that motion continues longer in proportion as we remove resisting 
forces; hence we infer, that if all resistances could be removed, it would continue forever. Butin 
practice we cannot remove all resisting forces, such as those of gravity, wt, friction, the air, &e. 
These strain against, or resist the forces which we may impart; so that with the exception of a 
body falling by the force of gravity, in a vacuum, it is difficult to mention a case of motion alone, 
without strain. 

Rem 2. Matter, in itself, cannot resist force. See Arts 5 and 11. When 
force produces motion alone, without strain, it is often called Dynamic force; when 






308 


FORCE IN RIGID BODIES. 


continuous strain without motion. Static: when a sudden strain for an instant only, 
as a blow of a hammer, Impulsive; and its action, an impulse. See Art 10, p 310. 

Rem 3. The L«i viiiST Force, or via viva of scientific writers, is nothing more ; 
than an expression referring to the quantity of work (motion and strain combined) 
which the force in a body at .any given instant, could perform, if left to itself, with- i 
out afterward receiving any additional force. See Art. 24. 

Art. 4. When force is imparted in any direction which passes through the cen !' 
of grav of a homogeneous rigid body, perfectly free to move , it quickly diffuses itself 
equally through every part; and gives to every atom composing the body, the same ! 
tendency to move onward in the same direction. 

But although this diffusion takes place rapidly, especially in compact bodies, still 
it requires some time; generally so short as not to be appreciable without close ob- jl 
servation. If we put the body B, Fig 1, into motion by striking it a blow near the top 
or bottom, the struck end will for an iustaut move faster than the other; after which i 
both will move with equal velocity ; because the great force which for an infinitely | 
short time acted upon the top only, becomes equally diffused throughout the body. 
We are not, however, now treating of cases of this kind, in which the direction of the 
force does not pass through the cen of grav of the body; for the body may then ro¬ 
tate, or whirl arouud as it moves forward. 

This equal diffusion of force is owing to the inherent cohesive force of bodies,which 1 
holds their atoms together, somewhat as lime holds together the grains of saud in a 
piece of mortar; and will not permit one grain to move unless the others move with ! 
it. In producing this diffusion, the cohesive forces, or the lime, do not diminish the I 
quantity of the imparted force; but merely cause that instead of continuing to give a great velocity 
to that part only at which it was imparted, it shall give a slower, but equal one, to every part. 





But if the rigid body to which we impart force in any direction, is not free to move, being prevented 
from so doing by opposing force, then our imparted force strains against the opposing oue ; and thus 
produces strain, instead of motion, in the body. 

Art. .1. We must clearly (listinguisk between merely mov- 
in}? a body, and lifting}' it. The smallest imaginable force would, if unre¬ 
sisted by other force, move the greatest imaginable body; but if we apply force to ] 
lift a body, then its wt, or force of gravity, resists. Force applied merely to move | 
things without lifting them, is often called traction: as in the case of horses or men I 
towing a canal boat, or hauling a vehicle along level ground. The same word, how¬ 
ever, is generally used for hauling, even when the thing is partially lifted ; as when 
a horse is hauling up hill; in which case he is lifting both himself and the vehicle 
with its load. See Traction. 

Matter always appears to resist force, but this deception is caused by its gravity, or by friction, or 
some other resisting force acting upon it at the same instant. 

A man cannot lift a wt of *20 tons ; but if it be placed hor upon proper friction rollers, he can move 
it, as we see in some drawbridges, turntables, &c; and if friction aud the air could be entirely re¬ 
moved, he could move it by a single breath ; aud it would continue to move forever. It would, how¬ 
ever. move very slowly, because the force of the single breath would have to diffuse itseir among 20 
tons of matter. He can move it, if it be placed in a suitable vessel in water, or if suspended from a 
long rope. A powerful locomotive that may move 2000 tons, cannot lift 10 tons vertically. 

If we imagine two bodies, eaoh as large and heavy as the earth, to be precisely balanced in a pair 
of scales without friction, it is plain that since all of their two opposing gravities strain against each 
other and thus neutralize each other's motions, a single grain of sand added to either scale-pan, 
would give motion to both bodies. 


The inherent cohesive force of matter, by which its particles are held in close union, frequently 
.lusts le mattei itself to appear to resist straining force; thus, a cake of ice may sustain a great 
? estro V ts , cohe - siv e force by converting it into water, it will yield readily. So with 
< netal and stones if reduced to dust. It is not the material that resists being broken; but the 
inherent cohesive force which holds the particles of the material firmly together. 


. Tlie fact that matter in itself opposes no resistance to force, 
is calico its inertia, or inertness; but the term “ inertia” is often used to 
denote a supposed resistance exerted by matter itself against force. We see no reason 
lor assuming the existence ot such a resisting property in matter, and we prefer to 
apply to inertia its literal definition of inertness or the absence of all tendency 
either to move or to resist moving force. 

In most kinds of machinery, nearly all the parts are supported by either pivots 














FORCE IN RIGID BODIES, 


309 


or gudgeons, which resist the action of gravity of the parts: so that the working 
force, or power, lias only to overcome the friction ; and not to lift the weight of the 
parts. The expression, “ overcoming- the inertia of a hotly.” is founded 
upon radical error; for there is no such resistance as inertia to be overcome. Inertia 
is the name of a fact; not of a force ; and that fact is simply, that matter does not 
resist force. What is called overcoming inertia, is simply putting in force ; as when 
we pour water into a glass, we do not overcome any resistance of the glass against 
being filled ; we simply apply the water to the glass, which receives it, and retains 
it; until, by applying force, we take it out again. “Overcoming the inertia” of a 
railroad train, &c, therefore, means nothing more than that if we wish the train to 
move, we must put moving force into it. It is inert, and cannot move of itself. 


Art. 6. Care must be taken not to confound motion with vel. ‘Velocity is 
the rate of speed of motion; or, in other words, it is the dist moved over in 
a given time , without any regard to the wt of the body. Thus we say that it moves 
w ith a vel, or at the rate, of 10 ft per sec; 20 miles per hour, &c; w-hether it weighs 
an ounce or a ton. 


Art. 7. We impart force to a body, by applying to it (that is, by 
! bringing into close contact with it) a second body in which the force already is. We 
must discriminate between applied, and imparted force; and the precaution is the 
j more necessary because writers generally appear not to recognize the distinction. 


Applied force is merely carried to a body ; imparted force is put into it. A heavy body containing 
i great force may be applied to a small body containing none; and yet the former may impart but 
j little of its force to the latter. If we place a body on, that is. apply it to, a steep inclined plane, we 
apply also the wt of the body, or its force of gravity : yet only a portion of this force is imparted, or 
put into the plane; while another portion remains in the body ; and being unresisted by the plane, 
produces the motion of sliding down it. Force may be applied to a body in any direction whatever; 
but can be imparted to it only at right angles to the surface, except as explained Art 19. 

That point of a body to which force is applied, is called the point of application; but in fact we 
cannot in practice apply force to a point, according to the scientific meaning of that word ; but have 
| to apply it distributed over an appreciable area (sometimes very large) of the surface of the body; 
{ still, as explained in Art 57, it may, as regards its action upon the entire mass of a rigid body, 

| be considered as applied at one point. The expression, direction of a force, refers to the direction 
iu which it is applied. 


Art. 8. When diff forces act upon a body, it is absolutely essential, in consid¬ 
ering their effects upon it, to know' whether they all act ill the same plane; 
for if they do not, their effects become totally diff. 


A flat piece of paper is a plane, and if on it we draw any 
number of straight lines, in any direction whatever, they will 
represent so man}' forces all acting in that same plane; that 
is, the same flat surface coincides with the directions of all of 
them. It will evidently do the same in whatever position this 
plane surf may be placed, whether hor, vert, or inclined. 

Straight lines drawn on the floor of a room, will represent 
forces in that same plane; lines on the ceiling, forces in that 
same plane; which of course is not the same plane of those 
on the floor; so with lines on the sides of the room. All the 
lines of. it, at. ct. Fig 2, are in the same plane t oic. Although 
•; t is in the plane toic: and t e at the same time in the plane 
tcge. aud in the plane tone; and * e in the plane sneg; and 
t s in the plane i t e s, still all of these, namely, it, te. s e, and _ 

t s, are eviden tl v in the same plane i t e s. Any two lines which meet, or would meet or intersect each 
other if sufflcientlv extended in either direction, are in the same plane; as of, if; or s f, if; ot ts, 
es Still two line's may be in the. same plane, and yet not meet if extended; as, for instance, the 
parallel lines c f. g e. in the plane tcge. The lines at and g e, being in parallel planes, o i c t and 
n s e g, cannot meet if extended. 

Rem. We must not confound acting in, with acting on, upon, or against the same plane. The 
floor of a room is a plane; and upon or against that same plane, forces in a thousand diff planes may 
act. The distinction is so self-evident, that a bare allusion to it will prevent mistake. See Fig 53}$, 
Art 55. 










310 


FORCE IN RIGID BODIES. 


Art. 9. The quantity, or amount, of any motion, of any body, is 
measd by mult tlxe wt of the body by the dist through which it is moved ; thus, il a 
body of 4 ftis is moved 3 ft; or one of 6 lbs, 2 it; or one of 12U It s, of a it; there is 
in each case an entire motion of 12 foot-pounds, without reference to the length of 
the time during which the motion takes place. Most authors apply the teim “ quan¬ 
tity of motion ” to momentum (see below); but we see no necessity for thus employing 
two terms to express one idea ; and we therefore use the above definition of “ quan¬ 
tity of motion ” as being at once mord'rational and more convenient. 

Momentum, scientifically speaking, is the product of the mass of a body (see j 
foot note p 317) multiplied by its velocity. But for practical purposes the term is ap¬ 
plied to the product of the weight and the velocity; the weight being expressed in 
pounds, and the velocity in feet per second; and we so employ it. Thus, if three 
bodies weigh respectively 4, 6 and 12 pounds, and move with velocities of (respect¬ 
ively) 3, 2 and 1 foot per second, the momentum of each is 12 foot-pounds per second. 
Momentum, as thus defined, is of course “ g ” times (or about 32.2 times) the true 
scientific momentum, and is plainly the same as the “ quantity of motion ” per 
second. 

Forces are estimated or measured by. or are in proportion to, the j 

momentums which they can respectively produce in a given time. Thus a 1 pound I 
weight and a 2 pound weight, after falling for one second from a state of rest, have 
each a velocity of say 32.2 feet per second. Their momentums at the end of the 
second are therefore respectively 32.2 and 04.4 foot-pounds per second, or in propor¬ 
tion to the gravity forces (1 pound and 2 pounds respectively) acting upon the two 
weights. 

It follows from the above that forces are proportional to the velocities which they 
can in a given time impart to a given body. 

In thus studying the relation between force and velocity, it is important to bear in 
mind the difference between the total force acting upon a body in a given direction, 
and the unresisted or net or resultant force acting in that direction. Thus, suppose, 
for convenience, that the resistance of a certain railroad train remains =■ 1 ton at all 
speeds (see Art. 20, p 374 e); that is, that a pull of 1 ton on the draw-bar would in 
all cases balance the resistance and maintain the speed uniform. Suppose the train ; 
to be pulled first by an engine exerting a constant pull of 2 tons on the draw-bar ; 
and then by one pulling 3 tons. Strictly speaking these engines impart to the train, 
in a given time, velocities which are to each other as 2 to 3, or as the respective pull¬ 
ing forces of the two engines ; but in both cases the supposed uniform resistance of 1 
ton would, in fhe same time, cause a retardation (or cowwfer-acceleration) of 1. Hence 
the net or resultant velocities (i e, the observed velocities) would be = 2 — 1 and 
3 — 1, or = 1 and 2; or in proportion to the net or resultant forces. 

Art. lO. An impnet, blow, fetroke or collision takes place when a moving 
body encounters another. If a body of 100 pounds has a velocity of 3 feet per second ; 
and another body of 5 pounds has a velocity of 60 feet per second, the momentum of 
each is 300 foot-pounds per second ; and if they meet each other when moving in op¬ 
posite directions, they will bring each other suddenly to rest. 


“ In some careful experiments made at Portsmouth dock-yard, England, a man of 
medium strength, and striking with a maul weighing 18 lbs, the handle of which was 
44 ins long, barely started a bolt about % of an inch at each blow; and it required a 
quiet pres of 107 tons to press the bolt down the same quantity: but a small addi- 
tional weight pressed it completely home.” 


Working force. When the force imparted to a body, not only 

moves it, but does so in spite of resisting forces; or in other words, when one part 
of it strains against and balances the resistances equal to itself; wdiile the other 
part of it moves the body, it is called working force. The quantity of working force 
expended, as well as the quantity of work thereby accomplished, is estimated by 
mult the resistance overcome, (which frequently consists of wt lifted.) bv the dist 
through which it is overcome. Thus, if the resistance be a friction of 4 lbs, over¬ 
come at every point along a dist of 3 ft; or if it be a wt of 4 lbs, lifted 3 feet high 
,,!' en 7 ( ; 1 i/ ,er f be working force, or the work done by it, amounts to 4 X 3 = 12 ft-lbs! 

He lifting ot wts; and the friction encountered in merely moi'ivg them by sliding 
oi lolling, constitute the principal sources of resistance, and work, in practice. The 
quantity of any work may evidently be considered by itself, without regard to the 
J upu' w 1 u* perfo . rm '*5 but (as in motion) we generally require to know the rate at 
h h woik can be done; that is, how much can be done within a certain time. 










FORCE IN RIGID BODIES. 


311 


Thus, in selecting a steam-engine, it is important to know how much it can eloper 
minute, hour, or day. ' We therefore stipulate that it shall be of so many horse¬ 
powers ; which means nothing more than it shall be capable of overcoming resisting 
forces, at the rate of so many times 33000 ft-lbs per min. See Arts 22 to 25. 

Working force is often called power; horse, steam, water power, &c. Since work, in a strictly 
scientific sense, involves motion and strain combined, a man who is standing still is not considered to 
be working, any more than a post or a rope sustaining a heavy load ; although he may be supporting 
an oppressive burden; or holding a car-brake with all his strength; for iu both cases his force is 
strain only. The work done by a horse drawing a heavy load on a level road, consists entirely in 
overcoming the friction at the axles and rims of the wheels ; but iu drawing it vp hill, he partly lifts 
the load , and /timsef/also. In the first case his w ork (scientifically) is not so much wt of load moved 
a certain dist; but so many pounds of rolling and axle-friction overcome through said dist. Up hill, 
his work consists of friction, overcome through one dist; namely, the length of the hill; and of 
the wt of the load, vehicle, and hitnself, lifted to another dist; namely, the vert height of the hill. On 
a perfect level road, and without friction, the horse would encounter no resistance in hauling his 
load, however great it might be. He would have nothing to do but impart motion to uuresisting 
matter. Therefore, in such a case his hauling would not be work, in the scientific sense. 

Tile ordinary unit of worli. or standard for measuring it, is one !b of wt 
lifted one foot high; or one ft> of any kind of resistance, overcome in any direction 
whatever, for the dist of one foot; and is called & font-pound. Or, when more conve¬ 
nient, we may use foot-tons, Ac, &c: as we use a two-ft rule, a yardstick, or a tape- 
line, as may best suit our purpose. Tl»e unit of KATE of worlt, or the quan¬ 
tity done in a given time, is one ft-lb per sec. 

The same quantity of force which will overcome a given resistance through a given dist, in a given 
time, will also overcome any other resistance through any other dist, in that same time, provided 
the resistance and dist when mult together give the same amount as in the first case. Thus, the 
force that will lift 50 lbs through 10 ft in a sec, will lift 500 lbs, 1 ft; or 25 lbs, 20 ft; or 5000 lbs 
yfy of a ft in a sec; and in all these cases the amount of force expended, as well as of work done, is 
precisely the same. In practice, the adjustment of the speed to suit diff amounts of resistance, is 
usually effected by the medium of cog-wheels, belts, or levers. By means of these, the engine, 
water-wheel, horse, or other motive power, may be made to overcome small resistances rapidly ; or 
great ones slowly, by the same working force. 


Art. 12. When vel undergoes no change, it is said to be uniform ; so with 
force, motion, strain, and work. When any of them becomes gradually greater, it is 
said to be accelerated; when gradually less, retarded. If the acceleration, or retarda¬ 
tion is in exact proportion to the time; that is, when during any and every equal 
interval of time, the same degree of change takes place, it is uniformly accelerated, 
or retarded. When otherwise, the words variable, and variably are used. 

Gravity is a uniformly accelerating force when it acts upon a body falling freely ; for it then increases 
the vel at the uniform rate of .322 of a foot during every hundredth part of a sec ; or 32.2 ft in every sec. 
Also when it acts upon a body moving down an inclined plane; although in this case the increase 
is not so rapid, because it is caused bv only a part of the gravity ; while another part presses the 
body to the plane; and a third part overcomes the friction. It is a uniformly retarding force, upon 
a body thrown vert upward; for no matter what may be the vel of the body when projected upward, 
it will be diminished .322 of a foot in each hundredth part of a sec during its rise ; or 32.2 ft during 
each entire sec. At least, such would be the case were it not for the varying resistance of the air 
at diff vels. It is a uniformly straining force when it causes a body at rest, to press upon another 
body ; or to pull upon a string by which it is suspended. The foregoing expressions, like those of 
momentum, strain, push, pull, lift, work, &c, do not indicate diff kinds of force; but merely diff kinds 
of effects produced by the one grand principle, force. 

The above 32.2 ft per sec is called the acceleration of g ravity; and by 
scientific writers is conventionally denoted by a small g: or, more correctly speak¬ 
ing, since the acceleration is not precisely the same at all parts of the earth, g de¬ 
notes the acceleration per sec, whatever it may be, at any particular place. See 
note to Art 25. 

Art. 13. Reaction. Strain. Strain, as before said, in Art 2, is either a 
pull or a push. The term may’be applied equally to the act. or to its effects; that is, 
it may be said to be either the action of opposing forces against each other; or the 
effects which that action produces upon the particles of the body in which they act. 
A single force cannot produce strain : for since nothing but force can oppose, resist, 
pull, or push, against force, there must he at least two, to strain by pulling or push¬ 
ing against each other. This mutual opposition or straining is called also the reac¬ 
tion of the forces. It can take place only between equal forces, or equal portions of 
unequal ones. When the forces are equal, and meet from diametrically opposite direc¬ 
tions; that is, in the same straight line, but in opposite directions along it, they 
become entirely converted into reaction, or strain. If they are unequal; or are not 




312 


FORCE IN RIGID BODIES 


in the same straight line; but meet obliquely ; then only equal portions of them will 
react against each other; while the remainder will continue as motion ; unless some 
third force is present to prevent it; as when friction is generated. The reaction of 
the equal wholes or parts consists in their mutually resisting, opposing, arresting, 
balancing, equilibrating, straining, pulling, pushing, counteracting, or destroying 
each other. 

All these words are equally applicable. As a matter of convenience only, we often say that the 
bodies themselves react. If a cauuou-ball in its flight cuts a leaf troui a tree, we say that the leaf \ 
has reacted against the hall with precisely the same degree of force that the ball acted against the 
leaf. That degree of force was sufficient to cut off a leaf, but not to arrest the ball; for, after a small 
portion of the moving force of the ball had been converted into straining force to react against the 
resistance of the leaf, the remainder was sufficient to carry it onward in its course. It, has, how- a 
ever, lost precisely as much force as that w r hich the leaf opposed to it. A ship of war, iu running 
against a canoe, receives as violent a blow as it gives; but the same blow that will upset or sink a 
canoe, will not appreciably affect the motion of a ship. The fist of a pugilist striking his opponent j 
in the face receives as severe a blow as it gives; but the blow which may seriously damage a nose, 
mouth, or eyes, may have no such effect upon hard knuckles. The pain received iu the one case, aud ! 
not in the other, is of course no measure for force. 

Rem. We have just said that strain is the mutual expenditure of two equal amounts of force. This 
mat readily be conceived in cases where two bodies come into sudden contact, and arrest each other's I 
progress bv a mutual blow; for then their forces no longer produce either motion, strain, or work. 
But iu continuous strains (pulls, or pushes) the principle is not so evident, at first. For instance, f 
when a wt rests upon a table, and continues to strain it day after day, it may be asked where is the j 
loss of gravity force in the weight; inasmuch as it weighs as much after pressing for a long time, I 
as it did at the beginning ; or where is the loss of inherent cohesive force in the material of the 
table, which is as strong as at first? 

The reply is that gravity, and the natural strengths, or inherent forces, of matter, are being inces¬ 
santly maintained or renewed, by unceasing streams, as it were, of those forces. The gravity of a 
wt w hich presses a table, or pulls on a rope, at one moment, is not identically the same that pressed 
or pulled it the moment before; and so with the cohesive force of the material of the table. If it ! 
were not so, a post or a rope which would be broken by a single force of 10 tons, would also be broken 
by sustaining one ton 10 consecutive times ; for the one ton would each time counteract or overcome 
one ton of the inherent resisting force of the post or rope, and in ten applications would overcome it 
all. We must therefore conclude that these natural forces are being unceasingly supplied from that 
inexhaustible source of power “ by which all things are upheld.” When we lean forward against a 
strong wind, we are continuously exerting new force agaiust the continuous stream of force which 
assails us; and in the same way does a post or a chain sustain its load. Continuous strains produced 
by the force of water, steam, animals, &e, we well know can only be maintained by a continuous sup¬ 
ply of said force ; to be as continuously counteracted or overcome by whatever opposing force of grind¬ 
ing, pumping, &c, it is directed against. The strain of an impulse, blow, or stroke, lasts only for an 
instant, because new force is not supplied to make it continuous. Two equal forces, straiuingagainst I 
each other, do not even keep a body at rest; but the body rests merely because the two forces balance 
each other, and therefore cannot prevent it from resting. As a matter of convenience only, we may, 
however, say they keep it at rest. 

Art. 14. While a horse is hauling a boat on a canal, it is not the boat and its 
load which react against his force ; because matter cannot react against force; it is 
the force of friction of the boat against the water; and the resistance of the wa¬ 
ter in front of the boat, produced by its cohesive force. So with an engine and 
train on a level railroad; the only resistance to the steam force of the engine, is the 
force of friction at the axles and tires of the wheels, and the pres force of the air in 
front. But on an up grade, the engine has also to partly lift the train; or, in other 
words, to react against its forces of friction and gravity combined. 

Neither the horse nor the engine exerts any more force upon the opposing forces, than these last ' 
exert upon them; but both the horse and the' engine possess a surplus of power beyond what is ne¬ 
cessary to strain against or destroy the forces opposed to them: and this surplus, as moving force, 
enables them in addition to move forward, as in the case of the cannon-ball just alluded to; and since 
the unresisting matter of the boat, load, and train, is attached to them by the tow-rope and coupling- 
links, they of course must follow. 

The resistance which an abutment opposes to the pres of an arch ; or a retaining-wall to the pres 
of the earth behind it. is no greater than those pres themselves ; but the abut and wall are, for the 
sake of safety, made capable of sustaining much greater pressures, in case accidental circumstances 
should produce such. 

Art. 15. The mere fact that a hotly is subjected to great 
Strains from equal forces reacting upon it in opposite directions, does not of it* 
self render the body more difficult to move than if it were free from strains; but 
in the cases which usually present themselves in practice the straining forces gener¬ 
ate friction, which does oppose motion. 

Thus let B, Fig 3, be a block resting on a hor support, and acted upon by a downward force d of say 
100 tons, produced say by an immense block of grauite resting upon B. Now it is plain that this 100 
tons dowuward force will be met and balauced by a 100 tons upward force w. being the resistance of 
the hor support. Hence these two equal reacting forces produce in the body B a strain of 100 tons; 
but evidently do uot impart to it as a whole any tendency to move in any direction whatever ; nor 
do they tend to prevent it from being moved in auy direction. The body therefore remains as be¬ 
fore a mere inert mass incapable of resisting the slightest moving force. 

Now suppose no friction to exist at either the base or the top of B. 

Then the slightest hor force A, a mere breath, would slide B along the hor support, moving it from 







FORCE IN RIGID BODIES. 


313 


under the 100 ton block on Its top. No matter how heavy B might be, the same 
smallest force would slide it, the only difference being that the heavier it was the 
less would be its velocity. 

l he heaviest bodies resting upon the surface of the earth, as well as ourselves, 
would be swept along by the slightest breeze if it were not for friction. 

If the screw of a vise be worked until it produces a great strain in the jaws of 
the vise, the vise is not thereby rendered more difficult to move. 


FigS'k 


Again, if a strain of thousands of tons were produced by the jaws of a vise, in a body weighing an 
ounce, this immense strain would uot prevent, nor even tend in the smallest degree to prevent, tha 
ounce body from falling down from the jaws of the vise. It is prevented by the third force, friction, 
which compels the one ounce of gravity force to become vert strain, instead of motion. The two forces 
of thousands of tons each, which produce the strain of the vise, are thereby entirely destroyed, as 
regards their action upon the body as a whole. Hence they could not prevent the one ounce from pro¬ 
ducing motion in it; nor could they affect it as a whole in any way ; for all their action is actually 
against each other. It is on this principle alone that strains'do not interfere with motions. 

If a body H, Fig 4, of 10 tons weight, is suspended from a long rope, its reaction 
against the equal opposing force at the other end of the rope, produces a continuous 
strain among all the particles which compose the rope; but which does not in the 
least affect the rope considered as a whole; inasmuch as it does not tend to move it 
in any direction. Now, in this case, there is no friction to 
be overcome; and we know from daily experience that it is 
therefore easy to move the unresisting body a little dist, by 
applying a very small hor force/. We cannot move it far, 
as, for instance, to m, because we then have not only to move, 
but to Lift it, or overcome its gravity force, through the vert 
height v c. In doing this, it is true our force does not have 
to sustain the entire wt of the body; because most of it is 
sustained by the rope. Still, if we move it at all , we have to 
overcome some of its weight; otherwise, a mere breath would 
move it, although very slowly. If we attempt to move it by 
an upward force u, we shall have still more of its wt to re¬ 
sist us; and if by a downward one d , we shall be resisted by 
the cohesive force of the rope. Therefore, in this case, we can move it more readily 
by the hor force f. 



Art. 16. If two unequal forces, which for illustration we will call 10 and 12, 
are imparted, either as pulls or as pushes, in precisely opposite directions, to a rigid 
body on which no other force is acting, then the small force 10, and 10 parts of the 
large one, will destroy each other as strain ; after which, of course, they can produce 
no effect of any kind; but the remaining two parts of the large one, meeting with 
no opposing force to react against, will continue onward as motion, in the same di¬ 
rection as before; taking the unresisting body along with them. In such a case, 
the large force is said to overcome the small one ; and such an expression is very con¬ 
venient, in reference to the entire original forces. But in a strictly scientific sense, 
one force cannot overcome another. 


Thus, in the foregoing case, so far as the strain is concerned, the large force must be considered as 
separated into 10 straiuing, and 2 moving parts. Neither of the two 10 forces which strained against 
each other overcame, or gained any advantage whatever over the other; for the two reacted equally 
on each other, and mutually arrested each other. And this is plainly the most that one foie,: oau do 
to another. 


Since two equal opposing forces, or equal portions of unequal ones, thus bring each other to a 
stand-still , or equilibrate each other, they are called Static; from the Latin "Sto, I stand; ' aud that 
branch of the science of force which treats only of cases in which all the applied forces keep each 
other at rest, is called “Statics," or “Equilibrium.” 

Art. 17. When motion has once been given to a body, it can only be taken out 
again by the reaction of some opposing force. The same identical portion of any 
force cannot produce both motion and strain at the same time. When continuous 
force is applied, as in mills, Ac, to do both, (or, in other words, to work,) it must be 
considered to divide itself into two parts for those separate purposes. If, while at 
work, the resistances to be overcome become less, the strain also becomes less, and 
the motion becomes greater; and vice versa. Motion is diminished by converting 
part of it into strain; and strain, by converting it into motion. 


21 











314 


FORCE IN RIGID BODIES, 


Art. 18. If force f be imparted to any rigid body, as N, Fig 5, at any point c; 

and if fo represent the direction in which 
it was imparted, whether as a pull or a 
push, then the force would produce the 
same effect upon the body considered as 
an entire mass, as if it had been im¬ 
parted as either a pull, or a push, in the 
same direction, at any other point of the 
body in said line; as at i, t, s, o, &c. 

Under Composition and Resolution of 
Forces, it will be seen to be sometimes 
necessary to consider a push f c, to be 
changed to a pull oh; and vice versa; 
when we wish to ascertain the joint 
effect or resultant of a pull and push imparted to a body at the same time. See 
Remark 1, Art 29. 



The foregoing important principle holds good, no matter how many diff forces may be acting upon 
the body at the same time, in diff directions ; or how much the direction of their joint effect, or re¬ 
sultant. may differ from that of any one of them : the action of each force, considered separately, 
may be regarded as just stated. The tendencies of several forces, acting at the same moment, may 
therefore frequently he first investigated one by one; and these teudeucies then combined into one; 
or the forces themselves may first be combined' iuto one or more resultauts, as directed under Com¬ 
position and Resolution of Forces, aud the effect of these resultauts considered. The engineer has 
generally to divide all the forces acting upon his structures iuto two classes: namely, those whose 
teudency is to secure the stability he requires : and those which tend to impair that stability. He , 
therefore first finds the resultant, or joint effort of each class separately; and then compares these , 
two resultants with each other. The mode of doing this will be shown further on. See Arts 35, 72, &c. , 

It is plain that if. instead of regarding the body as rigid, we considered it as elastic, or as breaka¬ 
ble, an entirely diff course would be necessary, as the question would then become one on the strength ; 
of materials .- for the force /. applied at c or t, as a push, might break off the pieces c and t ; and so , 
with the same force as a pull at s or o. Although masonry, iron, timber, and other building materials | 
are by no means absolutely rigid, yet- generally they may be assumed to be so when we are investi- , 
gating the effect of force to overthrow, or derange the structure as a whole. 

Art. 19. The fnll amount of a given force cannot (theoreti- \ 
cally) be put into a resisting body li. Fig: 6. except when ; 
applied in a direction at right angles to the surf of the body 
at the point of application. If it be applied in a direction at all oblique 
to the surf, then only a portion of it will (according to theory) enter the body, and 
produce any effect on it. The remainder will continue to produce motion in that 1 
other body in which the applied force was carried to the body B; unless some third 1 
force be present to receive and react at right angles to that remainder also. Thus, ^ 
the forces mo and ne, being at right angles to the surf fit the points of application 
o and e, will all enter B, if the resistance of B is as great or greater than they ; if not, 

they will move B. But the force Fg is not at , 
right angles to the surf at its point of applica- 1 J 
tion g ; therefore, only a portion of it will (by i s 
the theory) be imparted to the body B, however | 
great may be the resistance of B. 

Under the head Composition and Resolution of Forces, 
it will be seen that when a force is thus applied obliquely [ 
to a surf, its action at the point g is precisely the same as J 
that of two entirely separate forces ; one of which, v g, is ' 
at right angles to the surf at g; and the other, s g, par- I 
allel to the same surf. Only vg will enter the body H- , 
while sg will remain in the body F g, which carried the j * 
entire force to B ; and (if F g also is rigid) will (by theory) ! ' 
cause it to move in the direction yt , unless some third C 
force be opposed at right angles to it also. The quantity « 
of each of the forces v g and s g, is very readily found thus : On the line F g measure off by any con- a 
venient scale a dist. gi, to represent the amount in B>s. tons, Ac, of the force F g ; then on gi as a K 
diag draw a parallelogram i v g s, having two of its sides perp to the surf at g. and the other two 
parallel to it. Then i s, or vg, rneasd by the same scale as g i, will give the force imparted to the body v 
B; and i t or sg will give that which remains in the body F g. and will move it from g toward t. a 
unless prevented by some other force. If F g is not thus prevented from moving toward f, it is plain t 
that the force vg, which is transferred to the body B, cannot remain stationary at g; but must move 
along with F g, and thus press upon the surf at every point between g and t. 



This will appear in a clearer light on referring to a body placed upon an inclined 
plane so steep that the body will slide down. We know that a part of the force of . 
gravity, or wt of the body, presses, as strain, upon the plane, and at right angles to : 
it; while a part remains in the body, as motion; and causes it to slide down the 













FORCE IN RIGID BODIES. 


315 


plane; see Art 60.. As it slides, it is evident that the pressure part of the force also 

™id V over° n Tl^ lth ^ l 811 ? i ^h arted at every point of the plane along thedist 
s id over The case is identically the same with the pres force v g in Fig 6 - this will 
slide up the plane from g toward t ; being carried along by the sliding force sg 

t u e ? ry <loeS ,,ot 1,0,41 S°° 41 because it ig- 
catioi. 1 01 *, nctlon ’ vvluch Pressures always generate at points of appli¬ 

cation. Ill is friction always acts in the direction of the pressed surfaces, and con¬ 
sequently always opposes more or less resistance to that component, sg, which tends 

“ l0 " B the “ u,face a " d »° much of ,g as ta thus SteTta 
thereby changed from motion to strain, and enters the body B. If it were not for 
ti c mn a body, Fig 64, would slide down an inclined plane wx, no matter how 

slight its inclination might be; but we know that friction often prevents such slid¬ 
ing even when the plane forms a considerable angle, yxw, with a hor line yx. 
When tins angle becomes so great that the body is just on the point of starting to 
slide down it, it is called the angle of friction ; and in Fig 6, if the force Fo 
does not form with the perp vg an angle vg F, greater than this angle of friction, 
^. lH o.PP 08 e all the sliding force s g, no matter how great it may be ; 
!°udy A t rt 6 (& ntire l01Ce Wl11 tll6Q enter B at g ' and in its original direction F g. 

. This remark is particularly applicable to the case of the masonry 
joints in the abuts ot stone arches ; especially those of large span, with 
small rise. The pres which such an arch exerts upou its abuts is very 
great; and its liue of direction changes at each joint, as at 
r.'S '• It therefore becomes necessary first to find the position 
o this line, (see Art 72.) so as to know how to draw the varying incli- 
nations of the joints nearly at right angles to it; otherwise," the upper 
courses, if hor, may slide outward upon the lower ones, as shown by 
the arrow. In small arches of considerable rise, the sliding portion 
. this force may be safely resisted by good mortar or cement, if suffi- 
»ii. en j^ a H° w ed for it to harden properly ; but in large ones 
the direction of the joints must be relied on, unless we increase the 
expense by making the abuts unduly thick. 



in' 


Fid 7 

O 




The angle of friction of masonry on masonry (see table, p 373) is about 32°. Therefore if at any 
on ^n S a t fr y ’ aS m Vj. e Fig 7 ’ the resultant that cuts said joint at the line of pressures, 

to sHde d at that joint 1- “ 32 fr ° m a perp t0 said there will be no unresisted tendency 


Fig 8 is added merely to illustrate more 
strikingly the necessity for clearly distin¬ 
guishing between applied, and imparted 
forces. Here the great force a o is applied 
to the body B B at the point o ; but all of 
it that is theoretically put into, or enters 
the body; or produces any kind of effect 
upon it, is the very small amount represented by co, at right angles to the surf of 
B B at o, but we have seen that in practice friction makes it more. 

All this will be better understood after studying Comp and Res of Forces, Art 28. 


; -—^o 

B Fief. 8 B 


Art. 20. It rarely happens in practice that we impart very 
great force, or velocity, to a heavy body instantaneously; 

or by a single effort. In the machinery of mills, in railroad trains, in steam¬ 
ships, &c, it is done very gradually; and indeed, few water or steam powers 
possess sufficient force to do it otherwise. The principle explained in the 
following example, applies to the others. Thus, no locomotive, or marine en¬ 
gine. has sufficient power to start a heavy train, or a steamship, at once at a rapid 
speed. The first stroke of the piston imparts, or puts into the train, a quantity of 
force sufficient not only to react or strain against, balance, or destroy (and be de¬ 
stroyed by) all the resisting forces of friction, the air, grade, curvature, &c, which 
present themselves along the short portion of the road passed over during the first 
stroke; hut a small excess besides, which, being unresisted, is not destroyed; but 
continues in the train, as motion ; giving a slow movement to the unresisting matter 
of which it is composed. This last portion would remain in the train, and move it 
at the same very slow vel forever, if only we could remove all resisting forces from 
before it The next stroke in the same manner furnishes a new instalment of force ; 
which, like the first, divides itself into two parts; one of them as strain to destroy, 
and be destroyed by the new resistances met with in the next short dist along the 
track; and the other as motion, to remain stored up in the train, united with the 
motion put th'ere by the first stroke: thereby imparting increased vel to the unre¬ 
sisting matter of the train : and this process is repeated during a great number of 
strokes : the train moving faster and faster. But after a while it is found that no 
matter how powerful the engine may be, or how light its train, the speed no longer 
increases. 










316 


FORCE IN RIGID BODIES, 


The reason of this is that the resistances of the air, friction, Ac. increase with the speed, but at a more 
rapid rate; so that after a certain vel has been attained, they require all the power the engine is capable 
of, in order as strain to react against, balance, or destroy them alone; without leaving any more sur¬ 
plus to be stored away, as motion, in the train. After this point is attained, the power of the engine 
no longer draws the train ; but barely suffices to remove all the resistances which would otherwise im¬ 
pede it; and thus permits the moving force (which has previously been stored, aud accumulated in the ( 
train, by small instalments) to continue to move the unresisting matter of the train at an uudiminished i 
vel. The train might now be said to “ move itself;" and it would do so forever, without requiring 
any additional moving force, if the engine would only continue to destroy the resistances in its way. 
Should a stiff up-grade, sharp curvature, or other resistance present itself, so that the opposing forces j 
actually exceed the entire power of the engine, for a short time, the train will actually come to its 
assistance, and by converting part of its own motion into strain, will, as it were, lend pressure-force | 
to the engine, to help it push through its increased work ; performing iu fact the part of a fly-wheel 
in machinery. Frequently, when au engine appears to be pulling a heavy train up a short sharp 
grade, it is actually, so to speak, the train that is pushing itself up; the engine probably could not 
do it. This state of affairs can, however, continue for but a short time; otherwise all the motion 
of the train would be converted into strain ; and being thereby destroyed, the whole would come to rest. 

When a train is going at speed, and it becomes necessary to stop at a station some dist ahead, 
steam is shut off, so that the steam force of the engine shall no longer counterbalance, or destroy 
the resisting forces in front of it; and the number of the resistances themselves is increased by add¬ 
ing to them the friction of the brakes. Against all these combined, the train has now to “ work” 
its way unaided; and it docs so by the gradual conversion of its previously accumulated motion, or 
moving force, into straining force. Work, as before stated, consists in motion and strain combined. 
When the conversion has been completely effected, the train stops. Thus, we see, that up to 
the time of its slopping, the force which had gradually accumulated in it as motion before it had 
reached its greatest vel is gradually taken out of it as work, after steam is shut off. We may there¬ 
fore speak of it as well under the head of accumulated, or stored-vp work; a3 of accumulated motion , 
when we intend, in any kind of machinery, to convert such motion into work. Thus, the motion 
gradually accumulated in a fly-wheel, is also accumulated work, held in reserve until some extra 
strain ou the machinery calls for its aid. 

Art. 21. Up to the time that the vel ceased to increase because all the power 
•f the engine became required as strain to react against resisting forces, and conse¬ 
quently could no longer spare any as motion to the train, the work of the engine was 
what is termed variable; being gradually accelerated. Also, when steam was shut 
off, the work of the train was variable; being gradually retarded. Such is the case 
in almost every kind of machinery, during the interval between starting, and ulti¬ 
mately attaining its maximum speed. When the latter point is reached, supposing 
the resistances afterward to remain the same, the work is termed uni form, or steady, j 
All these remarks, as well as those of Art 20, apply alike to all kinds of heavy ma- ' 
chinery; no matter by what kind of force or power it is driven; the machinery 
takes the place of the train just spoken of; and the friction of the cog-wheels, gud¬ 
geons, pivots, the grinding, sawing, or whatever the work may be, takes the place 
of the grades, curves, and friction of the train. All alike are simply cases of force 
in its various shapes of motion, strain, and work. 

Art. 22. The quantity of any work, considered by itself, without 

reference to the time reqd to perform it. is plainly to be measd by mult together the 
resistance in lbs, by the (list through which it is overcome in ft, as stated in Art 11. 

After work becomes uniform, that is, when neither its strain nor its motion un¬ 
dergoes any change; its rate is measured by mult the resistance, or strain, - 
in lbs, tons. Ac, by the vel, or dist, in feet, Ac, through which the resistance is over¬ 
come in a given time, as a sec, min, hour, Ac. 

Thus, if the resistance is 3300 lbs, and is overcome through a dist of 10 ft in every min ; or if the 
resistauce is 33 lbs, aud is overcome through a dist of 1000 ft per min the rate of the work is iu each 

lbs vel lbs vel 

ease the same, namely, 33000 ft-lbs per min, or one horse-power; for 3300 X 10 — 33 x 1000 — 33000 
ft-tbs per min. The quantity of motion of a body (Art!») is also estimated in ft-Tbs ; and under the 
head Levers, it will be seen that the tendencies (called moments) which the power and the weight re¬ 
spectively have to commence motion about the fulcrum as a center, are measured in the same term. 
Work, mere motion, and moments, are, however, effects of force so diff from each other, that confu¬ 
sion is no more likely to occur, than in applying the same measure, one foot, to materials as diff as 
cloth, bar iron, boards, Ac. 

In scientific phraseology, work is either useful or prejudicial, the 

latter being the quantity of force lost by friction, by the resistance of the air, Ac. Thus, in pump¬ 
ing water, part of the applied force or power is lost'in the friction of the diff parts of the pump- so 
that a steam or water power of 100 lbs, moving fi ft per sec. cannot raise 100 lbs of water to a height 
of (5 ft per sec. Therefore machines, so far from paining power, according to the popular idea ac¬ 
tually lose it, in one sense of the word. In the practical application of all machiueri, the object is 
twofold; namely, to enable us conveniently to applv straining force, to balance, react against or ' 
destroy, the resisting forces of friction, and the cohesive forces of the material which is to be operated 
on ; and moving force, to cive motion to the unresisting matter of the machine, and of the material 
operated on, after the resisting forces which had acted upon them have thus been rendered ineffective. 

Art. 23. Tho total quantity of work that will he performed by the 
moving force that is in a hotly at any given moment, provided t.liat after chan trine 
from mere moving force into working force, it is left to expend itself in uniformly 
retarding work, without receiving any additional force to aid it, (as in the case of the 
moving force in the locomotive in Art 20, after steam is shut off; and when said 
force begins to work against the resistances of the road,) is as. or in proportion to 





FORCE IN RIGID BODIES. 


317 


(not. equal to,) the wt of the body, mult by the square of its vel at the moment it 
begins to work. For example, if a train at the time steam is shut off, lias in it an 
accumulated moving force due to a vel of 10 miles an hour; and if that force will 
by itself work the train against the resistances of the road for a dist of one-quarter 
of a mile, before coming to a stop; then, if steam is shut off while the train is 
moving at 2, 3, or 4 times that vel, and consequently with 2, 3, or 4 times the moving 
force, it will work through 4, 9, or 16 times the dist of the first case, before coming 
to rest. If bullets of equal wt be fired with vels proportioned to each other as 1, 2, 
3, they will respectively penetrate a plank to depths as 1, 4, 9. If an engine, water¬ 
wheel, Ac, works steadily in a mill, grinding at 2, 3, or 4 revolutions per min, it per¬ 
forms only 2, 3, or 4 times as much uniform work per min. as when at but 1 rev per 
min. But if steam or the water be suddenly shut off at 2, 3, or 4 revs per min, then 
the 2, 3, or 4 times quantity of moving force accumulated in the machinery at that 
moment, will, as working force, run the mill through 4, 9, or 16 times as many revs 
before stopping, as if shut off at 1 rev. If a rolling ball, started against a row of 
bricks, will overcome their resistances, and knock them down for a distance of 4 ft; 
then, if it be started at a vel 3, 4, or 5 times as great, it will overcome and knock 
them down for dists of 9, 16, and 25 times 4 ft; and in but 3, 4, or 5 times the time. 

But in all these cases the rate of the work done, that is, the quantity done in any given time, as 
one sec, is directly as the vels. Thus, the locomotive whose steam Is shut off ut 20, 30, or 40 miles 
an hour, will require but 2, 3, or 4 times as many seconds for running its 4, 9, or 16 dist before it 
comes to a stop; in other words, when its moving force is 2, 3, or 4 times as great, it will overcome 
but 2, 3, or 4 times the resistances in the same time , although the total amount of resistances over, 
come will be as 4, 9, and 16. And so with the other examples. 

Rem. We know that the dist through which a body must fall by the unif ac- 

cel force of gravity in order to acquire any given vel and moving force, is as the square of said vel; 
but directly as the time of falling. Also ii' a body is thrown vert upwards with any given vel or force, 
grav will retard it unif, and the height to which it will rise by the time that grav destroys all the 
force with which it was thrown, will be as the square of said vel: but the time will be directly as said 
vel. And so with anybody moving in any direction, and acted upon by any unif accel or retard force 
whatever. It will either acquireor part with its moving force within dists proportionate to the square 
of its vel, and in times proportionate to its simple vel. See Caution, Gravity, p 362. 

Art. 24. Vis viva, or living 1 force. The preceding article serves as an 
introduction to this subject; of which we shall endeavor to give some idea in plain 
language. The term itself is merely one of those absurdities to which savants re¬ 
sort, in order to impart an air of mystery to their writings. We might with the 
same propriety speak of a brickbat viva, or a living liod of mortar. 

We have seen in Art 23, that if that portion of force in a body which is occupied in giving motion 
alone to the body, be suddenly converted into working force, the quantity of work which it would 
perform against uniformly retarding resistances, before being entirely destroyed, or coming to rest, 
would be in proportion to the square of its vel at the time of beginning to work. Now. if “ vis viva,” 
or •• living force.” were merely the name given to this force; or to the quantity of work done by it, 
(as measured in ft-Ibs, by mult the resistances in lbs, by the dist in ft through which they were over¬ 
come,) the expressions, although silly, would still convey nn idea readily understood by practical 
men.’ We could then say, for instance, of a moving body, that its vis viva was 100 foot-lbs; mean¬ 
ing thereby tiiat it would overcome a uniform resistance of l lb through a dist of 100 ft; or a resist¬ 
ance of 100 lbs through a dist of 1 ft, &c. But scientific writers apply the terms to a purely imagi¬ 
nary quantity, equal to twice this; and which does not exist in any body, under any circumstances. 
The reason they do so is that it facilitates some of their calculations. But the practical engineer 
need not concern himself with either this reason, or vis viva itself; the simple statement of facts 
contained in the preceding and following articles, probably contains all that it is essential for him to 
know on the subject of moving force, converted into uniformly retarded working force. 


Art. 25. The amount of worlt, in foot-pounds, that can be ac¬ 
complished by a given moving body, by virtue of its momentum alone, is 


Work = Weight of moving body, in lbs X 


square of its velocity in feet per second 


2 g 


= Weight of moving body, in lbs X fall in feet required to give the velocity 
(g- = about 32.2. 2g = about 64. 4). See “ Falling bodies ”, p 362. 


An imaginary force equal to double this, will be the vis viva, or living force of the 
savants; now seldom used: 


VU viva = of My x 1 °L /Ij^ «ee* 

in lbs g 


* For the purposes of abstract science, it is not sufficiently exact to measure the quantity of matter 
„ i,» • h»*«i.»e the wt, or srriivitv of a body, as shown by a spring balance, varies somewhat in 


the acceleration or gravay oi tnat piacv,; ..... ..... ......... ... ,-- - - ... . „ 

quantity of matter undergoes no change at diff places, the measure of that quantity should likewise 








318 


FORCE IN RIGID BODIES. 


Art. 26. To perforin a given amount of work, we must expend an equal 
amount of energy. A moving body is said to have a kinetic energy 
equal to the amount of work which the body can perform by means ot its 
momentum alone; i e, 


kinetic energy _ weight of moving vel 2 in f t per 
in foot-pounds — body, in lbs 2 <7 


sec 


weight of moving v fall in ft required 
— body, in lbs A to give the velocity 


(g’ = about 32.2. 2 g = about G4.4.) 

For instance, a train weighing 1,120,000 tbs, and moving 22 ft per second, has 
a kinetic energy of 


222 

1,120,000 tbs X = 8 . 400 > 000 ft4bs - 


That is, if steam be shut off, the train will perform a work of 8,400,000 ft-tbs in 
coming to rest. Thus, if the sum of all the resistances (ot friction, aii, giades, 
curves etc) remained constantly = 5000 tbs, the train would travel 

8,400,000 ft-lbs _ 

5000 lbs 

Energy, thus expended in work, is not destroyed. It is either transferred to 
other bodies, or else stored up in the body itself; or part may be thus trans¬ 
ferred, and the rest thus stored. But, although energy cannot be destroyed, it 
may be rendered useless to us. Thus, a moving train, in coming to rest on a 
level track, transfers its kinetic energy into other kinetic energy; namely, the 
useless heat due to friction at the rails, brakes and journals; and this heat, 
although none of it is destroyed , is dissipated in the earth and air so as to be 
practically beyond our recovery. 

Potential energy, or possible energy, may be defined as stored-up energy. 
We lift a one-pound body one foot by expending upon it one foot-pound of 
energy. But this foot-pound is stored up in the body as an addition to its stock 
of potential energy. For, in falling through one foot, it will acquire a kinetic 
energy of one foot-pound, and will part with one foot-pound of its potential 
energy. 

Familiar instances of potential energy are—the weights or springs of a clock 
when fully or partly wound up, and whether moving or not: the pent-up water 
in a reservoir; the steam pressure in a boiler; and the explosive force of powder. 
We have mechanical energy in the case of the weights or spring or water; heat 
energy in the case of the steam, and chemical energy in the case of the powder. 

In many cases we may conveniently estimate the total potential energy of a 
body. Thus (neglecting the resistance of the air) the potential energy of a 
pound of powder is ==• the weight of any given cannon ball X the height to 
which the force of that, powder could throw it, = the weight of the ball X (the 
square of the initial velocity given to it by the explosion) 2 ff. But in other 
cases we care to find only a certain definite portion of the total potential energy. 
Tims, the total potential energy of a clock-weight would not be exhausted until 
the weight reached the center of the earth: but we generally deal only with 
that portion which was stored in it by winding-up. and which it will give out 
again as kinetic energy in running down. This portion is = the weight X the 
height which it has to run down = the weight X (the square of the velocity 
which it would acquire in falling freely through that height) -r-2g. 

We thus see that kinetic and potential energy are measured in the same way. 



I 








undergo none. Therefore scientific men adopt ——-- E —— to denote the quantity of matter in a 

, acceleration 

body ; and they call the resulting quot the “mass ” of the body, to distinguish it from mere wt. Thus 
at auy place where the acceleration of gravity is 32.2 feet per sec, and where a body weighs 20 lbs by 

20 

a spring balance, the body’s mass, or scientific quantity, is equal to ■■■ -■- — .621. To the prac- 

o 2 *. 

tlcal man, this mass or quantity conveys no idea whatever; but it is plain that the ordinary measure 
by weight cannot be perfectly correct, because the weight changes at diff places, while the quantity 
remains the same; and the measure by size would be equally incorrect, because the size varies with 
the temperature. 

Since, therefore, the only absolutely correct measure of quantity in a body is the scientific mass ; 
and since the imaginary vis viva is the quantitv. mult by the square of the vel, we have vis viva 
represented strictly bv. Mass X Vel*; or the M.V* of scientific writers. In science, the mass of 100 
tbs of iron is equal to that of 100 lbs of cotton. The greatest discrepancy that can occur at various 
heights and latitudes, by adopting wt as the measure of quantity, would not be likely to exceed 1 in 
300; or, under ordinary circumstances, 1 in 1000. 















FORCE IN RIGID BODIES. 


319 


Art. 27. As explained in Art. 9, p 310, forces are proportional to the veloc¬ 
ities which they can produce in a given body in a given time. From this it fol¬ 
lows that a greater force is required to impart a given velocity to a given body 
in a short time than in a longer time. For instance, the forward coupling links 
of a long train of cars would snap instantly under a pull sufficient to give to the 
train in five seconds a velocity of twenty miles per hour, supposing a locomotive 
of sufficient power to exist. In many such cases, therefore, we have to be con¬ 
tented with a slow, instead of a rapid” acceleration. 

A st ring may safely sustain a weight of one pound suspended from our hand. 
If we wish to impart a great upward velocity to the weight in a very .short time, 
we evidently can do so only by exerting upon it a great force; in other words, 
by jerking the string violently upward. But if the string has not tensile 
strength sufficient to transmit this force from our hand to the weight, it will 
break. We might safely give to the weight the desired velocity by applying a 
less force during a longer time. 


Art. 28. Composition and resolution of forces. We have already 
said that when dill' forces are imparted* (whether so applied, or not) in the same 
direction to a rigid body free to move, unresistedly, they all act as motion alone, in 
that same direction. If two equal forces are imparted in diametrically opposite 
directions, they mutually destroy each other entirely, as strain (pull or push) against 
each other; thereby producing strain among the particles of the body ; but having 
no tendency to move the body, as a whole, in either direction. If unequal, and in 
diametrically opposite directions, the whole of the small one, and a part of the large 
one, equal to the small 
one, destroy each other 
as strain; while their diff 
remains as motion, (or 
a moving of the whole 
body,) in its original di¬ 
rection. But if two 
forces, a o and b o, Figs 

9, whether equal or un- -J 

equal, are imparted at 
the same time to an un¬ 
resisting rigid body o, 
in directions either con¬ 
verging toward; or di¬ 
verging from, the same 

point o. at any angle whatever; then the body o cannot possibly be kept at rest by 
them ; or in other words, equilibrium cannot exist between them; or they cannot 
balance, or completely react against each other; the body must move. Equal parts 
of each of the two forces will mutually destroy each other as strain among the par¬ 
ticles of the body; while the remaining portions will unite to constitute a single 
force r o, wffiich will move the whole body in a direction o d, in the line r o extended; 
and which direction o d will always be somewffiere between those in wiiicli the separate 
forces would have moved it. 

If we lav off c o and l o bv anv convenient scale, to represent, respectively the amount of the forces 
a o and b 6. and then complete the parallelogram ocr t; the diag ro. measured by the same scale, will 
represent both the direction and the amount, of the single remaining force. 



* It, is absolutely necessary to keep distinctly in mind the diff between applied and imparted force. 
Writers carelessly confound the two very frequently. See Art 19. 









320 


FORCE IN RIGID BODIES 


The same process will answer also for forces which instead of motion, produce strain, not only in 
the particles of the body, but iu the body itself considered as a whole ; or, in other words, a tendency 
to press or pull the entire body in a certain direction. Thus, suppose that two men were either pull¬ 
ing or pushing with the forces co and to; trying iu vain to detach a piece o of rock, from a cliIt or 
which it forms a portion ; and which, by its inherent force of cohesion, to theclifi. duties their efforts. 
Here we have a case of extraneous forces, resisted, or reacted against, or balanced, by strength of 


As^in the case of motion, the two forces partly destroy each other as strain among the particles of 
the bodv ; and the remainders combine to form the single force r o, which tends to move the whole 
body toward d. The rock resists this siugle force, by a cohesive force precisely equal, and diametri¬ 
cally opposite to it; and so long as it does so, there is strain but no motion. The piece of rock may 
have strength enough to oppose a much greater resistance; but cannot actually exert it unless the 

men also exert more force. . , , , . . . 

In the matter of comp aud res of forces, it must be remembered that when force is applied to a 
bodv in order to produce motion, care must be taken that there is no other force to prevent it: but 
when the force is intended to produce strain, it is equally necessary that other force should be present 
to oppose it; for strain is the opposition of forces. 

The fig ocrt, Figs 9, is called the parallelogram of forces. The two 
original forces co, to, are called the components of the force ro\ which results liom 
their joint action; and the force ro is called the resultant of the original ones which 
compose it. The principle of the parallelogram of forces, than which there is none 
more important in the whole range of mechanical science, may be expressed thus: 
If any two forces, (both motions, or both strains,) whose directions either converge 
toward, or diverge from, the same point, be represented both in quantity and in di¬ 
rection l>y two adjacent sides of a parallelogram : then will their resultant be simi¬ 
larly represented by the diag of the parallelogram.* 




Rem. 1. If one of the forces, as c, upper Fig 9^, is 
a pull, ami the oilier a push, then to find their result¬ 
ant ot we must, before drawing the parallelogram of forces, move (or imagine 
to be moved) one of the forces to the opposite side of the point o, so as to change 
it from a pull to a push, or vice versa, so that both shall be pulls, or both pushes, 
as shown by the two lower figs. Otherwise we should obtain a wrong resultant 
n o of the top fig. Either a push or a pull equal to o t, if applied at o, would be 
equal in effect to the push a and the pull c. The remark is of frequent use when 
finding strains in bowstring aDd crescent trusses ; as iu maDj other cases. 

Rein. 2. When any three forces as a, b , c, form¬ 
ing only two angles axb and bxc, balance each other 
at any point x, then a straight line as o e can be 
drawn through that point so that all three forces 


shall be on one side from it; then also a parallelogram xn can be 
drawn on the three lines a, b c,, having the middle line b for its 
diagonal; and this diagonal will be of a different character from 
the two outer forces a and c; that is, if they are pulls, it will be 
a push, and vice versa. But if as in the three balancing forces 
t, i, s, three angles as sxl, txi, sxi, are formed, neither such a line, 
nor such a parallelogram can be drawn; and the three forces will 
all he alike, all pulls or all pushes. Ail this is evident from the two 
figures. 

Rem. 3. We have alluded to equal parts of each component as being lost, or de¬ 
stroyed, by reacting against eacli other; thus producing within the body a straining 
of its particles; and therefore having no tendency to move, push, or pull, the body 
as a whole, in any direction. 

Let h a and c a be any two components, and no their resultant. From 
the two angles ft and c, opposite to the diagonal, draw bo and c i at right 
angles to the diagonal; or to the diagonal extended, if necessary, as in 

li' i n (13/ Ttineil t UIA 1 i lino )i rt -1st n- ill « 1 urn IT o V\. > Aniwil nn a nn.tl, . 



Fig 9%. These two lines, bo, ic, will always be equal to one another; 
whatever may be the lengths and direction's of the components ha, ca. 
When two forces, as b a, ca, are imparted at a, there occurs a loss of force 
equal to what would result from the reaction of two forces equal to b o and 
c i. It is lost by becoming strain against the cohesive forces of the parti¬ 
cles which compose the body a. In anticipation of what is said in Art 31, 
we will state that the force b a may be regarded as made up of the forces 
bo, oa; and the force ca, of cf, ta; which act also in those directions, 
when b a and ca converge toward a, as in Fig 9^ ; or in the directions 
ao, ob, and at, i c, when the forces diverge from a, as in Fig 9?£. In either case, however, these 
forces, 6 o, ao, ci, i a, &c, must be considered as being imparted at a. This being supposed. It be¬ 
comes plain that when ft a and ca meet at a, inasmuch as 6 o and ci destroy each other as strain 

against the internal cohesive forces of the body, there remains nothing to act upon the body consid¬ 
ered as a whole, except oa and ia; which, being together equal to na, (as seen in the fig.) are. in 
other words, equal to, or actually compose, the resultant n a of the two components ft a, c a. See Rem 5. 


* Components and Resultants may be calculated by the form¬ 
ulas in Art 45, when a diagram is not considered sufficiently accurate. 















FORCE IN RIGID BODIES 


321 


We conceive that each of the original forces endeavors as it were to compel the other to leave its 
own course, and follow that of its antagonist; and the struggle continues uuti) they have succeeded 
in forcing each other into the same direction. This is of course effected by their reactions against 
each other; and, as occurs in all cases of reaction, they expend equal parts of their forces on each 
other. When the two forces act in diametrically opposite directions, where there is no neutral diag 
direction that can be adopted, there is no alternative but for the larger force to react against or de¬ 
stroy the smaller one entirely ; thereby losing an equal amount of its own force. Its remaius totter 
on slowly in their former unchanged direction. The writer can see no difference of principle between 
the reaction of opposite forces; that of oblique ones; and that of those at right angles to each other. 

Rem. 5. When the direction a b. Fig 9%, of one of the forces, forms an angle b a n, 
greater than 90°, with the diagonal, the shape of the parallelogram of forces becomes 
such that the two equal lines bo and ci, cannot be drawn at right angles to the diag 
art itself; or within the parallelogram; in which case the diag must be extended 
each way, as to o and i; and the lines bo, ci, must be drawn at right angles to the 
extensions. 

When this occurs, the component forces a o, ai, cannot as in Fig 
9}4 be measured on the diag an of the parallelogram ; because they 
will be greater than it; but must, like bo, ci, be measured outside 
of the fig. And here it must be remembered that ao and a i no 
longer measure forces acting (like those in Fig9J4) in the same di¬ 
rection. Thus the strain along a b may be considered (see Comp 
and Res of Forces, Art 31) to be made up of two forces imparted 
at a ; namely, a hor force equal to o b, and a vert one equal to a o. 
acting upward. And the strain along «. c, as made up of one hor 
force equal to i c, and a vert one ai. (greater than the whole diag,) 
acting downward; both of them imparted at a. Hence, the re¬ 
sultant an we find is equal to thediff between the two vert compo¬ 
nents a o and a i. Thus it is seen that this shape of the parallelo¬ 
gram in no way affects the principle laid down in Remark 3. 



XV 


Art. 39. According to Art 18, the force we, Fig 10, may be considered as im¬ 
parted to the rigid body B at any point whatever in its line of direction wc; also, 
the force xi, at any point in its direction x d ; conse¬ 
quently, both of them may be considered as imparted at 
the same point a; inasmuch as it is situated in both these 
lines. Hence, it is immaterial, so far as regards the effect 
of those two converging forces upon the body considered 
as one entire rigid mass, whether they are actually im¬ 
parted like zo and yo, at the same point o ; or like we and 
xi, at diff points i and e. For in either case their result¬ 
ant, or joint effect upon the body as a whole, is precisely 
the same; namely, a tendency to move the body in the 
same line of direction oat. This tendency will actually 
produce motion if no opposing force prevents; otherwise 
it will produce strain in the body. 

Rem. 1. Hence the resultant R, of two converging forces F f, Fig 101^; or of two 
diverging ones F f. Fig 10^, acting in the same plane, but imparted at diff points 





















322 


FORCE IN RIGID BODIES 


of a rigid body W. may be found as readily as when imparted at the same point; a-‘ 
at o, Figs y, or Fig 10. 

Thus, produce their lines of direction, either forward as in Fig 10*4 • or backward as in Fig 10J4 
as the case may require, until they meet, as at h. Make It a by any scale, equal to the force/; auc 
b c equal to the force F. From a and c, draw lines respectively parallel to b c and b a ; thus complet 
ing the parallelogram of forces, baic. The diag hi of this parallelogram, measured by the samt 
scale, will represent the reqd resultant R. both in quantity, and iu direction. It is thus seen that il *, 
is not necessary that the point b shall be in the body itself. 

Rem. 2. It is perhaps almost useless to again remind the young student that the bodies are all along 
assumed to be rigid ; or inelastic, and incapable of being broken or bent by the imparted forces. For 
otherwise the force /, in Fig 1054, might split off the top of the body ; or F might crush to dust its 
toe<; or both might penetrate it. Hut, assuming that the material is sufficiently strong to resist 
such splitting, crushing, and penetration, we at present confine ourselves to the effect of tbe forces, 
whether as motion, push, or pull, upon the body as a whole. The splitting, crushing. &c, is a mat¬ 
ter that must be considered under the head of Strenyth of Materials. It is of course quite as neces¬ 
sary iu practice to pay attention to these effects as to the others, but it must be done by a separate ' 
process. 

Art. 30. Since the effect produced upon a rigid body (con¬ 
sidered as a whole) by the resultant (ac, Fig 11) of any two forces! 

(b c, dc) tending to or from the same point, is the same as the joint 
effect of those two forces themselves, it follows that if we oppose 
to those two forces a third one ( nc) equal to the resultant (ac), 
and diametrically opposite to it, that this third force will com¬ 
pletely react against, balance, or destroy said two forces; or rather 
their remains. It is frequently necessary to consider such a third 
force, (n c,) equal and opposite to a resultant (a c); and inasmuch 
as we do not know that any specific name has been applied to it, 
although one is needed, we suggest anti-resultant. Re¬ 
sultant (ac) may be defined to be a single force which wiil pro¬ 
duce upon a body considered as a whole, the same result that its 
components (be, dc) produce. Or as a force which, if its direction 
were reversed, (thus making an anti-resultant,) would balance its components. 

Iu the preceding Figs, the arrows represent pressures; if all the arrows be reversed, thus indi¬ 
cating pulls, the principle and processes remain precisely the same; for force is still only force; and 
its effect upon a rigid body, considered as a whole, is the same whether it act as a pull, or as a push. 
See Art 18. 

When the forces diverge from the same point, their strain is a pull, or a tension; when they con- t 
verge toward it, a push, or pres, or compression. 

Art. 31. By a process the reverse of that in Art 28, any single force, nd, Fig 12, 
may be resolved into two component ones, « d, in d, one on each side of it, and in 

the same plane with it; 
which would produce the 
same effect as it upon a 
rigid body, d, (considered 
as a whole,) by merely 
drawing from d, 2 lines 
dg, dt, showing the di¬ 
rections of the two forces; 
and then, drawing from 
o two other lines on,o in, 
respectively parallel to dg, d t ; thus completing the parallelogram (dno v<) of forces, 
upon nd as its diag. Then measure d n, and d m, by the same scale as od\ and thej 
will give tbe amount of each of those forces. 

It is plain that an infinite number of differently proportioned parallelograms, such as dnom.dsoa, 
&c, may be drawn upon any line o d as a diag: and in any one of them, two adjacent sides will rep¬ 
resent components equal in effect to the single force od. represented by the diag. Thus the forces \ 
n d, m d, are equal to o d, as regards their effect upon a rigid body d, as a whole. So are also the 
forces s d and a d ; consequently the effect of s d, and a d, is equal to that of n d, and m d. It will be 
observed that the longer any two components on the same diag are, (as nd. md, longer than s d. a d,) 
the more uearly in a straight line, and more directly opposed to each other, do they become: and 
consequently the more nearly do they mutually destroy each other; leaving smaller portions of each 
to act upon the body. Thus the portion of the great forces nd. md, left to act upon the body d, is 
no greater than that of the small forces sd, ad; this remainder being in both cases represented by 
the resultant o d. 



11 . 




Rem. Hence, if we have two forces, as the two pulls 
ab, ac. Fig 1214, whose amounts and directions both are 
given ; and which are counteracted, or held in equilibrium, 
by two other forces such as the two pulls af, a e, whose 
directions alone are known, it becomes easy to find the 
amounts ad and a o of these last, thus; Complete the 
parallelogram bact; and draw its diag at. Make at 
equal to at, and in a line with it. Complete the paral¬ 
lelogram adio; then plainly ad will be the amount of 
the force in the direction a/; and ao that in the direc¬ 
tion a e. 










FORCE IN RIGID BODIES. 


323 


Art. 32. It follows from the foregoing articles, that a single force cannot be 
resolved into two components, one of which only is in the same direction as that 
lorce itself; for if a line representing that force be taken as a diag. it is self-evident 
that no parallelogram can be drawn upon it which shall have any of its sides par¬ 
allel to said diag. 

Therefore a rope, as ab. Fig 15, sustaining a wt w, so long as it remains perfectly vert, that is, pre¬ 
cisely in the direction of the force of gravity of the wt, will jeceive no assistance in upholdiu’g the 
wt by having added to it a single rope as ob. or by; or one extending from the wt itself in any in¬ 
clined direction. In other words, a perfectly vert rope cannot sustain one part of a load, and one in¬ 
clined rope another part. All this, iudeed, is a result of the fact stated in Art 15; that any force, 
however great, (as the vert force of an immense suspended weight w, Fig 15.) will be turned out of 
its direction by any other force, however small, (as a slight pull from a rope ob, or by.) unless there 
be some third force to prevent it. lu the present instance, this third force might be a third rope • 
for the rope a b will be relieved, and still remain vert, if we employ two oblique ones to assist it, pro¬ 
vided they be exactly opposite each other ; or, in other words, that all three ropes, or forces, be in 
one plane. 



So also in the case of a vert post sustaining a load ; the pres from the load cannot pass rert through 
the axis of the post, if the load at the same time is partly sustained by a single oblique brace pressing 
against the post Indeed, such a brace, by turning away the direction of the strain from the axis 
\ >f the post, may very materially diminish the power of the latter to sustain the load; for it will be 
found under Strength of Materials, that if the strain along a post or column does not pass directly 
through its axis, the column may in some cases lose two-thirds of its strength. The principle of 
course applies to force in any other direction, as well as vert. 

A resultant may be greater or less than either one of its two oblique components; 
but it can never be greater, or even quite equal, to both of them : on the plain prin¬ 
ciple that any two sides of a triangle are greater than the third side. If the com¬ 
ponents are equal, and inclined to each other at an angle of 120°, the resultant will 
be equal to one of them; therefore, the same weight that would break a single vert 
rope, or post, would break two ropes each of the same strength as the single one. or 
two posts, inclined 120° to each other. If the angle o a b, or y ab, which either of 
the forces form with the diag a b, exceeds 90°, see Rems 5, of pp 321, 327. 

Art. 33. The principle of the parallelogram of forces is of constant applica¬ 
tion in constructions of every kind; for instance, bridges, centers, roofs, retaining- 
walls, &c. Figs 13, 14, 15, 16, show a few of the most simple cases of force (the load 
w) applied to produce strain ; by reacting against opposing forces y a, na, presented 
by the walls. In all these, the load w, applied at a, is a single force of gravity ; and 
consequently acts in a vert direction downward. It is to be resolved into two com¬ 
ponent forces in the direction a to, an, in order that we may find the strains which 
it produces (according to the ordinary phraseology) along the pieces a to, an, so that 
we may proportion their dimensions to resist those strains; which strains are in 
fact produced by the reactions of the three forces, of the load, and the two walls. To 
do this, in all the figs, from a draw a vert line a b. to represent the direction of grav, 
or of the force in the load w. On this line, lay off by any convenient scale, the dist 
ab to represent the amount in ftis, tons, <fcc, of the load w. Also, from a draw the two 
lines am, an, in the directions of the reqd component forces. Then complete the 
parallelogram of forces, by drawing lines bo, by, from b, respectively parallel to 
am, a n. Then will a o, measd by the same scale as a b, give the amount of strain, 
whether push or pull, which the load w produces along the piece a to; and in like 
manner will ay give the amount which it produces along the piece an. 

It must be especially borne in mind, that we here speak only of the amounts and directions of the 
strains produced by the extraneous load w alone; without reference to those produced by the weight 
of the pieces themselves. If the force acting at a is not vert, but oblique, then the direction of a t 
must of course be drawn oblique; but if the force at a is gravity or wt, it must be vert. 







324 


FORCE IN RIGID BODIES, 


Caution. See foot-note, p 556. 


Fig 16}4 shows that the strains e t, e s, are really due to the action and reaclinr 
of the wt and the walls ; although we often speak of them as due to the load l atone 
which is represented by the diag e i. We have said that a force cannot produce straii 
unless there is opposing force to strain against. Now, when we place the force ot tlni 
load l at the point e, it is evident that it is upheld by the walls at A and B; or ii 
other words, that it reacts against these walls; and the walls against it. The wall A 
furnishes the force indicated by the arrow A ; and which may be considered as tin 
resultant of the hor force c; and of the vert one o. So also the torce B; as the re 
sultaut of m aud n. 





Now these force! 
A and B are ap 
plied at the poiul 
e, just as well a! 
the load l is; foi 
they pass up a! 
pushes, along th« 
rafters; as the 
force of l passes up 
as a pull, along the 
rope. The rafters 
and rope are mere 
ly the mediums 
through which the 
three forces reach 


1 


e; and the forces in passing through them from end to end, of course produce in 
them strains respectively proportionate to the forces. Now, the forces e t and e s, 
which are usually said to be produced by the load, are nothing more or less than the 
two forces A and B, produced by reaction of the walls; and which, for convenience 
of drawing the parallelogram of forces in practice, are laid otf each way from e. We 
have then three forces t e, s e, and e i. all acting at e, to produce sti-ain alone; aud 
this they must do by straining against each other. 

The following is the manner in which they do so. The two hor components m and c, (which will 
always be equal to each other; no matter how different the slopes of the two rafters may be,) beiug dia¬ 
metrically opposite in direction, react or strain against, or balance, each other ; thereby producing 
a hor strain, equal to one of them, throughout every part of each rafter. The two vert components o 
and n, (however unequal they may be.) will together be equal to the load f; or to its representative 
e i ; and having a direction exactly opposed to it, they react agaiust. or balance it; thereby producing 
in every part of the rafter e s, a vert straiu equal to n; aud iu the rafter e t, one equal to o. There¬ 
fore, since n, is here greater than o, the rafter e s bears more of the load l, than the rafter e t does; 
and in the same proportion. 

Thus, we see that every part of each of the three forces e i, e t, e s, produces strain, by balancing 
an equal part of one of the others. The walls really oppose to the load no force greater than its own ; 
namely, o and n, against e i. With the hor components m and c, the walls react only against each 
other. 

As it is difficult, however, to introduce a new phraseology, in place of one which, although errone¬ 
ous, is in universal use, we also shall speak of component strains like e f, e s, as if they were really pro¬ 
duced by the resultant, or load, e i. And in alluding to resultant motion , we shall probably often say 
they are the effects of components, instead of effects of their remainders, after the components have 
partially destroyed each other's moving forces by straining against each other to produce change of 




direction. 
Rem ’2 


• vc... The truth of such examples as Fig 14, with a rope or string, may easily be 
Rhosvn by means of two spring balances, to which the ends m and n of the string may 
be fastened. Suspend a weight w from the string, and the balances will show the 
strains along a in and a n. The balances must be held in inclined positions. 

The student should try all such experiments. This one will show that in proportion as the two parts 
am, an, of the rope, approach nearer to one straight line, the greater will be the strain produced 

upon them by any given load, or force w, and so great will 
this be, that if the weight w be only one pound, two of the 
strongest men cannot strain the rope perfectly straight be¬ 
tween them. Or if they stretch the rope alone to as nearly 
a straight line as possible, and if then a weight of a Tew lbs 
be suspended from it, this small weight will pull the men 
closer together. Or if the rope be stretched nearly straight 
between the two spikes so firmly driven as to require a great 
force to draw them, it will be found that a much smaller 
force applied as at w, will draw them readily. In other 
words, a rope so situated, and with force, or power w, applied 
to it in this manner, between its ends, and oblique to its di¬ 
rection, becomes a machine; for by it power may, (to use the 




















FORCE IN RIGID BODIES, 


325 


ordinary incorrect expression,) be gained. Itis called the funicular UiacllinC ; or some¬ 
times simply the COrtl. Fig 16>2 shows the principle on which this machine is frequently employed 
for overcoming a great resistance, r, through a short distance, by a small power p. One end c, of a 
rope c d r, is firmly fixed. The rope passes over a pulley d ; and its other end is tied to the resist¬ 
ance, or load r. By applying a small downward force p. at the center of the rope, drawing it down 
to s, the load r is thereby raised a short dist; for the same great strain w hich the small force p pro¬ 
duces from s to d , extends also dow n the rope, from d to r; except a slight loss produced by the 
friction of the pulley. Thus, the strain aloug the back-stays of a suspension bridge, is equal to that 
on the main chains just inside of the suspension piers; supposing the cables to rest upon rollers. In 
the theoretical, consideration of ropes and chains, they are in most cases assumed not to stretch; to 
be perfectly flexible; without weight; and infinitely thin. 

In such a machine the two parts s c, s d, Fig 16!^, are to be considered as two entirely distinct lies , 
in the same manner as am. aud a n. Figs 13 and 16, are two distinct struts Each of these ties may 
have to sustain a different amount of strain, depending on their respective inclinations to s p. Tims 
if the loadp. Fig 16pj, be suspended from a perfectly frictionless pulley or slip knot resting on the 
perfectly flexible cord c s dr, and if this pulley or knot be at first placednear c or d, it, with its load p 
will descend by gravity along the cord until it comes to rest at s. w hich is the lowest point thfr the 
cord admits of its attaining and at which alone the angles of Inclination of s c and s d to s 
p become equal; and the strains on the two parts will then ue equal. But if as in Fig If tne 
short string which sustains W is tied fast to the cord (so as not to move as the pulley did) at any 
point a, such that the angles of inclination of a m and a n to the diagonal a h shall be ditlereut, then 
the strains or pulls along a m and a u will also be different. 

It Is immaterial whether m and n. Fig 14, or c and d. Fig 16M, arc at the same 
height or not. 

For more on the funicular machine see p 344. 

Let the end g of the rope g c o n be fixed ; a power of 9 tons at »; the rope passing over a pulley 
at P ; aud bent out of line at c by a fixed pin. Make c g and c o by scale each equal to the power 9 
atn: aud complete the parallelogram ; the diagonal exof which is then found to be. say 6; oraresult- 
aut of 6 tons. Now, in this case, theoretically the strain lengthwise of the rope is everywhere equal to 
the power n, or 9 tons; and we have found that it produces also a strain cx, against the pin at c, of 6 
tous. It also produces a pushing strain on 
the pulley P. Its amount may be found in the 
same way, by measuring 9 tons by scale each 
way from o toward c and n; completing the 
parallelogram; and measuring its diagonal 
resultant. But now let us use this rope as a 
funicular machine ; and apply a power x c of 
6 tons at c. We find that this 6 tons produces 
a strain c g or c o, of 9 tons along the rope; 
and this strain along co will pass along to n ; 
and thus the power of 6 balances a resistance 
of 9 tons acting at n iu the direction n o. 

The diagonal cior any other will plainly be vertical only when the angles of 
Inclination of c g and c o, witli the horizon are equal. If they differ, both the di¬ 
rection and the length of the diagonal will change. 

All will remain the same if the end g instead of being fixed, is passed over a 
pulley as at P, and a load or a pull equal to that at the other end is applied to it. 

Rem. 3. Thesnrfaces of contact of pieces used in construction, are called 
joints. When a piece is intended to resist compression, or push, it is called a 
strut ; or if inclined, it is often called a brace; or if vertical, a post, pillar, 
or column. When to resist tension or pull, a tie. When to resist both tension 
and pull alternately, a tie-strut, or a strut-tie. A strut should be still - or 
inflexible; but a rope, chain, or thin rod, may answer for a tie. 



Rem 4. To disting’iiisli a tie from a strut, at a glance is sometimes 
difficult; but it may he done thus. From the point a, Figs IQ%, at which the force 
acts, draw a line a c, in the direction in which the force, if at liberty, would move 


away from that point. On any part, 
a o, of that line as a diag, draw a paral¬ 
lelogram of forces. Through the point 
a draw a line i i, parallel to the other 
diag 11. Then all the pieces which 
are on the same side of that line, that 
a c is, are struts; while those on the 
opposite side, are ties. We may also 
frequently determine, by imagining 
the piece to be a rope or chain, in¬ 
stead of a beam ; and seeing whether i 
a tie; if not, a strut. 



would then bear the strain. If it would it is 


When a piece of material is used to resist forces which tend to bend or break it crosswise, or trans¬ 
versely of its length, as in Figs 47. 48, 49. 50, it is called a beam; such as joists, girders, &c. The 
same piece, however, frequently acts at once, as a beam, and as a tie, &c. Its own weight strains a 
beam transversely ; but in our present illustrations of comp and res of forces, this strain, although 
frequently the most important one, could not be well considered at the same time. 






326 


FORCE IN RIGID BODIES, 


Art. 34. Since any single force may be resolved into two oblique ones in the 
same plane with it, and which shall produce the same efiect upon a ligid bod\ con- 
sidered as a whole, it follows that the single strain along any piece a m or a n, ot the • 
four figs on p 323, may bo thus resolved. In practice, it is frequently necessary to do ; 
this; and especially so for finding components at right angles to each other, in lior 

For instance, the joint o d, Fig 17, at the foot of the beam A, if made 
at right angles to the resultant r r of all the pressures along the beam, , 
of course receives the whole of these pressures; which consequently arc j 
all imparted to the abutment; leaving no portion uuresisted, so as to pro 
duce sliding; or even a tendency to slide along the joint o d. Lonse- 
quently. this joint is perfectly adapted to its duty. See Art 19. Butajomtr, ' 
of the "form of 6 i c, which is equally effective, is sometimes reqd for re- 1, 
ceiving a single strain (like that along A) along a piece E; and in order I, 
to properly proportion the vert and hor faces b i, and c t, of the joint, we j 
must find the proportion existing between the vert, and the hor compo- j 
nents equal to the single strain r r along E. To do this is very easy; for 
we have only to lay off by scale, any length e n along r r, to represent 
the single strain in that direction; and on it as a diag, from e and n 
draw vert and hor lines e t, n t, meeting in t. Then e t measured by the 
same scale, will give the vert strain ; while n t will give the hor one. The 
parts b i, ic of tiie joint, must consequently have the same proportion as 
these two components have to each other; bearing in mind,however, that 
since joints should be at right angles to the forces they hav e to sustain, 
the vert part b i must bear the hor strain; and the hor part i c, the 
vert one. 

When, by Art 33, we are finding, by 
means of "the parallelogram of forces 
o n y <], Fig 18, the total strains o n, o g, 
which an extraneous load F produces 
along two beams FR. Fg. iti« easy at the 
same time to find the vert and hor compo¬ 
nents also ; hv drawing the two hor lines 
n (, gj , and measuring them by the same 
scale used for the diag oy. Likewise , 
measure ot. and oj, for the correspond- k 
ing vert forces at the joints; because j 
when nt and gj maybe drawn inside j 
of the parallelogram, (which is not ' 
always the case; as see Fig 18^.) the 
component forces in the direction of any 
diag, whether vert or not. are measured 
respectively from the point o, where the 
extraneous force F is imparted to the beams; to those points t and j, where the diag is met by the 
equal hues n t, gj. 

Rem. 1. It is an important fact that however diff may be either the inclinations, 
or the lengths of the two beams; or how dilF the total strains in the directions of 
their respective lengths; the hor strains, caused both by the extraneous load and 
by the weiglits of the beams themselves, will always he equal on both of them. 
Thus, in Fig 18, n t is equal to gj ; and in Figs 13 to 10, if hor lines be drawn from o 
and y, to a b, those in any one fig will be equal to each other. 

Rem. 2. It is plain that each beam may be considered as receiving from the load 

F, either one force or its two components. 

The vert component oj, of the triangle o gj, being longer than o t, of the triangle otn, shows that I 
the beam o g bears more of the vert force or weight of the load F, than o U does; and in the same ! 
proportion as oj is to o t. The two components on the diagonal, (when inside of the parallelogram,) i 
will always together equal the length of the diag, or the weight F. But as n t and gj are of the sanu I 
lengths, they indicate that both beams are pressed sideways, or lior, to the same extent. 

\\ hen we come to treat on trusses, we shall find this method of obtaining vert components, by 
means of ot and oj, very useful. The lengths of the beams o R, og, do not affect the amount of 
strains produced upon them by the load F at their summits; but ns their own wts must increase with 
their lengths, the strains arising from them must increase also; hut we have not yet taken their own 
wts into consideration; neither are we yet prepared to do so. 


and vert directions. 





Rem. 3. If, as in Fig 1SJ4, one of the beams, as n r>, is 
lior, it is plain that all of the diag oy , that is, all of the 
weight F or IV, is borne by the other beam og; and n o 
sustains hor strain only. The beam og of course bears 
an equttl hor strain also, as shown l»y y g, equal to n o. 

Rem. 4. It is immaterial (Art 13) whether the load 
rests on top, as F; or is suspended below, as W ; for in 
either case it is simply vert force imparted at o. 

















FORCE IN RIGID BODIES 


327 



Fiql8^ 


Rem. 5. When one of the forces, as 
n o, makes an angle noy, greater than 
90°, with the diag o y, the positions of 
the beams on, og, become as in Fig 183/£, 
and we have a case like Fig 9%; that is, 
the hor lines nb, ga, from the angles n 
and g, and at right angles to the diag, 
cannot be drawn inside of the parallelo¬ 
gram. Therefore we must extend the 
diag both ways, to a and b. If we wish 
to consider each of the forces on and og 
as made up of two components; then 
for those of on, we have bn, and ob; 
and for those on og, we have ag and 
o a. Hence when the angle n o a ex- ^— / M 

ceeds 90 the vert strain on o g is greater than the load w, 

which (according to the ordinary phraseology) produces it. But the part a y of the vert o o, has 
no reference to the load; but represents an upward vert force produced by the wall M, to balance a 
downward vert one equal to 6 o from the wall P. This excess ay over the diag, occurs only when 
one of the beams forms an angle greater than 90° with the diag. \Ve call the attention of the student 
to this case, because we do not remember to have met with it in any book. It will perhaps be a new 
idea to many, that the vert pres on the wall M, can be greater than the entire load. 

Art. 35. As a simple practical example of very common occurrence, of the ap¬ 
plication of the foregoing principle of finding the resultant of two forces in the same 
plane, and tending to one point; let S, Fig 19, represent a block of stone weighing 
3 tons; .and standing on a hor base mn\ but not attached to it in any way by ce¬ 
ment, Sic., but with a stop at n, merely to prevent sliding toward b. 

In this case there will be no force acting upon the body in such 
a way as to prevent its being overturned around its toe n as a 
turning point, except its wt, or force of gravity, which always 
acts vert downward. By Arts 56,57, all this force maybe con¬ 
sidered to be concentrated at the cen of grav i of the stone; and 
as acting at any point whatever in its vert line of direction l g. 

Now suppose a pres fh, of 2 tons, (which may be either one 
simple force, or the resultant of many.forces,) to be imparted to 
the stone, in the same plane with the force of gravity; which it 
will evidently be only if its direction /c meets the direction of 
gravity Ig, at some point; as, for instance, at a; because then a 
plane surface would coincide with both directions. Art. 8. The 
question is, which of these two forces will prevail; the two tons 
of fh. to overturn the stone ; or the 3 tons of gravity to prevent 
its being overturned? By Art 29, both forces may be considered 
to be imparted to the rigid stone, at the point a, where their lines 
of direction meet; aud we may make a c equal to 2 inches, ft, &c, 
to represent the amount and the direction of the 2 tons of pres of the 
force /h : and a v, by the same scale, for the direction, and 3 tons 
of pres of the weight of the stone. From c draw a line c d paral¬ 
lel to an; and from v draw vd parallel to ac; these will meet at _ 

d : thus completing the parallelogram of forces a c d v. The diag 

a d of this parallelogram, measd by the same scale, will give about 2% tons for the single resultant 
force, which would by itself produce upon the rigid stone the same effect as gravity and fh com¬ 
bined. This force by Art 18, may be considered as imparted to the stone at any point in the line of 
its direction (w o) through the stone ; as a push at w, as shown by the arrow z w ; or as a.pull at o, 
by means of a rope h o. fastened to the stone at o. Since this resultant is supposed to take the place 
both of gravity and of fh, the two last must of course be considered as annihilated ; so that the stone 
becomes as it were an unresisting body of matter without weight; and acted upon by the force ad, 
or z w, which must of course move it, and thus compel it to overturn around n as a pivot. 


'-■A 


1 

1 

1 

/ ' •.*;& 

x i 

i aix 

i 

/ -t 

\° fi 

i 

» 

, : / 

1)1'' . 




11 


18 

IV19 


111 


Rem. 1. Had the direction of the resultant ad struck the base of the stone at n, 
instead of striking outside of the base as at b, the stone would barely have stood; 
because then the resultant, on leaving the body at n, would have encountered the 
resisting force of the ground on which the stone stood, acting upon the body at that 
point. Ilad the direction struck within n, that is, between n and m. the stone would 
stand still more firmly; and the more firmly in proportion as 

it strikes nearer g, where the direction Ig of the gravity of the stone meets the base. 
The direction of a resultant may strike within the base; and the body remain firm, 
so far as regards overturning; but yet may slide. See Art 63; very important. 

This example shows also the necessity for assuming at times that bodies are rigid, or unbreakable. 
For in the case of stones of but little strength, the application of the great force of the resultant so 
near to n, would break the body at that point; and might, besides, mash n into tlio yielding earth on 













328 


FORCE IN RIGID BODIES, 


■which it stood. A knowledge of the direction of the resultants of forces acting on bridge abutments.- 
retaining-walls, &c. is therefore of use also bv enabling us to guard against such accidents, by select 
ing the strongest stoues for the most strained parts of the structure; as well as by adopting extra 
precautions iu preparing those portions of the earth foundations, upon which those parts rest. 

It must be remembered, however, in such cases as the foregoing, that with the exception of the poini 
o, at which the resultaut leaves the body, ad is not the direction which the resultant actually followi 
in the body ; but is oue which we may assume it to have, so long only as we assume the body to b<- 
practically rigid: that is, that it canuot be iu any way brokeu, beut, or have its form changed, b>> 
the forces actually imparted. Frequently we canuot safely assume a mass of masonry to be thus,, 
rigid, for it may be composed of many separate pieces merely placed in contact with each other 
without mortar, as in dry masonry ; or even if mortar or cement be used to uuite these pieces, it may 
not have time to set or harden properly, before the deranging forces are brought to bear upon it. It 
that case, although the resultaut might fall entirely within the body, aud within the base, thus de¬ 
noting perfect security to a rigid body; yet the structure might be completely destroyed by the 
sliding or other derangement of its parts among one another, under a force much less than would be! 
required to overturn it. On this account, if we wish to obtain security at the least expense, we must 
frequently trace the actual curved direction of the resultant through its entire course; so that wd 
may at every point of it place the joints of our masonry at right augles to it, as in Fig 7; or adopt 
other precautions to prevent the parts of the structure from separating. See Art 72. 

Rkm. 2. If in Fig 19 we suppose strong mortar or cement to exist between the base of the stone 
and a rigid foundation of masonry or rock, upon which we may assume it to stand, then it may nob 
be overthrown, although the direction of the resultant a d falls outside of the base m n. For then a 
third force, namely, the cohesive strength of the mortar, is brought to act upon the stone ; and the 
resultant of ail three forces may fall within the base. In all cases where a body remains at rest, not¬ 
withstanding that the resultant of the forces falls outside of its base, (whether the base be lior, vert, 
or inclined.) we may be certain that it is because some other force, which we have neglected, is acting , 
upon it at the same time; for when the direction of all the forces passes beyond its base, and is con¬ 
sequently force unresisted, the body must move. See Rem, Art 65; also see Art 72. When one body 
is thus cemented to another, the two become iu fact one body, so long as the cement does not give : 
way under the imparted forces : so that a problem which is one in Statics, if there is no cement, may i 
become one in Strength of Materials, when there is cemeut. The cement takes the place of natural 
cohesive force between the bodies which it unites. 

i 

Art. 36. When the number of 
forces in the same plane, whether 
tending to or from the same point 
or not, is greater than two, their result- , 
ant may he Ionnd in the manner! 
already given for two. Thus, with the*| 
tlifee forces b a, ca,oa, Fig 19J4, first find they 
resultant of any' two of them ; as, for instance, , 
the resultant na, of o a, and ca. Then consider) | 
on and ca as removed and na as taking tlxeiri | 
place; and then find the resultant to «,of n a and 
1 ab; then is via the single resultant that will ' 

i produce upon the rigid body W, the same effect' 

as the three forces oa, ca, ba. If the three , 
forces are imparted at diff parts of the body, proceed as in Figs 10^ and 10U". If 
the number of forces be greater than 3, the process is precisely the same; find 1st 
the resultant of two of them ; then the resultant of the 1st resultant and 3d force 
then the resultaut of the 2d resultaut and 4tli force; and so on to the end. 




X 



Fig 19^ illustrates a ease in 
which three forces a, b, and c, in 
the same plane, do NOT tend to¬ 
wards the same point. We may be- 1 

gin with any two of the forces at pleasure. We! 
will take b and c ; and. as at Fig 10J^, 
prolong them backwards to ft, and find their 
resultant hi. Then prolonging hi, and the 
third force a to meet at n, we lay off from n 
two sides ot the parallelogram equal respec¬ 
tively to a and to hi, and complete the paral¬ 
lelogram no. Then the diagonal on is the re¬ 
quired resultant, to be applied to the body 
J at e, as shown by xe. This resultant would 
then by itself produce upon the rigid body 
considered as a whole, the same effect as would 
the three forces a, b, c. Or if its direction were 
inverted, so as to pull, instead of push at e, it 
would become an antiresultaut to the three 
forces, and thus hold them in equilibrium. 

The same process applies to any number of 
forces. 












FORCE IN RIGID BODIES. 


329 


Art. 37. It sometimes happens, after having 
found the resultant of all the forces except the last 
one, that said resultant and remaining force are in 
the same straight line. Thus, with the forces u, v, iv, 

Fig 20; the resultant r of u and w, is in the same 
straight line with the last remaining force r; and of 
course no parallelogram of forces can be drawn which 
shall have v and r for two of its sides. When this 
happens, if v and r are of diff lengths, we have the 
case given in Art 16, of two unequal forces meeting 
in the same straight line; but in opposite directions 
along it. Consequently, the small one, and an equal part of the large one, mutually 
destroy each other; and the remainder of the large one is the resultant of the two. 



But if v and r are of the same length, then we have the case of two equal opposing forces, which 
mutually destroy each other entirely; and the body remains at rest. Consequently, there is no result¬ 
ant in this case; for no single force can have the effect of keeping a body at rest; but will always 
more it. In other words u, v, and w, are then in equilibrium. 


Art 3S. Tile polygon of forces. The resultant of any number of 
forces in the same plane; may he found by means of the polygon of forces, 
thus: Let a, b, and c, be three 
such forces; whose resultant R is to 
be found. Begin with any one of 
them, as a, and draw a', parallel, and 
equal to it; and place an arrow-head 
at the proper end of it, to show its 
direction. From this arrow-head, 
draw b', equal and parallel to b ; 
placing an arrow-head at its end. 

From this second arrow-head, draw 
c', equal and parallel to c; and so on 
with any number of forces ; taken in any order. Finally, from the arrow-head d of 
the last of the forces, draw a line A to the butt-end, n, of the first one; thus closing 
the figure; and place an arrow-head as on the others. Now, this closing line A, or 
d n, with its arrow, represents both in quantity and direction the antiresultant of 
i the three given forces; or if its arrow be reversed, it will represent their resultant. 
Consequently, we have only to draw from o a line, o R, parallel to A; and to make 
R equal to A; but pointing in the opposite direction. Then is R the resultant. 
This process will give different figures, according to which force 
we begin with; or whether we take the forces in right, or left-hand order; still, A 
will always come out the same in all of them. If the three given forces (or any 
greater number, as the case may be) had been in equilibrium with each other, that 
is had mutually destroyed each other’s tendency to cause motion, they of course 
could have no resultant, or single force that would produce an equal effect; because 
a single force, if the only one acting on a body, must produce motion. When this 
is the case the forces will of themselves form a closed polygon. In either case some 
of the lines may cross each other as do a' and c' at A, or not, as at N below. If the 
forces do not all act through the same point, see Art. 40, p 332. 

W r hen any number of forces are in equilibrium, 
any one of them is the anti-resultant of all the 
rest, because it keeps them all in equilibrium; 

Also, any number of the forces balance all the 
rest. 

If any number of forces, as a' b' c'd’ Fig 22, 

(whether they act through one point, as s, or 
not) are in equilibrium; and if we know the 
directions of all of them, .and the amounts of all 
but two ; then the polygon N will give us the 
amounts of those two. Suppose we know the 
amounts of a' and b' and require those of c' and 
d'. First draw by scale the known ones a and 
b (see polygon N) parallel to their actual direc¬ 
tions, and then complete the polygon with lines c and d , respectively parallel to c' 
and d'. Then the lengths of c and d will give the amounts of c' and d' respectively. 

Any diagonal across a polygon of forces, represents the resultant 

of all the forces on either side of it. 

If the forces are not all in one plane, the polygon is not in one 
plane, and cannot be drawn on a flat surface, but is twisted, or “ gauche,” as is 
o c t y, Fig 36, p 333. 




22 










330 


FORCE IN RIGID BODIES, 


Rkm. 1. It must not be inferred because forces balance each other, that therefore they balance the 
body to which they are imparted; for the body might move under the influence of other forces. Thus, 
the forces a', ft', c\ d\ may be supposed to be ttie balancing forces of several persons holding a body 
at rest iu a railroad car moving with great speed. Their forces prevent each other from giving motion 
to the body ; but do not prevent the steam force of the engine from doing so. It is only when all the 
forces imparted to a body, including its own weight, are iu equilibrium, that the body itself is also at 
rest Or if we hold a book, ruler, &c, vert between our thumb and forefinger; the opposite and 
equal pressures of the thumb and finger, hold each other in equilibrium, so that they cannot move 
the book either to the right or left, and the friction between them and the book, holds the gravity or 
weight of the book iu equilibrium, so that it does not fall. So that so far as these forces are concerned, 
they produce no motion in the book; but we can move it vertically up and down, by introducing 
the third force of our wrist; or hor by stretching out our arm; or by walking; none of which w ul 
interfere with the equilibrium of the other forces; they only prevent each other from producing any 
motion. * 

Rem. 2. A triangle being a' polygon of 3 sides, if any 3 forces which form a trianglo, 
be applied in one plane, to a body, and in directions parallel to the sides ot the tri¬ 
angle ; and tending either to or front one point; they tVill hold each other in equili¬ 
brium. And, vice versa, when we see a body kept at rest solely by the action of 3 
forces which are not parallel to each other, we may be sure that those forces are pro¬ 
portional to the sides of a triangle drawn parallel to them; that they are in one 
plane; that they all tend either to or from one point; and that any one of them 
acts in the direction of an antiresultant to the other two. Moreover, each of the 
forces is proportionate to the sine of the angle included between the other two; so 
that if we know one of the forces, we can readily find the others if we have the 
angles. This is very often of use in practice; as in finding hv calculation alone, the 
line of pressures through an arch ; the pres of earth against retaining-walls, Ac. It 
must be remembered that the wt of a body usually constitutes one of the forces to 
be considered as acting upon it. This rem Is very important. 

Ex. 1. Let a c. Fig 22J4, be a beam ; its foot rest¬ 
ing on of; and its head c merely leaning against s. 
smooth vert wall; and whether a c be unloaded ; or 
whether it supports a load placed in any manner 
upon it, or suspended from it; let the vert lino 
which passes through the cen of gray of the beam 
and its load (both of which are supposed to be 
known) be represented by pg. The beam audits 
load may be regarded as a single body, acted upon, 
and kept at rest, by three forces ; namely, its own 
gravity or wt; the force h, at c; and the farce/, at 
a. Moodier forces act on it. Now, gravity acts vert 
only ; and in the case before us it may all be re¬ 
garded as acting in the line p g. The force at c can 
act only at right angles to the surf or joint at that 
place, (see Art 19;) aud since the joint is vert, the 
force ft must be hor, or along ftp. The question 
now is, how to find the direction of the third force /. 
To do this we must avail ourselves of the principle 
that when three forces, not parallel to ea^h other, 
hold a body at rest, or iu equilibrium, as these 
three forces hold the beam a c, their directions all tend to or from one point; which is either at the 
cen of grav of the body ; or iu a vert line passing through said cen. Hence, since the vert direction 
p g of the force of gravity of the body ; and the direction ft p of the force ft, meet at p , therefore, the 
direction fp, of the force /, must also meet there. Hence we have only to draw a line fp, in order 
to find the reqd direction. A post intended to support the end a of the beam, should have the position 
fa; and the joint ot should be at right angles to fa; and not to a c, as might at first be supposed 
from Figs 13 and 16, in which the wt of the beams is not considered, and in which the extra¬ 

neous weight w is applied only at the joint, a. 

Having found the directions of the three forces in Fig 22)^, it only remains to find their amounts. 
To do this, we already have one of them given, namely, gravity, or the wt of the beam and its load; 
aud we know that they must be in proportion to the sides of the triangle drawn parallel to their di¬ 
rections. Consequently, if on the vert direction p g, we lay off by scale any portion whatever, as p d, 
to represent the force of gravity, then will the hor side of the triangle, p db, represent by the same 
scale the hor pres at c; and the side ftp, the oblique pres at a. The hor pres at the foot is equal to 
that at the head of the beam. It is of course included in the oblique pres /; which is compounded 
of said hor force, and of the vert force at a. The vert force is equal to the weight of the beam aud 
its load; none of which is sustained at c; nor can be, so long as the joint at the head and wall is 
vert. See Fig 51, p 552. 

Ex. 2. This is very similar to the preceding. Let a bij, Fig 22% he the half of 
any arch bridge, loaded or unloaded equally throughout; and of which we know in 
either case the total wt; and that the cen of gray of said wt is somewhere in the vert 
line# g. Now this half bridge is, like the preceding beam, kept in equilibrium, or 
at rest, by three forces only; namely, the wt; a hor pres //, at the crown, arising 
from the other half of the arch; and an oblique pres o, at the springing, or skew- 
hack./ i. To find the directions and amounts of these forces, from a draw a hor line, 
meeting the vert gg at c. From c draw a line to the center of ji ; this is the direc¬ 
tion of the oblique force o. From c measure down by scale any dist cs, on the vert 
direction gg, to represent the weight; and from s, draw s t hor. Then s t, measd by 
tlie same scale, will ho the hor pros //; and c t , the oblique one o. The joint j ?, at 
the spring of the arch, hears all of cs ; that is, all the wt of the half arch, and half 














FORCE IN RIGID BODIES. 


331 


load. No vert pres or wt is sustained at the center In of the arch ; nothing but the 
hor pres. But j i also sustains this hor pres, lor c t is composed ol c s and s t. 

The oblique force c t constitutes the total thrust exerted by the entire arch, 
against each of its two abuts; and the line ct shows the direction in which this 
thrust enters the abut at the skewbacky i. After entering at that point, it begins to 
curve downward, on the principle explained in Art 72. Since ct is the hypothenuse 
of a right-angled triangle, of which the leg cs represents the half wt; and st the 
hor pres of the arch, it follows that the total thrust c t of an arch may be found thus: 
add together the square o f half its wt; and the square of the hor pres ; and take the sq 
rt of the sum. This applies also to arches of iron or wood. 

The joints of any arch which is a portion of a circle, are 
usually drawn toward the center of the circle; and, practi¬ 
cally, this answers every purpose; but it is plain that strict 
theory would require the joint ij to be at right angles to co. 

So also the other joints of the archstoues would be reqd to 
be perp to the pres which they have to sustain. 

Moreover, since the pres ct, upon the joint ji, is much 
greater than the pres st, upon the joint In, theory would 
require the joint ji to be proportionally deeper than In; 
whereas in practice they are usually made the same, except 
in very large arches. See Stone Bridges. 

The last few lines of Ex 1, respecting the hor pres at the 
foot, and at the head, apply equally here. The young stu¬ 
dent should familiarize himself thoroughly with the princi¬ 
ple illustrated by these two examples, as it is one of very 
frequent application in practice; as in retaining-walls, abut¬ 
ments, &c. 

Here, as in the preceding example of the beam, we do not 
consider the strains produced along the length of the arch 
Inji itself; but merely the two forces which, acting at its center in, and at its foot ji, keep each 
Other, and the wt of the half arch and its load, in equilibrium. For the others, see Stone Bridges. 

Art. 39. A third mode of finding the resultant R, Figs 23, 
of any number of forces E, F, G, in one plane; and acting' through one 
point, x. 

Draw two lines, H H, and V Y, at right angles to each other. From their point of intersection o, 
draw lines by any convenient scale, to represent the directions and amounts of the forces. By Art 
34, resolve each of these forces into two component ones parallel to H H and Y V. Thus F o is re¬ 
solved into u o and e o; Go, into m o and to; E o, into to and n o. Then measure by the scale, and 
add together, those components to and to, parallel to H H, and which tend to move the point o 




H 


toward the left hand. Also add together those (in this case only one) no, which tend to move o 
toward the riy/it haud. Subtract the least sum from the greatest: their diff, equal to s o, will be the 
resultant of the two sets of forces which respectively tend to move o to the right, and to the left. In 
this instance, this s o must evidently be placed to the right of o, because the components on that side 
give the greatest sum. 

Next, add together those components 6o % no, parallel to W, which tend to move the point o vp• 
ward. In like manner add together those (in this case only one) components mo, parallel to V V, 
which tend to move o downward. Subtract, as before, the least sum from the greatest; their diff, 
equal to a o, will be the resultant of the two sets of forces whioh respectively tend to move o upward 
and downward. In this instance, a o must be measd off below o, because the upward tendency is the 
greatest. By this process, then, we first reduce all the original forces to two components, so and ao. 
This being done, we have only to complete the parallelogram of forces osca, and draw its diag co; 
which will be the final single resultant of all the original forces. From * draw xy parallel to co, 
and make 6y equal to co ; then Is by, or R, the reqd resultant; and 6 the point for imparting it to 
the body P, so that its effect may be equal to that of the three original forces combined. 






















332 


FORCE IN RIGID BODIES. 


Art. 40. Even wlien any number of forces in the same 
plane do NOT lend to or from the same point, the principle of 

the polygon of forces, or of Art 39, may be used in precisely the same manner as at 
Figs 21 and 23, for finding the length and direction of their resultant. Or it they 
are in equilibrium, and hence can have no resultant, they will still form a closed 
figure as A Fig 21. or N Fig 22, as well as if they acted through one point. 1 here 
will however be this difference, that when all the torces, as a, o. and 
c Fig 21. tend to or from one point o , we know that their resultant, as R, must he 
applied parallel to A, and must tend to or from that same point o. In other words, 
we know where its point of application must be. And so with 
the resultant co, Art 39. But when the forces do not tend to or from one point, and 
we find their resultant by Art 38 or 39, we know only its amount and direction; 
but do not know where to apply it. In such cases we may use Art 
36. Fig 19U. If the lines, representing any number of forces in one 

plane form a polygon, then we know that those torces, if they all act 
through one point, are in equilibrium; and we know that if they do not all act 
through one point they are either in equilibrium, and hence have no resultant ; 
or else they have two resultants, equal and parallel to each other, and acting in 
opposite directions; forming a “couple” (case 3, p 351) and causing the hod} to 
Vevolve around a point midway between them; while said point, and the body as a 


CAUTION. 

Page 333, lines 30 to 41. Model representing four forces. 

When there are hut three forces, as in Fig. 30, the model, being a 
triangular pyramid, does not admit of change of form, so long as its 
sides remain flat, lienee, such a model, if correctly made, necessarily 
represents the forces in their true relative positions. But where there 
are four forces, as in Figs. 34 and 35, the completed model is a 
/owr-sided pyramid, and can be made to assume different shapes by 
changing the inclinations of its sides to each other. Each such change 
in the shape of the model gives a different representation of the rela¬ 
tive positions of the original forces, and thus makes a difference in the 
position of the resultant, relative to the positions of the original 
forces; and, in most cases, a difference in its amount also. Hence, 
when there are four forces, we must know, not only the angle between 
each two adjacent forces, but also the angle between two planes formed 
by three of the forces; and must take care to so hold the model that 
its corresponding planes shall form that same angle. 


Art. 41. Forces In different planes; but tending to or from . 
the same point. Such forces cannot, like those in one plane, be correctly rep¬ 
resented together 011 one flat surf, such as a sheet of paper. 
Thus, let F'ig 27 bo a cube; and t x, cx, ix, three forces 
acting in the directions of its edges; and all tending to 
the same point x. It is plain that the relative positions 
of these forces are not correctly represented; for txc,txi t 
and cxi, are in reality right angles; whereas, in the fig, 
txc appears to be an acute one; cx i, a right angle; and 
t x i an obtuse one. 



On this account the resultant of such forces cannot be had by 
measurement from a drawing. Recourse must therefore be had to 
calculation: which, however, will be facilitated by a drawing. The 
theoretical principle is very simple ; being, in fact, the Same as when 
the forces are all In one plane; namely, first find by Art 28 the re¬ 
sultant of any two of them, rfor any two are really In one plane;) 
then find the resultant of this resultant and the third force; and 
so on to the end. It is easy to find the first resultant; but the 
others are more troublesome. Instances are comparatively rare, in 
which the resultants of sueh forces are ieqd 10 be found: the attention of the engineer being gen¬ 
erally confined to those in ona plane; as When proportioning bridges, roofs, retaining-walls, 610 . 












FORCE IN RIGID BODIES 


333 


Art. 42. To find the resultant of forces In different planes, 
but all tending through one point. 


In cases where mathematical accuracy is not necessary, and the number of forces 
only three, or four, the writer will venture to propose a method by models; which, 
if open to the objection of empiricism, has the advantage of requiring less time than 
other processes; is sufficiently correct for most practical purposes; and shows the 
resultant in its actual position, which is done by no method of calculation. 

Let <io, 1> o, co. Fig 30, be the three forces, meeting at o; their angles with each other, aob, hoc, 
Co a, (which alone are necessary in this method,) being of course known. Draw on pasteboard the 



three forces ao, bo, co, as in Fig 31, with their actual angles a o b, b o c. c o a. By Art 28, draw 
the pai'allelogram of forces for the middle pair bo, co: and draw its diag w o, which will be the re¬ 
sultant of those two; leaving the resultant of it, and ao, yet to be found. Cut away neatly the 
whole fig, a o a cw b a. Make deep knife-scratches along ob. o c, so that the two outer triangles may 
be more readily turned at angles to the middle one. Turn them until the two edges o a, o a, meet; 
and then paste a piece of thin paper along the meeting joint, to keep them in place. Stand the model 
upon its side o b w c as a base ; and we shall have the slipper shape uobw, Fig 32 ; o w being the sole, 
and aob the hollow foot. 

We now have the fipst resultant w o, and the third remaining force a o, in their actual relative, po¬ 
sitions. Now, to find their resultant, also in its actual position, cut a separate triangular piece of paste¬ 
board of the size and shape of woo. Find the center i, of the edge wo, and draw a line to on each 
side of it. Finally, by meaus of the edges a o, w o, paste this piece to (he inside of the model, along 
its center-line wo. This done, to represents one half of the reqd resultant, in its actual position. 

The reason why it represents but one-half of it is plain ; for, as be¬ 
fore stated, we now have ao and w o in their actual positions in the 
model; consequently, if we complete the parallelogram of forces 
too an, and draw its diagonal no. this last will be their resultant. 

But since the two diags of every parallelogram divide each other into 
two equal parts, the diag aw thus divides the resultant: consequently 
i o is one-half the resultant. 

If there be four forces, as an. bn, cn, dn, Fig 34. draw them as in 
the fig, with their actual angles anb, b n c, &c. Draw also the re¬ 
sultants n v, of a n and 6 n ; and n w, of n c and n d. Then cut out 
the entire fig, as before; and paste together the two edges an, an. 

Then we have the two resultants av, aw, Fig 35, forming two simple 
forces, in their actual relative positions; and we have only to measure 
their dist apart from t> to w; and thence find their resultant ar, 
which w'ill evidently be that of the four original forces. 

Or, as in the preceding case, cut out a separate piece of pasteboard, y^V’ 1]r.'-V’ - 

avw. Fig 35, aud having drawn on each side of it a line from a to the _ 

center o of vw, paste it inside of the model. Then will a o represent \ 

one-half of the resultant of the four forces, in its actual position. W 

Should the model be exposed to hard usage by workmen, it should be 'lit O Ti 

made of wood ; the triangles anb. b n c, &c, being cut out separately ; J’lO 

the joining edges bevelled ; and then glued together. See also Art 43, 



i. 


Art. 43. Tlio parallelopipcd of forces. If any three forces, ao, bo, 
co. Figs 36, in diff planes, meet at one point o, whether they all be strains, or all 
motions, their resultant or joint ef¬ 
fect will be represented, both in c\___j, cl 

quantity and in direction, by the 
diag v o, of a parallelepiped acth i, 
of which three converging edges 
may be assumed to represent the 
three converging forces. 

This suggests another mode of showing 
the resultant of three such forces by a 
model; for it is only necessary to prepare 
a box A or B, as the case may be; and y o 
will represent the reqd resultant. 


-v-i 

7N 


A 























334 


FORCE IN RIGID BODIES. 


No three forces in different planes can be in eqnilibrlnm. 

Art. 44. Forces in different planes; and not tending- to or 
from one point. It is but rarely that such forces have a resultant, or anti- 
resultant; that is, no single force can usually be found either to produce an equal 
effect, or to balance them. It is so seldom that they present themselves to the engi¬ 
neer’s attention, and their solution is so tedious, except in very simple cases, that 
we shall confine ourselves to one of that kind. As in Art 41, the resultants cannot 

be had by measurement from a drawing. 

Let ao, ao, a o, Fig 37, be three such forces; aud suppose 
them all to act against (see Remark, Art 8) the same plaue 
ppp\ and against the same side, or surf of it; that is. none 
of them pointing upward against the uuder side of the plaue 
in the fig. Having the points o ot application, aud the rela¬ 
tive positions of the forces themselves, as well as the angles 
aoc which they form with the plane ppp, resolve each of the 
forces into two components; one of which, c o, coincides with 
the plane ; while the other (parallel aud equal to a c, but meet¬ 
ing co at o) is at right angles to c o, or to the plaue. We then, 
have two sets of forces ; one set in the plane, and the other at 
right angles to it. Since those in the plane do not tend to or 
from one point, their resultant must be found by Fig 19}^, 
while that of the several parallel components (equal to a 
c, but applied at o) may be obtained by Arts 56 and 59. 
These two resultants will rarely be in the same plane with 
each other, and consequently can have no joint resultant. If 
they should chance, however, to fall in the same plane, use 
Art 28 for finding their resultant. In simple cases, where the 
forces act against one plane, as in our fig, pieces of wire, cut to lengths to represent the forces: and 
stuck iuto a piece of smooth board, in their proper relative positions, will greatly facilitate the find¬ 
ing of the resultant approximately enough for most practical oases. 

The same geueral process must be used, no matter how great may be the number and directions of 
the forces. A plane must be assumed to pass somewhere through the system : and the directions of 
all the forces must be conceived to be so extended as to terminate at points of application in said 
plane. Kach force must then be resolved into two, as iu the foregoing example; and the resultants 
of the two sets of forces, as well as their joint resultant, if they have one. must he found as before. 
If any of the forces should he parallel to the assumed plane, but not in it, it evidently cannot be re¬ 
solved into two, one of which shall be in the plane; for (Art 32) no force can have one of its compo¬ 

nents parallel to itself. Hence, in such a case, the resultant cannot be found by this process. 

Art. 45. It is comparatively seldom that strict mathematical accuracy is reqd 
in finding the resultants of forces in engineering practice; therefore, the foregoing 
easy methods by measurement from a drawing, or model, w ill usually answer every 
purpose. Moreover, they appeal to the eye; and are therefore much less liable to 
serious errors than methods involving numerous calculations. But when more cor- 
rect results are needed, they may he had by means of a table of nat sines, tangents, 
A 1 "' 8 - m , the case °f two components and their resultant, calling the components! 
Fig 38, C and c: and the resultant R, then 




If the angle m b n between the conipo* 
no nts C . c is 90°, R will = /c* + c 2; and C = 




y IV“ —u- , ttiju c = v av *— L 

of a b n = c -r- cosine of a b m. And C will = R x“ C c 
sine of a b n ; and c = R X cosine of a b m 

Or whether the angle between theeoin 

P°i’’ < on tS ^ a,, ‘ l c b(1 90 °» or niore, or less, a 

In r lg 89, 


C will = 


R X sine of v r x sine of x 

sine of C be ; and c = sineof C be * 



Observe that v is used for finding C: and x for find¬ 
ing c. 


And R will = e x 81De of Cb c 1 C X sine ofCbc 
sine of x or sine of v 

an 8 le or either of the others exceed: 
90°, subtract it from 180°, and use the sine of th< 
remainder. 





























FORCE IN RIGID BODIES, 


OOk 

OOO 


Art. 46. Moments. Leverage. If a b, 

Fig 40, represent any force acting in any direction 
whatever; and if o, or i, be any point whatever, 
whether in or out of the body on which the force 
is acting; and if from said point a line o s, or i c , 
be drawn at right angles to said direction of the 
force, then said line o s, or i c, is called the arm, 
or leverage of the force a b, about said point. 

And if the amount of the force in lbs, <fcc, be mult 
by the length of the arm or leverage in feet, &c, 
the prod in ft-lbs, &c, is called the moment of 
the force about that point. Thus, if the force a b 
be 8 lbs, or tons; and the line o s, 6 feet, then the 
moment of a b about o is 8 X 6 = 48 ft-lbs; or ft- 
tons. A force has of course no moment about any point through which if passes. 

This moment represents the total tendency of the force to produce motion about the given point. 
We cannot hold hor. between the ends of a thumb and forefinger, a piece of stick a foot long, which 
has a 3 ft wt at the other end of it; because the tendency of the wt to produce motion is too great for 
the force of our fingers to resist; but we can in that manner hold a stick two feet long, with a 3 ft wt 
at each end, if we take it at the center. For although in this case there is twice as much moment as 
before exerted at our fingers; yet it is not now exerted against them ; because we now have two equal 
moments in opposite directions, reacting against each other; and leaving nothing for the fingers to 
react against, except the mere vert wt of 6 fts. 

Siuce the moment about o tends to produce motion at that point in the direction in which the hands 
of a watch move, or from the left hand, toward the right, it is called a right-hand moment. But the 
moment of the same force, about the point i. tends to produce motion at that point, from right to left, 
as shovvu by the arrow-head on the small circle ; hence it is called a left-hand moment. The moment 
of the force d y, with its leverage y i, about the point i, is a left-hand one; as is also that of x to 
with its leverage e i. 

When the arm o s, or i e, instead of being merely an imagined diet, is a rigid bar , at one end of 
which, as s or c, the force is imparted ; thus giving the bar a tendency to move around the point o or 
i as a fixed center, it is frequently called a lever; and the point o or i, the fdlcrum of the lever. 

If the lever, instead of being like c t, at right angles to the direction a m of the 
force, should be oblique to it, as in i m , or i a ; or should be curved, or bent in any 
way, as igt; this in no way affects the leverage, or moment of the force; for the. 

leverage is always tiie p«>rp (list, or in other words, < Rao shortest <list 
from the fulcrum. to ll»e direction of the force: and is entirely 
independent of the length of the lever itself. This is a grand funda¬ 
mental principle of all levers, and leveraarcs; and the young 
student should carefully impress it upon his memory, inasmuch as it is of constant 
application in practice. 

The fulcrum is not always at one end of the lever, but may he between the two 
ends ; so that there are two arms. Cog-wlieels are merely continuous circular levers, 
with the fulcrum at the center. 

Art. 47. As a further illustration,let 
afb, Fig 41, be a bent lever, turning on 
its fulcrum /; and m and n two wts sus¬ 
pended from its ends, constituting two 
forces acting in the vert directions a n, 
b c. Now,/c, at right angles to the di¬ 
rection be; and fa, at right angles to 
the direction a v, are the arms, or lever¬ 
ages of the forces m, and n, about the 
point f. 

In the fig these leverages are equal, say each 
is 6 ft; and let each wt. wandti.be 100 fts; then 
the right-hand moment of m. and the left-hand 
one of n. about /, are each 6X190=600 ft-fts ; and 
since both the forces, and /. are all in the same 
plane, it is evident that the two opposite mo¬ 
ments, balance each other; although the lever 
fb, is much longer than fa. 

Now suppose the wt w to be removed ; and that instead of it. a person pulls in the direction 6 s, by 
means of a string fastened at b. With what force must he pull in order to balance the wt n? First 
measure the leverage ft, from the fulcrum, and at right angles to the direction 6 s of the new force; 
and suppose it is found to be 9 ft. Now we have already found the moment of n about / to be 600 ft- 
fts ; and we require the same moment on the opposite side; so that all that is reqd is to find what 

number the 9 feet leverage must be mult by, in order to make 600. This is plainly —— = 66.66 fts, 

for the pull which the person must exert; because 9 X 66.66 — 600. 

So it is seen that with the same length of lever, fb, we can have diff powers, (so called,) or lever - 










336 


FORCE IN RIGID BODIES, 


ages, according to the direction in which we apply our force to the lever. This, however, evidently 
has its limit; for the greatest power is gaiued (to use the popular expression) when we apply our 
force in the direction by, at right angles to a line/6, drawn from the fulcrum to the outer end of the 
lever. If we apply it in the direction b d, we get ouly the leverage /It. 

On the same principle as in the foregoing example, if o t and ga, Fig 
42, be two beams of equal scantling, but of diff lengths; with one end 
of each firmly fixed in a vert wall, and both sustaining equal suspended 
wts w, x ; the moments of the wts about the points o and g will be equal, 
because the arms or leverages o e, y a, are equal. Therefore the wt to 
will have no more tendency to break off the long beam at o, than x has 
to break the short one at y. The wts of the beams themselves are uot 
here taken into consideration ; and this is always the case in speaking 
of levers, unless otherwise expressed. In very many cases, the wt of 
levers of two arms does not affect the result aimed at, provided the arms 
are so proportioned as to balance each other when unloaded ; no matter 
wbat their comparative lengths or wts may be. 

If, in Fig 42, we apply pulls to the beams, in the parallel directions 
tm, c n, at right angles to ot, then the leverages become changed from 
o e and g a, to ot and g c ; and since o t measures 6 times the length of 
g c. it follows that the beam ot would be broken off at o, by A part as 
much force in this new direction, as g a would; for the leverage being 
6 times as great, must be mult by only as much wt, in order to have 
an equal breaking moment. 

Art. 48. In ordinary phraseology, the load, or resistance of any kind, which 
we wish to move, overcome, or balance, by means of a lever, is called the weight.; 
while the force of whatever kind which we apply to accomplish this, is called the 
power.* Usually, but not always the power is applied to the longer arm. 
Equilibrium, or balance, or equal moments in opposite directions, will then plainly 
take place, when the long leverage (not lever) has the same proportion to the short 
one, that the wt has to the power; because then only can the long leverage mult by 
the small power, have the same moment as the short leverage mult by the great wt. 

This is seen in the common steelyard, Fig 43; which is merely an iron lever, turning on a fulcrum 
/, and having the wts of its two arms fa.fb, so proportioned as to balance each other wheu un¬ 
loaded. Here the power, FI, at the dist of two divisions from the 
fulcrum; balances the wt, W 2, at the dist of one division. If the 
wt W 2 were suspended at y, only half a division from /, it would 
balance the power P 1, suspended where W2 is in the fig, at a whole 
division from/. If the power is reqd to move, or overcome the wt, it 
is plain that either the power itself, or the length of its arm, must be 
greater than when mere equilibrium is to be effected; in other words, 
besides the two straining forces, which by their mutual action balance 
or equilibrate each other, we need some unresisted force to impart 
motion to the inert matter. 

In the two levers of the same length. Fig 44, the leverage fw of the 
wt w, is of the same length in both; namely, one division; but the 
leverage fp. of the power p, is but two divisions long in the upper oue, 
and three divisions in the lower. Therefore a power of 1 lb will balance 
only a wt of 2 lbs in the upper one, and of three lbs in the lower. In the 
upper one, the power will move twice as fast as the wt; in the lower oue, 
three times as fast. When the fulcrum is between the wt and the power, 
as in the upper one, the lever is said to be of the first class; when the ful¬ 
crum is at one end, aud the power at the other, second class; fulcrum at 
one end, and wt at the other, third class. In all cases it is assumed that 
the fulcrum is in the same plane (Art 8) with the directions of both the wt 
aud the power; otherwise the principles do uot apply. When two weights 
balance each other on two arms of a lever, as a steelyard, or common 
weighing scales, &e, their directions are the vert lines passing through 
their centers of grav ; and the same imaginary vert plane which coincides 
with those directions, coincides with, or passes through, the fulcrum also. When this is not the 
case, no equilibrium can exist. This may be readily proved by experiment; for we cannot balance a 
bow shaped piece of stick or wire, so long as the bow is hor; for it will turn on the fulcrum, of its 
own accord, until the bow becomes vert; so that the same vert plane that passes through the fulcrum 
shall pass also through the cen of grav of each half of the bow. If all the forces acting on the lever 
are hor, or oblique, the imaginary plane must be so too. 

Rkm. From what has been said respecting the lower Fig 44, it follows that when a load w Is home 
at any point of a beam /p supported at both ends, then the portion of the load supported bv each of 
these points/and p. is in the same proportion to the whole load that the respective portion's f w and 
p w of the beam, are to the whole span fp ot the beam ; but inversely ; that is, the smallest portion 
of the load is borne by the support at p ; and the largest portion bv /. In the fig, fw is one third 
and p w is two thirds of the clear length or span of the beam ; hence, p supports % of the load to; 
and/supports %. Or as/p j to j j/to : load atp- And as/p : to ;: p to : load at/. 


* The fact that by means of leverage a small power can be made to move a great wt, is in common 
parlance styled a train of power. In a scientific sense the expression is absurd, yet in practice it 
has by its universal use become very convenient, and we shall therefore employ it. When the lever, 
instead of merely balancing the power aud the weight, has to be put into motion, it is plain that there 
must be some excess of force applied at the power end, to produce the motion. 





















FORCE IN RIGID BODIES. 


337 


Art. 49. This example is the same as in Fig 19, 
where the question is solved independently of leverage and 
moments. 

Let S be a stone of 3 tons ; standing on a hard hor base n 
m ; let i be its cen of grav ; and let the dist ng from the toe 
n, and at right angles to the vert direction, ig, of the gravity 
of the stone, be 2 feet. Also let f h be a force of 2 tons, im¬ 
parted to the stone at h; and let the dist no from the toen, 
and at right angles to the direction fa of the force fh be 5 
ft. Will the fore efh upset the stone around the toe n, as a 
turning point? 



Here ng, or 2 ft. is the leverage of the force (3 tons) of gravity; consequently, the moment of the 
stone about the poiut n, is equal to 2 ft X 3 tons rr s foot-tons; and this moment (which is called the 
moment of stability of the stone, about n) alone tends to prevent the stone from overturning about 
n. Again, n o, or 5 ft, is the leverage of the force, (2 tons,) of /A; consequently, the moment of fh 
about the poiut n, is equal to 5 ft X 2 tons — 10 foot-tons ; and this moment alone tends to overturn 
the stone about n. Since the overturning moment is the greatest, the stone will of course upset. 
The foregoing case resembles that of an abutment resisting the thrust of an arch; or that of a re¬ 
taining - wall, sustaining the thrust of earth against its back ; said thrust being supposed to be con¬ 
centrated at its center of pressure. (Art 57.) It is analogous to a 
simple bent lever e/o, Pig 25%, supported at its fulcrum/; 
around which it may revolve. The short arm /e, of 2 ft, is acted 
upon by the 3 tons wt of the stone; moment = 2 X 3 = 6 ft-tons. 

The long arm fo, of 5 ft, is acted upon by the 2 ton force A; mo¬ 
ment 5 X 2= 10 ft-tons; which, being greater than the 6 ft-tons 
of the stone, equilibrium cannot exist; and motion must ensue. 

The wt of the body S, in Fig 25, constitutes one of the forces act¬ 
ing upon it; while its inert matter constitutes the two lever arms 
at Pig 25%. It frequently thus happens that a body is at the 
same time the resistance to be overcome; and the lever with 
which to overcome it. It is plain, that in the same manner as 
above, we may lind separately the moments of any number of 
forces acting in the same plane, upon a body ; and may afterward 
ascertain their united effects, by adding into one sum those which 

tend to overturn it; and into another sum, those which tend to prevent its overturning; the diff 
between these sums will be their joint effect. 



Rem. In Fig 25 we have supposed the body S to turn about a single point, n; 
but in practical cases they turn about edges, as d y , Fig 25%) the assumed turning 
edge of the body B ; which may be re¬ 
garded as a retaining-wall, or abutment, 

&c. In making calculations for the 
strength of such structures, it is usual 
to restrict ourselves, for convenience, to 
a supposed vert slice, one foot thick, 
like the shaded end of the fig; in which 
case the turning edge is one foot long 
instead of extending along the toe, dy, 
of the entire structure; (see Art 70.) 

This, however, causes no change in the 
calculations, which remain the same as 
for Fig 25; for we suppose all the wt of 
the slice to be concentrated at its cen 
of grav; and the forces to be imparted 
in the same vert plane with the direction of the gravity ; so that it amounts virtually 
to the same thing as if we assumed our one-foot slice to be infinitely thin ; but still 
to have the same wt as if it were one foot thick. We of course restrict ourselves, 
also, to the forces acting upon such 1 ft slice. 

But in fact, it is not absolutely necessary in such cases, to suppose our applied forces to be in the 
same vert plane with the gravity of the wall, provided that both our structure , and the base against 
which the edge d y bears, and revolves , may be regarded as practically rigid, or unyielding , under the 
action of those forces. For, if there be no yielding whatever along the edge dy. then it is immaterial, 
so far as regards overturning, whether the force be applied at o, or at t ; for d y then becomes analo¬ 
gous to the rigid axle c i, Fig 25%, with two lever-arms s e, and c a. The wt r, may be supposed to be 
that of the structure; and from that common machine, the wheel and axle, we know that it is imma¬ 
terial whether the applied force, and other lever-arm, are attached to the axle at c; or at any other 
point along its length ; so long as the axle is equally unyielding at every point. Art 53 will perhaps 
make this more clear. 



Caution. Although it is immaterial, as just explained, as regards overturning, 
whether the force, Fig 25%, be applied at t or at o ; it is plainly not immaterial as 
regards a tendency to swing the body around horizontally, or as regards pressures 
(and consequent friction) at the ends of the stick, Fig 25%, or in the bearing //, 
Fig. 45. 





























338 


FORCE IN RIGID EODIES. 




m 


Art. 50. Equilibrium of forces, and of moments. The stu¬ 
dent must distinguish clearly between those cases in which forces hold each 
other in equilibrium, and those in which the moments of forces do so. Thus, if 

two equal forces n and R, Fig Y, act against each 
other in (he same straight line n R, but in op¬ 
posite directions, we correctly say that these two 
forces are in equilibrium,or balance each other, 
or prevent each other from giving motion to the 
body. Rut also in tbe case of a lever, one arm of 
which is say 10 ft long, and the other only 2 ft, we 
usually say that 2 lbs of force at the end of the 
long arm, will balance or hold in equilibrium 10 
lbs at the end of the short one. But this is not 
scientifically correct; for a force of 2 lbs cannot 
possibly balance one of 10 lbs. It is actually the 
moment of the 2 lbs that balances the mo- 
ment of the 10 lbs. That is, 2 lbs X 10 ft leverage = 20 ft-lbs, the moment of 
the 2 lbs, balances the 10 lbs X 2 ft leverage = 20 t't-fbs, the moment of the 10 lbs. 
As to the two forces 2 and 10, they are balanced by the upward 12 lb reaction of the 
fulcrum. 

Rem. When any number of forces as m, n, o, j>, Fig Y, in the same plane, 
whether acting through one point or not, hold each other in equilibrium, then any one of them, as n, 
is equal to the resultant R of all the rest; aud will be in the same straight line with it, but will act 
in the opposite direction. And this is the most ready method that suggests it¬ 
self to the writer for (letcrmiiiin^ whether several given forces 
are in equilibrium or not. Art 30 may be used for finding the 

required resultant R. See Art 40. 

Art. 51. Equality of moments. This principle consists in the follow¬ 
ing : If any number of forces as a, b, 




c, d , Fig 26, all in the same plane, and 
acting upon a body N, in any directions 
whatever in that plane, hold each other in 
equilibrium, then if any point i be taken 
in that same plane, whether within the 
body, or out of it. the moments of 
the forces will hold each other in equi¬ 
librium around that point; that is, the 
moments of all those forces (6 and d) 
which tend to turn the body N in a right 
hand direction, (or like the hands of a 
watch,) around the point i, will together 
be equal to the moments of all those (a and c) which tend to turn it around i in a 


If the four forces a, b, c,d , mutually prevent each other from imparting to the 
body N, any tendency to move, as a whole, in any straight direction whatever, they are themselves 
in equilibrium: and this being the case, their right and left hand moments 
around the point i will also be found to be in equilibrium, as shown below. And so also with anv 
other point in the same plane. But in that case the arms or leverages would be changed. 1 


Forces. 

Arms. 

Right-hand Moments. 

Left-hand Moments. 

a=6 

3.9 

.... 

23.4 

c = 3 

5.8 

. . . . 

17.4 

b —4 

4.6 

18.4 


d = 7.2 

3.111 

22.4 

• . . . 



40.8 Total. 

40.8 Total. 


But moments arournl a point may hold each other in equi¬ 
librium even when the forces themselves do not balance 
each other; as in the case of the lever in Art 50. 


Rem. If any number of forces as a, b, c, d, Fig- 26. in the same 
plane, whether acting through the same point, or not hold 
each other in equilibrium, they will do the same if tliev 
all be made to act in the same directions through one 
point ; as A. B, C, D, through p. one 


But it does not follow, that because forces may balance each other when applied 
will do so if applied to a body N at any points whatever, in the same plane. 


at one point p, 


they 




















FORCE IN RIGID BODIES. 


339 


Art. 5—. Virtual velocities. Whenever the power and the wt balance 
each other, either in a single lever, or in a connected system of levers, or leverages 
ol any kind whatever, then if we suppose them to be put into motion about the ful¬ 
crum, their respective vels will be in the same proportion or ratio as their leverages- 
that is, it the leverage of the power is 2, 5, or 50 times as long as that of the wt'Yhe 
power will move 2, 6, or 50 times as fast as the wt. Therefore, by observing these 
vels, we may determine the ratio of the leverages. The wt and the power are to 
each other, therefore, inversely as their vels, as well as inversely as their leverages- 
and this is based upon the principle of virtual velocities ; and is very important. * 

Art. 53. Neither the amount, nor the effect of leverage is changed, if the arms 
ot the lever, (whether straight or crooked, or in whatever relative positions they may 
be.) instead of being in one piece, and supported by a single point or edge as a ful¬ 
crum, as in Fig 44, p 336; should consist of two separate pieces m, n, Fig 45, firmly 
united to a straight rigid axle a x, 
of any length; (usually placed at 
right angles to the levers m,«,) and 
supported at two or more points 
f,f. The moments are then about 
the axis, or longitudinal center-line 
of the axle; any point of which 
may be regarded as the fulcrum. It 
is immaterial at what points along 
the axle, the lever-arms m, n, may 
be attached to it; both may be be¬ 
tween f, /; or both outside; or 
one in each position. But see 
Caution, p 337. This is illustrated by the common wheel and axle. The 
rad of the wheel, and that of the axle, measd from their common axis, constitute 
two continuous levers. Also by series of cog-wheels, which we see placed indiffer¬ 
ently at any points along extended shafts, or axles, whether vert, hor, or inclined. 

If the levers m,n, are not at right angles to the length of the axle, then their lever¬ 
ages, and not their actual lengths, (measd from the center line of the axle,) must evi¬ 
dently be used in calculating the moments of forces acting upon them. See Remark, 
Art 49. 



Rem. The assumption of an imaginary axle, or 
cases as this. The entire load being known which i 
tion of its cen of grav; to find how much of it is 
sustained by each of the supports: Draw by scale, 
a triangle a b c, Pig 46, showing in plan the correct 
position of the supports; and of the cen of grav g, 
of the sustained load. Assume any side, as a c, to 
be an axis ; and from it draw the two perps : ib, to 
the opposite angle b ; and e g. Now, it is plain that 
e g, and ib, »iay be considered as two leverages 
from the supposed axle a c. At g is placed the en¬ 
tire load, whose moment about the axis of the axle, 
is equal to the load, (say 5 tons,) mult by the meas¬ 
ured (by scale) length of e g, say 8 ft. And this 
moment (5 X 8 = 40 ft-tons) is balanced by that 
of an upward force at b ; which upward force must 
he equal to that share of the load which rests upon 
b. inasmuch as th« two are in equilibrium. To find 
the amount of the force at b, we have only to div 
the moment (40 ft-tons) of g, by the leverage t b. of 


axis, as in Rem to Art 49, enables to solve such 
s sustained by three vert supports; also the posi- 



reqd force l. Suppose i b is found by the scale) 


to be 25 feet; then the upward force at b, is = 1.6 tons ; and we have its moment about the axis 
= 1 6 X 25 — 40 ft-tons, the same as that of g , J Therefore 5 supports 1.6 tons or the load. Tn pre¬ 
cisely the same manner we may assume each side in turn to be an axis ; and find how much is sus¬ 
tained at the opposite angle. The pressure on each of four legs cannot be calculated. 

The Fig T, in which a represents one end of the axis ; a g the leverage e g of the load p; and a s 
that (t b) of the force b, will make the principle more apparent. 




vv 

JQ 


a 


t- 


¥ 


Art. 54. Ex. 1. The condition of a beam 
i b. Fig 47, may often be examined on the 

9 rincig>le of a lever. Suppose it to lie of uniform 
leptb and thickness; its length a b. in the clear between its supports. 

0 ft; its wt 600 lbs ; and its position hor. In this case we know that 
ne-half its wt. or 300 fbs, is borne by each support. To prove this, we 
nav consider its entire weight to be concentrated at its cen of grav, 
ehich in this case will be at its center t. See Arts 56, 57. Then we suppose •—■ 

me of the supports, as o, to he removed ; and an upward force/to be 

icting upon a lever a b, 20 ft long, without wt; and sustaining a load of 600 tbs to their 

rum a. Since the force and the load both act vert, and the beam is said force* and 

lirections. therefore the dist a t and a b from the fulcrum, are the true — ftnoo ft lbs • and to 

oad. Now, the moment of the load, about the fulcrum, is 600 lbs mult by 10 ft - 6000 ft-Bs, aud to 


Fid 47 f 





































340 


FORCE IN RIGID BODIES 


6000 


— 300 lbs, or one-1 alf of the wt of the beam, 



balance this the upward force/must be equal to 

th ff, P » r n addition to its own wt, the beam had actually sustained a load to at its center, we must add 
this load to the wt of the beam ; and then proceed as before. So also if it sustains a load unifoimly 
distributed over its length. 

Ex. 2. If the cen of grav of the beam be at any point 

y Fig 48, not at its center, we use the leverages a y and a s, instead of 
a t aud <i b , and if in addition it sustains a load z at any point n what¬ 
ever, we first find as before the force reqd at F for the beam alone; aud 
afterward, by usiug a n, aud as as leverages, we fiud the force reqd by 
the load; aud add the two forces together for the total F. 

Rem. To find the portions of z borne by a and by s 

say, as the whole span s a is to the whole load z , so is n a to the load 
on s; aud as s a is to z so is n 8 to the load ou a. 

Ex. 3. If the beam sustains several loads at diff points, 

as in Fig 49, calculate for each of them separately, using the leverages 
a i, a c, a o, &c ; and add all together for total F. For portions of each 
of these loads borne by a and v see above Rem. 

Ex. 4. If the beam in any such case, is inclined, as in Fig 50, the 
hor dist a o, a g, &c, must be taken as measured from the fulcrum a, 
instead of of, a i, &e; because, since all the forces are vert indirection, j 
only a hor line can be at right angles to them, and serve to measure j 
their leverages from the fulcrum o. If the beam be rigid, and its 
ends cut hor, as shown in this fig, it will have no tendeucy to slide; 
because all the forces which through it are applied to the bodies m 
aut p are vert; aud since the joints are at right angles to those ! 
bodies at those points, the entire forces will be imparted also; no por ; 
tion of them remaining unresisted, to act us motion, so long as the 
beam remains rigid, and consequently straight. But if it bends un- I 
der either its own wt, or that of its load, new forces come into action, 
which will tend to push the supports outward from each other; so 
also in the foregoing cases. 

It is only where we may practically regard a beam as rigid, or uncnangeable under the forces, that 
the foregoing concentration of entire weights or forces at the cen of grav, can be safely assumed. 

It will uot apply when we are investigating the strength, and deflections of beams; see Art 58. 

After having thus obtaiued F, in any of these cases, or in other words, having found how much 
of the entire wt of beam and load bears upon one support, we have only to subtract it from the eutire , 
wt, to obtain that on the other support. It is plainly immaterial which end of the beam is assumed 
to be the fulcrum in any of these cases. 

Ex. 5. Let a o, Figs 51, be a hor beam 10 ft long, projecting from a vert wall a c ; and resting at one 

end on a step a; the other end being sustained by either 
a strut, or a tie o c, 12M ft long. The beam, and its uni- | 
form load, weighing together 3 tons, what will be the push- . 
ing strain aloug the direction of the strut; or the pulling 
strain aloug the tie ? Draw the Fig to scale; and meas- ; 
ure a i (which will be found to be 6 ft) at right angles j 
to c o. Now, the weight of a rigid body, when considered i 
only with regard to its effect in moving the entire un- ! 
altered body, or in straining it bodily against another j 
body, acts the same as if it were all concentrated at its \ 
oen of grav; and since we are now about to consider it < 
in that light, and not as tending to bend or break the ! 
beam a o (in which case only half its uniform load, and 
wt must be assumed to be concentrated at its cen of grav; 
we consider the 3 tons wt to act at g , 5 ft, or half the I 
length of the beam from a. Now, the 3 tons, being a force 
of grav, will act in a vert direction; and since the beam 
is hor, a g is at right angles to this direction of the force 
exerted by the beam and its load. Consequently, if we 
assume the beam to be a lever, movable about a, as a fulorum, a g is the leverage of that force of 
grav ; aud the moment of that force about a, oonsequeutly, is 3 X 5rr 15 foot-tons. But this moment I 
is reacted against by that of another foroe in the direction c o; which aots at the point o of the levet 
o o, to uphold the bdam and its load. The leverage a i of this force, that is, the dist from the fulcrum 
a, and at right angles to the direction c o of the foroe, has already been found to be 6 ft; consequently, ! 
the foroe itself, in order to have a moment of 15 ft-tons a boat a (as the beam and its load have) must 
15 

evidently be = 2.5 tons, the reqd strain along the strut, or along the tie, o c; for 2.5 X 6 =■ 15. 






























FORCE IN RIGID BODIES. 


341 


A load resting 1 oit two props either at its ends or otherwise. When a 
load c of any shape whatever, rests in any position upon two props x and z, the 
portions of its wt borne by the respective props will be to each 
other inversely as the horizontal distances ox, oz, from the 
cen of grav c of the load, to the props. Thus if o z is two, three 
or four times as great as o x, then will x bear two, three, or four 
times as much of the entire wt of the load c as z does. There¬ 
fore to find how much each prop bears, first find 
the cen of grav c of the entire load; and its hor dist (say o x) from 
either one of the props, (say x.) 



Then as en ^ re ^ or dist x 2 • Entire wt 
h between the two props • of load 


Dist o x 


Wt borne by the 
other prop z. 


And this wt taken from the entire load leaves the wt borne by x. 

This all amounts to the same as if we consider x z to be the clear span 
of a beam without wt, and supporting a load equal to c, concentrated at the cen 
of grav of c. 

Conversely, to place two props x and z so that each may bear a giveil por¬ 
tion of the entire load c, take any two hor dists o x and o z from the cen of grav 
c, inversely as the two portions of the load to be borne by each. Thus if x is 
to bear two-thirds of the wt, make o z equal to two-thirds of x z. 


Ex, e. The following is very important in its application 
to arches of any material. 

sssssfaSiElSi 

:his, we may consider the half bridge Cti nrj to oe a iev<r a fh j« r -„.j on i n which it would 
wt?° n And l, iet e fts a, iever C ^ f.lbottt 'r.VeT ft? M of course being at right angles to the direction 






342 


FORCE IN RIGID EODIES, 



<7 s. Then is its moment about r equal to 80 X 6 = 480 ft-tons. (Art 46.) Now, whatever may b<» 
the amount of the bor force ft a, which acts at the 
end a of this lever, to counteract this moment of 
480 ft-tons, its leverage, ( Art 46) is plainly equal to re, 
measured from the fulcrum r, and at right angles to 
thedirection ft i of said force. Suppose we Bud by mea¬ 
surement from the drawing that re is 8 leet. Then the 

480 „„ , . u 

force itself must necessarily be —-60 tons ; which 

o 

is the hor pres which the opposite half of the bridge 
exerts against the keystone a, of the arch; for 
60 X 8 = 480 ft-tons of moment. 

Rem. 1. Rut so long as an arch is not deranged, 
but remains firmly in position, the half arch, in¬ 
stead of tending to revolve about the point r, presses 
equally over the entire surf r d of its skewback. 

Therefore, the leverage with which the hor force ft 
acts upon the skewback, is actually y o, measured 
from the center of rd, and in practice it must be 
used instead of r e. In the same manner, y m be¬ 
comes the leverage for the wt, instead of r t. 

Rem. 2. The cen of grav of the 
half arch, can he fouml by making 

a drawing end rj, about 4 to 6 ius long, on pasteboard, or on a stiff drawing-paper, to a scale. 
Cut out the fig; and balance it flatways on a sharp straight-edge, or over the edge of a fable, in 
two directions or positions. Where these two directions intersect each other is the cen of grav. It 
is not indeed this cen itself that is needed, but the line g s, of its direction ; which may be found 
at once by taking care that the straight-edge is parallel to the back n d, while balancing the fig. 


Rem. 3. Under the head leverage, may be classed the tread-wheel; windlass and lever; capstan 
and lever; and all axles turned by a winch or by a crank ; such as the drum and winch with which a 
water-bucket is raised from a well. &c. They are all merely continuous simple levers, of which the 
axis is the fulcrum ; the rad of the circle described by the power is one arm, aud the rad of that de¬ 
scribed by the shaft, drum, &c, is the other. 


a 


-O- 


b^G- 


10 


-©- 


Rem. 4. Compound levers, a a, bb, c c. Fig 52^. maybe used where 
there is not space for the arms of a single lever of sufficient power. They need not 
extend in one line; but maybe placed 

one over the other; or in such other po- ^ \ 

sitions as may be convenient. Their 
effect is much greater than the combined 
effects of the three simple levers, and is 
found thus; As the product of the weight- 
arms, 2 X 1 X 3 = 6; is to that of the 
power-arms, 10 X 8 X 7 = 560; so is the 
power to the weight; or, as 6 : 560 :: 

IM : W93*4. These arms are measd in 
all cases from the fulcrum ; which is 
sometimes at the end of one or more of 
the levers, when compounded ; see Fig 44. 

The combined effects of the three simple 
levers would be but 5 + 8 2% = 15*^; or P 1, W 15*4. 


Pi. 


II 


5 *! 


Pi 



A series or train of toothed pinions and wheels, working into each 

other, is merely a series of continuous compound levers. These are generally set in motion by a 
winch-handle, the rad of which is the first leverage of the series; while at the other end of the train 
the wt is usually suspended from a drum, the rad of which is the last leverage. To find the effect, 
mult into one prod the radii of the winch and of the wheels, and into another prod the radii of the 
pinions and of the drum ; then, as the last of these prods is to the first, so is the power to the wt, 
as in the preceding case. 

In both the foregoing cases of compound leverage, as in all other cases whatever of leverage, the 
vel of the wt is to that of the power, as the power Itself is to the wt; thus, in Fig 52J4, the wt will 

move only - ■part as fast as the power; or the power must descend 93 ^ inches, iu order to raise 

the weight 1 inch; on the principle of virtual vels. See Art 52. 


Rem. 5. TIlO SCPPW is a combination of leverage, with an inclined plane; a spiral inclined 
plane being formed by the threads of the screw. While the power applied to the lever which turns 
the screw moves around an entire circle, the body moves only the dist between the centers of two 
threads; and since in all mechanical contrivances, the wt is to the power as the vel of the power is 
to that of the wt, so in this case, theoretically, the wt is to the power, as the entire circumf of the 
circle described by the power, is to the dist between the centers of two threads; but in practice, the 
friction of the screw (which under heavy loads becomes very great) has also to be overcome by the 
power ; and this fact makes the calculations of but little use. 

ril<* Pulley, also, when a fixed one, is referable to leverage. In the fixed pullev A. Fig 
52*2, there is no gain of power; for here the diam a h is a lever of two equal arms, revolving around 
its fulcrum at the center of the pulley. Consequently, the wt aud the power have equal leverages; 



























FORCE IN RIGID BODIES 


343 



each equal to the rad of the 
circle ; and in order to balance 
a wt W of say 1 ton, the power 
P must also be 1 ton ; for if oue 
of them moves, the other must 
plainly move with the same 
vel. To raise the wt, the power 
must exceed the wt; because 
some unresisted force is re¬ 
quired to give motion, as well 
as to overcome the friction of 
the axie around which the pul¬ 
ley revolves, and the friction 
of the rope in the groove 
around its circumf. These fric¬ 
tions become so great when 
many pulleys are combined, 
that theoretical calculations of 
the power are of little value. 
Although a fixed pulley gives 
no gain of power, it is very 
convenient for allowing change 
of direction in applying the 
power; so that by pulling 
downward, or hor, &c, we can 
cause the wt to rise vert. 11 is 
plain that the rope in this 
pulley is equally strained at all 
points. Theoretically, this is 
the case with any one single 
rope, as rcdfge, Fig 52%, 
passing around any number of 
pulleys, whether fixed, as A or 
1), or movable, as B ; and all 
the theoretical calculations of 
the power may be based upon 
They will, however, be incorrect in practice, on account of the friction just 


Fig.53 


this principle alone. 

alluded to. In Fig 52%, where only one rope is used, the lower puliuy-block B s, to which the wt W 
is attached, is directly upheld by the two parts df and e g of the. rope. Consequently each of these 
parts is equally strained. isiuce both parts are vertical the strain on each is that due to a force 
equal to one half the w# ; and since the whole single rope is theoretically strained to the same ex¬ 
tent, that part of it to which the power is applied must be strained equal to half the wt W ; or, in 
other words, the power itself must be equal to half the wt, and will move twice as fast, and, of course, 
twice as far in the same time. 

In Fig 53 the lower pulley-block ty is sustained directly by the 4 partsccccof the single rope ; 
therefore, each part of the rope, and, consequently the whole of it, is equally strained by a force equal 
to % of the wt W: and the power P must be = the same %, and will move 4 times as fast as the wt, 
or 4 times as far in the same time, it is immaterial whether the two pulleys in the lower block. Fig 
58, be placed one above the other, as shown, or (as usual, and more convenient), side by side ; so also 
with those in the upper block. If there were 3, 4, or 5, &c, pulleys in each block, then there would 
be 6 , 8 , or 10 sustaining parts c c, &c, of rope, each stretched equal to A, A, or jL of the wt W; 

and the power would also be in the same proportions to the wt; in other words, to find the theoret¬ 
ical proportion of the power to the wt, when there is but oue rope throughout, as in Figs 52% and 
53, divide 1 by the number of parts cccc of rope which directly sustain the lower block. In fig 53 
the number of parts is 4. When more than one rope is used in a system of pulleys, the principle is 

the same as above, but the rule must be differently worded. See 
Fig 53%. 

Here the lower block y b, with its attached wt of say 4 tons, is directly 
sustained by the twrn parts a and c of one rope. Consequently, each 
part has a strain of % the wt, or 2 tons; which is uniform throughout 
that rope. But all the 2 tons strain on the part c is sustained bv the 
hook s ; while that on the part a is sustained by the two parts n and m 
of the other rope; each of which plainly Sustains one half of it, or 1 
ton, which is uniform throughout this second rope to its very end. 
Therefore the power also is 1 ton, Or % of the wt W. The mode of 
proceeding Is the same, whatever may be the number of movable pul¬ 
leys. To find the theoretical proportion of the power to the wt, mult 
together continuously as many 2 s as there are movable pulleys, and 
div 1 by the prod. Thus, here we have two movable pulleys, and 2 X 
2 = 4; and % = the answer. If there were 4 movable pulleys, we 
should have 2X 2X2X2 = 16; and yL- — answer. 

In all our figs, that end of the rope to which the power P1 is applied 
is represented as hanging vertically, and parallel to the other parts of 
the rope ; but this was done merely because the power is supposed to 
be a weight , and of course acting vert. But if the power is muscular 
force, or any other kind that may act in any direction whatever, then 
the power end of the rope, as mn. Fig 53, may have that direction in 
which it is most convenient. The amount of power required will not 
be thereby changed; for it is plain that leverage from the center of the 
pulley o s to m, when the power is at n, and acting in the inclined di¬ 
rection of the rope, is equal to that from the same center to o, when the 
power is at P, and acting vert. 



* If the two parts, d/and eg, of the rope were inclined, the strain on each would be greater than 
half W, as at ac, ao, F.ig 5, p 345. 




































344 


THE CORD OR FUNICULAR MACHINE. 


THE COED OR FUNICULAR MACHINE. 


Art. 1. Some allusion to this subject has already been made on page 325, | 
which see. Theory requires that the cord, rope or string, &c, shall be ab- 
solntely flexible, inextensible, friction less, without weight, and infinitely thin ; ! 
and that such pulleys, posts, pins or pegs, loops or rings as may be used with the 
cord shall also be absolutely frictionless; and at times devoid of weight unless I 
said wt is included in one of the acting forces. These assumptions cannot of 
course be realized in practice, which however will agree with theory in propor¬ 
tion as we approximate to them; and this we can frequently do so far as to render r 
the theory of great use. We know that all cords have wt and thickness, and can 
be stretched; and that so far from being flexible, they may require very con¬ 
siderable force to bend them around pulleys, posts, pins, &c. They also possess ! 
friction. Also that all pulleys, pins, sliding rings or loops &c, have more or less 
friction; which when there are many of them, may entirely vitiate all calcu¬ 
lations. A pulley lias, in itself, no advantage over a smooth 
cylindrical pin, or post (as the case may be)except that its friction being 
a rolling' one, is less than the sliding friction which takes place between a 
cord and a pin, &c. Therefore in what follows we may usually (so far as theory is 
concerned) substitute one for the other. 

Rein. It will be seen that the principle of the cord serves for finding the 
strains on the ropes of the different systems of pulleys. 

Art. 2. Assuming then such a theoretical cord, pulleys, pins, &c, all devoid 
of friction, tlie broad principle of the cord is that, any force/ Figs 1, 
imparted to either end of a cord (whether the cord be straight as fg or / c; or 

bent out of its course as f e by any num¬ 
ber of frictionless pulleys or pins as n r 
m s, &c, however placed) is all trans¬ 
mitted along the ent ire cord to its other 
end,straining it uniformlyin every part. 
In fg or fc this may be regarded as self- 
evident ; in / e it is proved by experi¬ 
ment. It is perhaps needless to add that 
a straining lorce cannot be imparted to 
one end if there is not an equal one at 

the other end to react or strain against it. 

If at any one of the pulleys, pins, Ac, as m , we make mo, ma, each equal to the 
force at either /or e, and complete the parallelogram m o da of forces, then the 
diagonal or resultant d m will give both the direction and amount of strain which 
the cord produces on that pin. This strain will differ at the several pins ac¬ 
cording to the angles formed by the two components; and may thus be greater, 
or less, or equal to the force at/ or e. 

Rein. With theoretical cords, pulleys, pins, Ac, the two components m o, m a 
at any pin will always form equal angles with tlie resultant md; and 
upon this fact the principle of the cord depends. 

Art. .*5. The principle of the cord applies also when as in Figs 2, 3, 4, a load 
or third force/ is imparted between the ends of the cord as at a, by means of 




a frictionless pulley, ring or loop which can move along the cord with perfect 
ease. It such a pulley, ring or loop be first placed upon the cord near m or n it 
will with its load or other force move down along the steepest part of the cord 
until it comes to rest at that point a at which (i m and ci n form equal angles 
b n c, b a o w ith the direction n b of the force/. If both ends in n of the corn are 
at the same height, and the forcey a load and of course acting vertically, 
then a will be at the middle of the cord ; but if the ends are at different 











THE CORD OR FUNICULAR MACHINE. 


345 


heights as in Fig 2, from either of them as n draw a vertical, asm; and from 
the other one, as m lay off the whole length miof the cord ; bisect n x at e, and 
draw e a horizontal; a will be the required point. 

If we draw a b to show both the direction and the amount of the force/, and 
complete the parallelogram b c a o of forces, then either a c or its equal a o will 
give the uniform strain which / produces from end to end (m to n) of the cord. 
From which it is plain that a force equal to a c or a o may be considered to be im- 
arted at each end m , n, of the cord, and there to react against a c and a o, there- 
y straining the cord uniformly throughout. 

Rem. 1. Precisely the same result will follow if one or both of the ends m, 
n, instead of being fixed or fastened at those points as is supposed in Figs 2 and 3, 
are continued as at m. Fig 4, over a frictionless pulley or pin, and prolonged either 
vertically as m l , in which case it will sustain a load or vertical force l equal 
to a o or a c; or in any other direction as m s or m e, in which case it will sustain 
at s or e a pull equal to a o or a c acting in said directions. And the same will 
take place if one or both ends as n, Fig 4, be extended over or under any num¬ 
ber of such pulleys or pins, (no matter what angles they form) say to v. The 
strain at v will still be equal to a c or a o, and will be uniform from v to l. The 
strain on any pulley or pin may plainly be found as in Art 2. 

Rem. 2. If we know the angles b a o, b a c, and the force or resultant a b, 
we can calculate the strains or components a o, a c ; or knowing the 
augles and components we can calculate the force a b, all by the formulas in Art 
45, p 334, which are based upon plane trigonometry, which enables us to find un¬ 
known parts of a triangle when certain other parts are given. 

Rem. 3. If the angle m, a, n, is 120°, each component ao,ac, will be 
equal to the resultant a b ; if it is less than 120°, each component will be less than 
the resultant; and vice versa. 

Art. 4. A little reflection will enable the student to see that this Art is 
merely a farther illustration of the preceding ones. In Fig 5 a load s, or any 
equal vertical pull will strain the 
strings s a and z v each to an amount 
equal the load,becausethey both 
act in its own direction. If 
the parts m a, g a of the cord a in n g 
a which passes over the l'riction- 
less pulley or pin z were also vert¬ 
ical, each of them would be strained 
equal to one-half the load s; but 
they are not vertical, and if we make 
a b to represent s, and complete the 
parallelogram b o a c, we shall find that 
the resulting strains a c, a o are each 
rather more than half of s; and 
they will increase if the angles at a 
increase; and may be calculated 
by the formulas in Art 45, p 334. 



In Fig 5 we had the force s or a b given as a resultant, and from it we found its 
two components or strains a o, n c. In Fig 6 we have the two components h m, h 
g given (each evidently equal to the load s) and from them we deduce the result¬ 
ant h b'. which represents the strain on the pulley, and on the string s w. It is 
plain that this string cannot be vertical as zv is, but must (as z v does) adjust it¬ 
self to the direction of the force or resultant which pulls it. If the load s is the 
same in both Figs, we see that the pull on z w will be about twice as great as on 
zv; and the one along the cord smgp about twice as great as that along am ng a. 
This difference will of course vary with the angles. 

Rem. 1. Let s, Fig 5, instead of the load there represented, be a suspended 
tnan weighing 2001bs, and holding m a, g ci together at a with both hands, or let 
m n and ga both hang vertically from the pulley, and let him hold their lower 
ends apart by one hand at each end. Now if each part ol the rope will bear but 
a little more than 100 lbs, and zv a little more than 200 lbs, he will be safe from 
falling. But if he lets go one end of the rope, putting it into the hands of an- 

23 





346 


THE CORD OR FUNICULAR MACHINE. 


other man standing by to hold it; or if he hooks one end to a projecting spike in 
a wall close at hand, the rope will break, and he will fall, because he at once ! 
doubles the strain along both the ropes amng a, and z v. 

Rent. 2. Fig 7 shows a device by which boatmen sometimes haul a boat out 
of the water, and up on to the beach, at a landing, when it is too heavy for their , 
unaided efforts. The rope ernxnb is to be considered as horizontal in this case, t 
One end b being fastened to the bow of the boat, the rope is carried past one 
smooth post n, to another at m, around which it makes a whole turn ; and a man 
stands at that end e to take in the slack while the others, taking hold of the rope 
midway between m and n, pull it into a position man in which, if the angl e 
man exceeds 120°, (see Rem 3, Art 3) each component an, am of their 
force exceeds said force itself; and a strain equal to one of these components (except 
so far as it may be reduced by the rigidity of the rope and by its friction against the 
post n) is transmitted uniformly to the boat b, drawing it a short distance up the 
beach. The rope is then straightened again from m to n by taking in the slack 
at e, and the operation is repeated as often as necessary. 

To find the strain a m or a n, divide the force by twice the nat cosine 
of the angle x a n, or x a m. Or to find how many times the strain exceeds the 

force divide the distance a n by twice the distance 
a x. Here a x represents only half the force, or half 
the diagonal or resultant of a m and a n. 

The force of a man pulling by jerks at a; or a 
will average between 30 and 80 lbs, 

Art. 5. We will now dispense with the fric¬ 
tionless pulley required by the principle of the 
cord, and which would of itself roll along the cord 
until it came to rest at a point which would cause 
equal angles and consequently uniform strain throughout. We will 
substitute for it a tight knot which cannot slide, at a point a. Fig 8, 

which causes unequal angles b am, b an, be¬ 
tween the direction a b of the force /, and the two 
parts a m, a n, of the cord. Here, completing the 
parallelogram b cao, we find that the part a not the 
cord is strained to an amount denoted by ac\ and 
that same strain would affect any extension of the 
cord on that side, like that to v, Fig 4. The 
part a m would be strained equal to a o; and that 
strain would continue to the end m, or to the end 
of any extension of that part. Hence the 
strain is not uniform from end to end of 
the cord, as it would have been if the knot had been 
frictionless; in which case it would have slid 
along the cord until the angles would be equal. 

Rem. 1. But even in this case of a tight knot at a, there is always one 
direction, as tas, in which the force can be imparted so as to cause equal an¬ 
gles s am, s an, with the direction as of the force t, and then the strain will be 
uniform from end to end, as if a frictionless pulley had been used. 

Rem. 2. With a tight knot at a it is plain that a force may be imparted 
there from any direction as fa, t a &c. If the direction coincides with 
either part a m, or a n of the rope, that part will bear all the strain ; 
the other part remaining entirely free. 

Rem. 3. From Rule l,p 150, for drawing an Ellipse, it will be seen that at 
whatever point as a, Fig 8, we apply force to an inextensible cord nam with fixed 
ends, that point will be in the circumference of an ellipse, the foci of which are 
at the ends m, n. 








PARALLEL FORCES 


347 


Art. 55. Parallel forces are those whose directions, as 
in Fig 5334, (whether opposite to one another, or not; or whether 
in the same plane, or not.) are parallel. In Fig 53%, the forces, 
although acting upon one plane, 0000 , are not in one plane, but 
in several. 

It is a peculiarity of parallel forces in one, plane, that all their 
arms, or leverages with respect to any given point in the same 
plane are in the same straight line. Thus, if l , m, n, 0 , Fig 54, 
be in the same plane, then their leverages p q, p r, p a, about the 
point p, are all in the same straight line p a. The point p 
is supposed to be in the same plane as the forces. pr 



Two parallel forces are evidently always in the same plane; that is, 
the same flat surf could coincide with both of them ; and their re¬ 
sultant or antiresultant will be in that same plane: which amounts 
to saying, in other words, that if 3 parallel forces hold each other in 
equilibrium, they are in the same plane. 




Art. 56. The resultant of any number of parallel forces, 

whether in the same plane, or in the same direction, or not, is always parallel to 
them. If they all act in the same direction, whether in the same plane or not, their 
resultant is equal in amount to their sum: or, in other words, an antiresultant 
force sufficient to balance them, must be equal to all 
the forces added together. But if they are in oppo¬ 
site directions, their resultant will be equal to the 
diff between those which act in one direction, and 
those which act in the opposite one ; and its direction 
will be that of the greater sum. Thus, in Fig 53%, 
if the forces pointing to the left amount to 10 tons, 
and those to the right 4 tons; then the resultant will 
be 10 — 4 = 6 tons ; and it will point to the left. 

The parallel vert downward forces of gravity, upon the innu¬ 
merable separate particles, situated in the infinite number of 
imaginary vert planes, in any body, as W, Fig 55, is an illustra¬ 
tion of this. If any such body be suspended by a string from a 
spring-balance B, the vert upward pull of the string will balance 
or equilibrate all these innumerable forces. Consequently, the 
string represents their antiresultant, which is equal to their re¬ 
sultant. We know that the vert pull on the string, as shown by 
the spring-balance, is equal to the wt of the body ; which wt is made up of the innumerable parallel 
vert forces alluded to. Thus we see that when any number of parallel forces, whether in the same 
plane or not, set in the same direction, their anti resultant is parallel to them, and equal to their sum ; 
consequently their resultant must be so also. The same principle applies to parallel forces in any 
direction whatever. 

When a body thus acted on by gravity is kept at rest, or balanced, as in the fig. then the direction 
of the resultant or antiresultant, or of the string in the fig, passes through a certain point, called the 
center of gravity of the body. This isacertain point, upon which when acted 

upon by gravity only, the body will balance itself, in whatever position it may be placed; and if the 
entire wt or grav of the body could be concentrated into that single point, its effect, whether regarded 
as moving the entire rigid body, or as producing strain (pull or push) between it and another rigid 
body, would remain precisely the same as it actually is with the grav diffused throughout the entire 
mass.* 



* In some bodies the cen of grav is also the center of the wt of the 
body ; but very frequently this is not the case. Thus, in a body 
a be. Fig 55hj, with its cen of grav at c, there is more wt on the side 
a c. than on the side c h. 

If a body W, Fig 55, suspended freely from any point n, is at rest, 
its cen of grav is directly under said point. If the body W be pnshpd 
a little tonne side, and then left to itself, it w;il plainly tend of itself 
to swing back to its first position ; and when this is the case, it is 
said to be in stable, equilibrium. But if the body, instead of being 
suspended, be balanced on top of a slim rod, and if we then push it 
a little to one.side, it will not tend to return, but will fall over; and 
therefore the equilibrium of a body so balanced is said to be unstable. 


d 






Also 


55 ^ 


in such cases as that of a grindstone supported bv its hor axis passing through its cen of grav. if we 
cause it to revolve a short dist, stop it, and then leave it to itself, it will have no tendency either t« 
return, or to keep on revolving; and its equilibrium is called indifferent. 



































348 


CENTER OF GRAVITY. 


CENTER OF GRAVITY. 


The ceil of grav of a square, rectangle, rhombus, or rhom¬ 
boid, is at the intersection of its two diagonals. 

Of a circle, ellipse, or regular polygon, in the center of the figure. 

Of a triangle, at the intersection of lines drawn from any two angles, to the 
middles of the sides respectively opposite said angles. 

Or, draw a line from any one of the angles, to the middle of the side opposite said 

angle ; the cen of grav is in this line at % of its length from the side which it bisects. 

Of either a trapezium, or a trapezoid, 

draw the two diags a c and b d. Div either of them, say 
a c, into two equal parts as at to. Take the longest part 
d s. of the other diag d b, and set it off from b to n. Join 
n to, and div it into 3 equal parts. The cen of grav will be 
at o, the first of these divisions from to. 




a 


X 



Of a trapezoid only. Prolong either parallel 

side, as b a, in either direction, say toward g; and make 
a g equal to the opposite side d c. Then prolong the 
other parallel side d c, in the opposite direction ; mak¬ 
ing c h equal to the side b a. Join g h. Find the centre 
e of a b; and the center f of dc. Join ef. Then o is 
the cen of grav dc + 2aft ef 

Or fo = -— -— X 

dc + a b 3 

Of a semicircle. Mult the height or rad a b by .4244; the 
prod will be a c; and c is the cen of grav. 

Of a cycloid. See “The cycloid,” p 154. 

Of a sector of a circle, adbe. Mult twice the 

chord a b, by the rad a c. Div the prod by 3 times the length 
a d b of the arc of the sector; (see Lengths of Arcs, p 141, &c.) 
The quot is c o; and o is the cen of grav. 

Of a quadrant, c o = c dX -6002. 

Of a segment of a circle, nob. Cube the chord 
a b. Div this cube by 12 times the area of the segment; (see 
Areas of Segments, page 146.) The quot will be c n; and n is 
the cen of grav. 

Of a circular arc n o b, (the line alone,) not exceed¬ 
ing a semicircle. As the length of the arc is to its chord, so is rad c o to 
c n ; n being the cen of gr. Or, quite approx, mult the rise s o by .65 for 
« n. Or more correctly, if the rise s o is .01 of the chord a b, or less, mult 
it by .666 forsn; if .1, mult by .665; it .15 by .663; if .2. by .660; if .25 
by .657 ; if .3, by .653 ; if .35, by .649; if .4, by .645 ; if .45, by .641 ; if .5, 
by .637. 

Of a parabola a b c, at o in the axis x b, gths of its length 
from x. 

Of a semi parabola abt. At n; on being 3-eighths of 

the half base a x, and o x being gths of x b. 

To find the common cen of grav. «y, of 
two figs, a and b. First find their separate cens 

of grav, s and t, and areas. Then 

Sum of the areas : s t:: area a : g l :: area b : g s. 

To find flic common cen of grav, j/, 
of any numberof figs, as a. b, and c. Having 

found the irea and cen of grav of each Dg separately, begin 
with any two of them, as a and b, as just explaiued. Then, 
taking any other one, as c, 

Sum of areas ..... sum of areas ... . c . c 
or a, b, and c • 9 c • • of a and b . V c . . a'ca c . y g. 

In tlie same manner proceed with Fig d, if required. 

To find the cen of grav, p, of a hollow 

fig. x. Find the common cen of grav, w, of the openings, 
as directed above for two or more figs. Find the cen of grav, 
/, of the entire flg as though there were no opeuiugs. Theu 























NOTES ON “ CENTER OF GRAVITY,” pages 348 and 349. 


Page 348. 


After “Of a quadrant,” line 30, add Of a sextant, co = cd x.63(*«i. 


Lines 35-37. The rule given for center of gravity of a circular arc (the 


line alone), viz.: cn 


radius cox chorda b 


arc aob 

whether less than, equal to, or greater than, a semi-circle. 


5 applies to all circular arcs, 


Lines 42-45. These rules, for parabola and semi-parabola, refer to 
surfaces, as do all other rules on page 348, except those for circular arcs, lines 
35 to 41, which refer to lines only. 


Page 349. 


Lines 9 to 47. Erase, and substitute the following, to the end of page iv. 

The curved or slanting surface only, of a right cone (circular or 
elliptic) or of a right pyramid. The center of gravity is in the axis (the 
line joining the apex and the center of gravity of the base) and at one-third 
its length from the base. 

The curved surface only, of a hemisphere or other spherical seg¬ 
ment, or of a spherical zone. The center of gravity is in the center of 
the axis. 


CENTERS OF GRAVITY OF SOLIDS. 


flie solids are of course supposed to be homogeneous, i. e., of uniform density 

throuj 

Sphere or* spheroid, (ellipsoid). The center of gravity is the center of 

the body. 

l 








Page 349.—Continued. 


Cube, parallelopiped, or other prism, regular or irregular, right or 
oblique; and cylinder, circular or elliptic, etc., right or oblique. The center 
of gravity is in the center of the axis joining the centers of gravity of the 
two parallel ends of the solid. 


lingula of a cylinder, circular, or elliptic (provided one of the axes of 
the ellipse coincides with the oblique cutting plane); right or oblique 




Let G be the center of gravity; O T the axis (joining the centers of gravity 
of the ends) and X G a line drawn parallel to the axis in the plane passing 
through the axis and through the uppermost and lowermost points C and I> 
of the oblique cutting plane. Then the position of G in said plane is found 
thus: 


♦ 


OX = 


O B a 

* 2 h+T 


X G - £ (2 h + a + £ 2T+ *) ‘ 


If the oblique plane C D meets the base at A, so that h = O, while C D remains 



11 















































Pag'e 349.—Continued. 


Cone, circular, elliptic, etc., right or 
oblique; or pyramid, regular or irregu¬ 
lar, right or oblique. The center of 
gravity G is in the axis O T, drawn from 
the apex, or top, T, to the center of 
gravity O of the base; and 




O G 


O T 
4 


Frustum of a cone, circular or elliptic, right or oblique; or of a pyramid, 
regular or irregular, right or oblique; in which the two ends A B and C D are 
parallel. 




Call the area of the large end A, that of the small end n, and the height O Z 
of the frustum, measured along its axis s h. 

The center of gravity, G, of the frustum is in the axis O Z, which joins the 
centers of gravity O and Z of the two ends; and its distance from the base, 
measured along the axis , is 


O G 



A -f 2 /a a + 3 a 
A + /A a + a 


In a frustum of a circular cone, right or oblique, with parallel ends, 


this becomes 


O G 


h v R2 + 2 R r + 3 r 2 

4 R ,2 + R r + r 2 * 




Sphere 

the body. 


P^j^ihe radii of the large and small ends of the frustum 














Page 349.—Continued. 


Frustum, A H C 1) of a cone, circular, elliptic, etc., right or oblique; or 
of a pyramid, regular or irregular, right or oblique; whether the ends are 
parallel or not. By the rule given above, find the center of gravity ]M of 
the entire pyramid (or cone, as the case may be) A B T, of which the frustum 
forms the lower part; and the center of gravity S of the smaller pyramid or 
cone D C T, = entire pyramid or cone, minus the frustum. Also find the 
rohnne of each: 



and the 

Volume of volume of volume of 

the frustum = entire pyramid — smaller 
A B C I) or conej A B T one, DOT 


Then the center of gravity G of the frustum A BCD is in the extension of 
the line S N; and 


N G 


S N 


x 


volume of smaller pyramid or cone 
volume of frustum. 


Segment of a sphere. Let O be the center 
of the sphere. The center of gravity G of the 
segment is in the axis O T. Let r = radius of 
the sphere, and h = the height or rise of the seg¬ 
ment. Then 


O G — | 


(2 r — h)2 

3 r—IT 


T 



In a hemisphere, this becomes 



OG = | OT radius r. 















CENTER OF GRAVITY. 


349 


Area of fig minus .... area of . . . sura of areas . . 

the openings •• entire fig • •• of openings "' 

1^ an irregular fig which may be divided into triangles, trapeziums, trapezoids, &c, the process 
Is the same as with the figs a, b, c, d, and x. 

Tlie ceil of grav of any plane fig may be found by drawing it to a 

scale on pasteboard ; then cut out the figure ; balance it in two or more positions over the edge of a 
table, or on a sharp knife-edge; and mark on it the directions of the edge. Where these directions 
intersect each other, will be the reqd point, near enough for most practical purposes. The paper on 
which the fig is prepared, must be so stiff that the fig will not bend when balanced. 

Of a cube, parallelopiped. cylinder, prison, sphere, sphe¬ 
roid, ellipsoid; the cen of grav is in the cen of the body. 

Of any frustum of either a right or oblique cylinder: or of a 

right or oblique prism; whether cut parallel to its base, or obliquely. In the center of the axis of 
the frustum. 

Of a right pyramid, or cone; in its axis, atj^of its length from the base. 
Of any pyramid, or cone; whether right, or oblique. In a line from its 

Vertex, or top, to the cen of grav of its base ; and at J4 the length of this line from the base. 

Of a frustum of a right cone, cut parallel to its base. Call the rad 

of the larger end R; and that of the smaller end r\ and the height of the frustum measd on its 
axis, h. Then, 

h ~ R 2 + 2Rr + 3r2 

— X — — :-—— = dist on axis from greater end, or base. 

4 R 2 R r r 2 

Of a frustum of a right pyramid or of a right cone, cut parallel 

to its base. Call the area of the large end A ; that of the small end a ; the height measd on the axis, h ; 
h A + 2yT a -t- 3 a 

X- - —— = dist on axis from greater end, or base. 

* A+VAa+a 

Of any frustum of any pyramid, or cone; whether right or ob¬ 
lique; or whether the ends of the frustum are parallel to the base or not; it is in a line t o, drawn 
•etween the centers of grav of the two ends of the frustum ; but its dist y c above the larger end 
an, and perp to it, 
must first be found 
upon the line t y, 
drawn from the cen 
of grav of the small 
end, to, and at right 
angles with, the large 
end an. Having first 
found the line t y, 

whether for a pyramid or a cone, use it instead of the height h. in the preceding formula, for a frus¬ 
tum of a right pyramid. The result will be y c ; and by drawing c s parallel to a n, we find the reqd 
point s, in the line t o. 

Of the curved or slanting surface only, of a right cone; in 

the axis; and at % of its length from the base. 

Of a hemisphere; in its axis, at % of its length from the base. 

Of a segment of a sphere. From twice the rad of the sphere, take the 
height of the segment. Square the remainder. Then from 3 times the rad of the sphere, take the 
height of the segment. Div the square just found by this last rem : and take % of the quot, for the 
dist from' the cen of the sphere to the cen of grav of the segment. 

Of the curved surface only of a hemisphere; of a spherical seg¬ 
ment; or of a zone: at the middle of its axis, or height. 

Of a paraboloid : in its axis, and at % of its length from the base. 

Of a hollow body, or of any number of solid or hollow 
bodies, use the last three paragraphs on p 348, except that weights must be taken 
instead of areas. 

Rem. We must not confound cen of grav with cen of weight. It does not follow when a body bal¬ 
ances on a knife-edge, that there is equal wt on both sides of it; but merely that the wts of the several 
particles on one side, when mult by their respective leverages, or dists, at right angles from the 
knife-edge, have a united moment about the knife-edge, equal to that of the particles on the other side 
of it, when mult by their leverages. See Leverage, «to. 








350 


FORCE IN RIGID BODIES 


Art. 57. That point through which the direction of a single antiresultant 
force must pass, in order to balance several other forces acting at diff points; or, in 
other words, that point through which the direction of the resultant of those forces 
must pass, is called the center ol‘ pressure, or of force, or of strain, 
of those forces, as the case may be. 

For instance, let S, Fig 56, be a common wooden box; but having 
one side, as oo, loosely titled, so as barely to allow of pushing it back¬ 
ward and forward. Fill the box with dry sand, (clean small gravel ‘ 
will be better,) and it will be found that there is but one single point, 
i, at which we can. by holding to it a thin rod ri, balance the pres of 
the gravel against the opposite side of oo. If we apply the rod at any 
other poiut, o o will give way before the sand ; if the rod is held above 
i, the bottom of o o will be pushed outward; if held below f, the top of 
o o will move outward. This point i is dist above the bottom of the 
sand one-third of the depth of the sand; in other words, the ceu of pres ■ 
of sand of any depth is. like that of water, at one-third of that depth 
from the bottom. In the case before us, the depth is supposed to be 
uuiform, so that the cen of pres is at the same height above the bot¬ 
tom, clear across the box. 

Now the balancing force applied through the rod at i, is the antiresultant of all the pressures re- ij 
suiting from the several particles of gravel against the opposite side of oo; and its effect upon the 
rigid body oo, (omitting of course any tendency to bend or break it, which comes under the head of 
Strength of Materials.) is precisely the same as that of all those forces combined; except that it is 
in the opposite direction. Its tendency to push oo bodily, or as an entire mass, toward the right y 
hand, is precisely the same as that of the gravel to push it to the left hand; or it is the same as 
would result were we to heap up sand in front of o o, so as to balance the sand behind it. , 1 

Rem. It is this important principle of the cen of pres, that enables us to adopt the convenient 11 
practice of representing, by a single line, the effect of force actually distributed over a considerable ! 
surf. Thus, iu Fig 52, the hor force ha, by which each half of the arch mutually prevents the other i 
half from falling, is actually distributed over an area whose depth is the depth cj of the keystone; * 
and its breadth, that of the whole bridge, as measd across the roadway. Yet the arrow ha, when 
drawn to a scale, perfectly represents the effects of this distributed force in upholding the half arch, , 
considered as an entire rigid mass. So far as regards splitting or cracking the stone immediately at > 
a, the effects would of course be diff; but as the whole force is only supposed, for convenience, to be 
applied at a, this diff is merely ideal iu this instance. See Arts 5 and 8, p 226, 227. 

It is evident from the foregoing that “cen of grav" means nothing more than “cen of force; ” ex- 
cept only that the former is a convenient term for denoting that the force is that of grav alone. 

I 

Art. 58. When either grav or other forces are to be considered with regard i ' 
their effects in bending, or breaking bodies; instead of moving or straining them as t 
entire masses, supposed to be rigid, or incapable of change of form, they cannot be 
assumed to be concentrated at either the cen of force, or the cen of grav. 




-Ill 



In the last case our applied extraneous forces are supposed to be 
brought to bear only upon other extianeous forces acting upon the 
bodies at the same time; the bodies themselves being regarded mere¬ 
ly as unalterable mediums through which said forces are enabled to 
act upon each other; but in bending, breaking, twisting, shearing, 
&c, our extraneous or mechanical forces must be considered to act 
against the inherent cohesive forces of the bodies themselves / 
therefore these forces, which before were entirely neglected, now 
acquire a primary importance; the question of strength of materials 
comes in, and the assumption of perfect rigidity must be altogether 
« ^us, in Fig 56J4, so far as regards either moving a 

rigid body c o, or straining it against the force n, it is immaterial 
whether we employ the two equal parallel forces a, b, or a single 
force m, equal to both of them, and acting at their center of force 
But since no bodies are absolutely rigid, but may all be bent 
or broken, it is plain that the two forces a. h. straining the 
body c o against the force n. would bend or break it much more 
readily than the force m would. 


An absolutely rigid hor beam s s, would sustain any amount 
or load l, without bending; and consequently would always 
press vert upon its upright supports u, u, without any tendency 
to press them sideways. But an actual beam n a, ir overloaded, 
will bend; thereby generating at its ends forces wnieh are not 
vert, but which will tend to overthrow the supports tt. 


Art. 59. To fiml tlie point of importation of the resultant 
oi parallel forces. Case 1. Two equal or unequal parallel forces, Fig 57, i 


























FORCE IN RIGID BODIES. 


351 


the same direction; and consequently in the same plane. Draw any straight line 
a o, uniting their directions 1 m, 3 n. 

Measure this line, and div it into two 
parts, to, io, proportioned like the 
forces; but placed inversely to tlio 
forces; that is, place the longest part 
near the smallest force, and vice versa. 

Through the point i draw the direc¬ 
tion of the resultant R, parallel to the 
forces. Then is t the point of imparta- 

tion. Make the resultant, ( 4, equal to the sum of the two T>..* JT 
forces. If the two forces are equal, Ii is evidently midway 
between them. 



The foregoing, as well as some other facts connected with parallel 
forces, will be more easily recalled to mind, by associating them with 
the idea of the common steel-yard, Fig 58. Here the two forces in Fig 
67 are represented by the wt 1 ft at c, and the wt 3 fts at d ; one of them 
3 times as great as the other. We know that these weights, when sus¬ 
pended at dists f a, fb, one of them 3 times as great as the other, from 
the fulcrum /, are balanced by their anti resultant fh, which is equal to 
3+1 — 4 fts, or the sum of the two forces. This antiresultant has 
precisely the same tendency to pull the steel-yard upward, that the two 
weights have to pull it downward. Also ft, equal to fh, but acting in 
the opposite direction, is the resultant of the two forces; its effect to 
pull the steel-yard bodily downward, when applied to it at/, is precisely 
equal to that of the two wts applied at a and b. The effect of the two 
wts at a and b, to bend or break the steel-yard, would plainly be very 
diff from that of their resultant «/, applied at/. See Art 58. 



a 




-r—i- 


d T 

-f- 

i 

y s 


4'i 

Fir,58 


Case 2. Two unequal parallel forces, as 
da, 3 tons, and hf, 4 tons. Fig 59, imparted 
to a rigid body in opposite directions, but 
not in the same straight line. 

Starting from the line of direction dm, of the 
i ll force, draw any line m n, passing through 
vt beyond the direction hf of the large one. Now 
und the amount of the resultant R. This, by Art 
56, is equal to the diff between h f and d a, since they 
are in opposite directions; and has the same direc¬ 
tion as the large one; therefore it is equal to 4 —3= 

1 ton; and its direction is the same as that of h /. 

Then say, as the resultant is to the small force d a, 

30 is the dist o to, to the dist o s, along the line to n. 

Through « draw s c, parallel to the two forces ; and 
c will be the reqd point of impartation of the resultant R; the tendency of which to move the entire 
rigid body, will be equal to the joint tendency of daa.ud/if. An anti-resultant of 1 ton at t would 
counteract both the revolving tendency of hf and da (supposing R removed), and their teudency to 
move the bodj' as a whole. 



This case, like the preceding one, is illustrated by the steel-yard, Fig 58; where a d, and fh. repre¬ 
sent the two forces in the last fig; white b c represents their balancing antiresultaot, corresponding to t. 

Case 3. Couples. Two equal parallel forces, a, and b, 

Fig 59 a, imparted to a rigid body in opposite directions, but not 
in the same straight line, are called a couple.. The force of a 
couple, means simply one of the forces ; the perp dist c, between 
the directions of the two forces, is called the arm, or leverage 
of the couple. If one of the forces be mult by this arm, the 
prod is the moment of the couple, in foot-lbs, &c. A couple has 
no tendency to move the entire body forward in the direction 
of either force; but merely to make it rotate around a point o, half-way between 
the points at which the two forces are imparted. No single foree can balance a 
couple. To do this requires a second couple whose moment is equal and opposite to 
that of the first one. Hence a couple cannot have a single force as a resultant. 

Case 4. Any number of parallel forces, whether in the same plane, or in the same 
direction, or not. The process in this case consists in a mere repetition of that in 
Case 1, Fig 57, as follows. 

Namely, commencing with those forces which point in the same direction, as a, b, c, Fig 60, which 
all point downward; between the directions of any two of them, as a and b, draw any straight line 
i, and div it into two parts j o, ji, proportioned like the forces, but placed inversely to the forces. 
'V rough j draw to «, parallel and equal to the forces a and b. Then isms the resultant of those tivo 
Ices. Next find the resultant of this resultant to s and any third force, as c, in the same manner. 
, l is, draw any line t w, uniting their directions; div it into two parts 11 and l to, proportioned 
the forces, but placed inversely. Through l draw n g, and make it equal to to s + c. Then is 



















352 


FORCE IN RIGID BODIES, 


ng the resultant of the three forces a, b, c ; and g is the cen of force of those forces, and consequently j 
is the point of application of their resultant. So do with any number in the same direction. Then 
proceed in the same manner with those which point in the opposite direction, as d, e, x, y; aud j 
having found their resultant, find by Case 2 or 3, or by Art 13 or Art 16, as the case may be, the re- 
sultaut of the two resultaut forces, now obtained, and acting in opposite directions. 

It is not at all necessary that the forces be supposed to act upon a plane surf, as in Fig 60; the pro¬ 
cess is precisely the same when, as in Fig 61. they are imparted at points x, y, z, not in the same 
plane. This is a consequence of the principle laid down in Art IS, namely, that the effect pro¬ 

duced upon a rigid body by an imparted force, remains the same, no matter at what point of the body 
it be imparted, so long as that point is in the line of the direction of that force. 


11 





Although Figs 60 and 61 serve to illustrate the principle, they plainly do not give the actual posi¬ 
tions of the forces and resultant; because they are necessarily drawn in a kind of perspective, so ; 
that all the parts cannot be ineasd by a scale. The amounts of the resultants are easily fouud by 
calculation; inasmuch as they are equal to the sums of the forces. 



The points for imparting them can be found correctly from 
a drawing in plan, like Fig 62; where the stars represent by 
scale the actual dists apart of the directions of the forces. ( 
The quantities of the forces, instead of being shown by lines, | 
are to be written in figures, as showu in the fig. This beiug 
done, it is easy to find the points s, g, &c, of the resultants. 


Art. 60. The Inclined I*Iane is a rigid straight plane surf, as a b , Fig 
63, not hor. If a vert line be be drawn from the top b of the plane, to meet a hot¬ 
line ac, drawn from its bottom a, then be is called the height of the plane; ac its 
base; and a b its length. The angle b a c, which the plane forms with the hor line 
a c, is called its inclination, slope, or steepness; which, however, is frequently ex¬ 
pressed also by the proportion which the 
base bears to the height; thus, if the length 
a c of the base be 1, 1%, 2, &c, times that of 
the height be, the inclination or slope is 
said to be 1 to 1, 1% to 1, 2 to 1, &c. The 
angle b ac is the angle of inclination of the 
plane. 

By Art 57, when one rigid 

body as N or M, Fig 63, is placed loosely upon an¬ 
other, as npon the rigid plane a 5, the effect produced 
by its wt is the same as if all that wt were concen¬ 
trated at its cen of grav g, and acted in the direction 
of a vert line g v drawn through said center. When 
we assume the wt to be thus concentrated at the 
point g, we must remember that all other parts of 
the body must be considered to be without weight; 
although still retaining their inherent cohesive force, or strength. 

If, as in N, this vert line gv, w hich now represents the direction of the entire wt of the body, 
passes beyond, or outside of the base, the body must fall; because this wt meets with no opposing 
force, in the direction vg, to react against it; and thus prevent it from producing motion. 

But if, ns in the body M, the line g v falls within the base r s, the body will not npset: but we shall 
have (Art 19) a force y v equal to the wt of the body, and applied obliquely to a rigid surface a ft, at 

























FORCE IN RIGID BODIES 


353 


the point y; and consequently resolvable into two components; namely, iv, perp to the surf ab, and 
therefore imparted to it as a pressure ; and xv, parallel to the surf, and consequently not imparted 
to it. All these lines may be drawn by scale, to represent their respective forces. When we consider 
a single force as g v to be thus resolved into two components, with a view to ascertaining their effects, 
it is plain that said single force must then be considered as no longer existing; but as being replaced 
by its components. Now the component force xv being parallel to the plane, it follows (Art 15) that 
the pressure or strain iv, no matter how great it may be, cannot in the slightest degree oppose the cross 
action of the moving force x v, uo matter how small it may be ; and xv must theretore produce motion in 
the body, causing it to slide down the plane; unless some third force, not yet spoken of, shall present 
itself, opposed to xv. But any forces which press bodies together, always produce a new force, fric¬ 
tion, at the joint, or surfs of contact of the bodies; and this friction acts in direct opposition to any 
force, in any direction whatever that is in the plane of that joint or surfs. Therefore 

the force iv produces fric between the surfs, r s, of the body M and of the inclined plane; and this 
fric acts in the direction rs, or diametrically opposite to that of the force xv. The amount of fric 
depends upon that of the pres; as also upon the nature of the bodies at whose surfs it is produced, 
upon the degree of smoothness of those surfs, and upon whether they are lubricated or not. If the 
fric is greater than the force x v in the opposite direction, the body of course cannot move; but if less, 
it will move, under the action of a force equal to the excess of x v over the fric. It must be remem¬ 
bered that the pres component iv, which produces fric on an inclined plane, is not equal to the wt 
of the body, but is less than it. It is equal only when the surf is hor, so that the vert force gv, rep¬ 
resenting the entire wt of the body, is at right angles to the joint, aud when, consequently, it all acts 
as pres. Therefore, the steeper the plane becomes, the less is the fric; because then less of the wt 
of the body acts as pres, and more of it as moving force. Hence, a locomotive has less adhesion on 
an inclined grade, than on a level; for the so called adhesion is in reality nothing but fric. But 
although both the perp pres and the fric become less in amount as the plane becomes steeper, yet they 
retain practically the same proportion to each other; until pressures become so great that abrasion 
of the surfs of contact takes place; the proportion of the fric to the pres then increases. 

Rem. It is evident that when we wish to push a body up an inclined plane, we must overcome 
both the fric, and the parallel force x v ; but in pushing it down, we are opposed only by the fric ; for 
the parallel force assists us. 

Art. 61. Experiment has determined the amount of fric which takes place be¬ 
tween the surfs of such materials as are employed in construction ; that is, it has 
determined the proportion (or, more correctly, the ratio) between the pres and the 
fric. Any person may easily do this for himself, thus: A body r.s M, Fig 63, is 
placed upon the plane surf ab; of which one end, as b, is gradually raised until the 
body is barely about to begin to slide. When this takes place, we know that the 
force, x v has become barely equal to the fric; and the angle b a c, which the plane then 
makes with the hor a c, is called the angle of friction, or limiting angle of resistance, 
or angle of repose, for the particular kind of surf experimented on. 

Now, a little reflection will show that whatever *iay be this angle, bac. of fric. the line which 
measures uot. only the parallel force, but also the fric existing at that moment, (and at no other one,) 
is to the line i v, "which measures the perp pres, (not the wt of the body ;) as the vertical height h c 
of the plane at the same moment, is to its hor base a c. That is, at the point of sliding, as Fric¬ 
tion : Perp Pressure :: Height : Base; or, as Base : Ht :: Perp Pres : Friction. Therefore, when 
a body barely begins to slide, measure a c hor, and b c vert; div the last by the first, and the quot 
will be the proportion which the fric of the bodies experimented upon, bears to the pres which causes 
it. Or, measure the angle bac in degrees. &c; the nat tang of this angle will be that same propor¬ 
tion. This proportion is called the coefficient of friction for those bodies; a table of which will be 
found on page 373. A hor line dg. drawn from g. and terminating in vi extended, will, when 
measd by the same scale as gv, iv, xv. give a hor force which, without the aid of friction, would 
react against the force xv, and prevent it from moving the body down the plane. Or if the length 
a b of the plane be taken by scale to represent the wt of a body, then 6 l, perp to a b, to meet a c pro¬ 
duced at l, will give that same hor force. 

Art. 62. If the length m n, Fig 63^, of an inclined plane, be taken by a scale, 
to represent the wt in lbs, tons. Ac, of any body placed upon it; then the base on 
will, by the same scale, give the perp pres in lbs, 
tons, Ac, which the body imparts to the surf ot 
the plane; and the height mo will give the 
amount of force parallel to m n, aud which tends 
to move the body down the plane, either by 
sliding or rolling. If the pres on be mult by 
the proper coeff of fric, the prod will plainly be 
the actual amount of fric in ft>8, Ac. II the trie 
thus obtained proves to be greater than the 
sliding force m o, then the body will remain at 
rest on the plane; hut if less, then sliding or 
rolling down the plane will he the result; and 
the amount of force which starts or begins the 
motion, will be equal to the excess of m o over the fric. 

As the motion continues, it will be accelerated by the accumulation of gravity. 

When a body is placed upon an inclined plane, whether it slides or not, the pres which it pro¬ 
duces at right angles to the surf of the plane, is equal to the wt X nat oosine of angle of slope: 
the sliding force parallel to the surface of the plane, - wt X nat sine of angle of slope; the actual 
amount of fric = wt X nat cosine angle of slope X coeff of fric. For the fric does not vary as the 
angle of slope of the plane, but as the cosine of that angle ; in the same manner as the perp pres 

varies. 

Ex. 1. Suppose we wish to slide a wooden box M, Fig 63, filled with stone, and weighing in all 






354 


FORCE IN RIGID BODIES 


1200 lbs, up the iron rails of an inclined plane, sloping 5°; what force must we use, parallel to the 
plane; assuming the coeti of wood on irou to be .4, or of the perp pres? Here we have to over¬ 
come the parallel force x v, and the fric. Now, as just stated, this parallel force x v is equal 
to, wt X uat sine of slope, — 1200 X .087 101.1 lbs. The fric is equal to, wt X uat cos of slope X 

coeff of fric: = 1200 X .096 X -1 — US. Consequently, 101.1 -(- 178 = 582.1 lbs, is the force reqd. In 
fact, however, this force merely balances the downward tendency of the box, together with its fric; 
thus rendering them incapable of resisting any additional upward force; but it is plain that we must 
apply some additional force, in order to impart motion to the now unresisting box. 

Now, suppose we wish to slide the box down the plane, what force must we use? Here nothing re¬ 
sists us but the fric, just fouud to be 178 lbs. The parallel force helps us to the amount of 101.4 lbs; 
therefore we need only to add 178 — 104.4 — 373.6 ttu. 

The box will theu be upon the point of moving. Any additional force will move it. 

For acceleration on inclined planes see p363. 

The following table will facilitate calculations respecting the draft required on grades, inclined 

planes, 4c. In practice, allowance tor friction must be made 
in the last 2 cols. Original. 


Inclination or Slope of the Plane. 


The slopiug length is 


vertical height 
sine, col 6 . 


Vert. 

Hor. 

Ft. per mile. 

I Deg 

Min 

1 

in 

3. 

1760.00 

18 

26 

1 

in 

4. 

1320.00 

14 

2 

1 

in 

5. 

1056.00 

11 

19 

1 

in 

6 . 

880.00 

9 

28 

1 

in 

8 . 

660.00 ’ 

7 

8 

1 

in 

9. 

588.66 

6 

20 

1 

in 

10 . 

528.00 

5 

43 

1 

in 

11.4 

161.94 

5 

00 

1 

in 

12 . 

410.00 

4 

16 

1 

in 

14.3 

369.23 

4 

00 

1 

in 

15. 

352.00 

3 

49 

1 

in 

19.1 

276.73 

3 

00 

1 

in 

, 20 . 

261.00 

2 

52 

1 

in 

23.1 

229.01 

2 

30 

1 

in 

25. 

211.20 

2 

17 

1 

in 

28.6 

181.36 

2 

00 

I 

in 

30. 

176.00 

1 

55 

1 

in 

32.7 

161.47 

1 

15 

1 

in 

35. 

150.86 

1 

38 

1 

iu 

38.2 

138.22 

1 

30 

1 

in 

10 . 

132.00 

1 

26 

1 

in 

15.8 

115.29 

1 

15 

1 

in 

50. 

105.60 

1 

9 

1 

in 

57.3 

92.16 

1 

0 

1 

in 

60. 

88.00 

0 

57% 

1 

iu 

70. 

75.13 

0 

19 

1 

in 

76.1 

69.12 

0 

15 

1 

in 

80. 

66.00 

0 

43 

1 

iu 

90. 

58.67 

0 

38 

1 

in 

100 . 

52.80 

0 

34 

1 

in 

114.6 

16.07 

0 

30 

1 

in 

125. 

42.21 

0 

•17% 

1 

in 

150. 

35.20 

0 

23 

1 

in 

175. 

30.17 

0 

19** 

1 

iu 

200 . 

26.10 

0 

17 

1 

in 

229.2 

23.01 

0 

15 

1 

in 

250. 

21.12 

0 

14 

1 

in 

300. 

17.60 

0 

11 ** 

1 

in 

343.9 

15.35 

0 

10 

1 

in 

400. 

13.20 

0 

8 % 

1 

in 

500. 

10.56 

0 

7 

1 

in 

600. 

8.80 

0 

6 

1 

in 

800. 

6.60 

0 

4 % 

1 

in 

1000 . 

5.28 

0 

3?* 

1 

in 

3137. 

1.54 

0 

1 


Level. 

0.00 

0 

0 


Pres, on 
Plane, in 
parts of the 
wt. Or, nat. 
cos. of angle 
of Plane. 

Pres, on 
Plane, in lbs 
per ton. 

Tendency 
down the 
Plane, in 
parts of the 
wt. Or. nat. 
sine of angle 
of Plane. 

Tendency 
down the 
Plane, in iba 
per ton. 

.9487 

2125 

.3162 

708. 

.9702 

2173 

.2125 

513. 

.9806 

2196 

.1962 

139. 

.9864 

2210 

.1615 

368. 

.9923 

2223 

.1212 

278. 

.9939 

2226 

.1103 

247. 

.9950 

2229 

.0996 

223. 

.9962 

2231 

.0872 

195. 

.9965 

2232 

.0831 

186. 

.9976 

2232 

.0698 

156. 

.9978 

2233 

.0666 

119. 

.9986 

2237 

.0523 

117. 

.9987 

44 

.0500 

112 . 

.9990 

44 

.0436 

97.7 

.9992 

2238 

.0398 

89.2 

.9994 


.0319 

78.2 

“ 

<4 

.03*1 

74.8 

.9995 

2239 

.0305 

68.1 

.9996 

44 

.0285 

63.8 

.9997 

2210 

.0262 

58.6 

“ 

<4 

.0250 

56.0 

<4 

44 

.0218 

18 8 

.9998 

44 

.0201 

15.0 

4 « 

4 4 

.0175 

39.1 

.9999 

44 

.0167 

37.1 

4 « 

*4 

.0113 

32.0 


*4 

.0131 

29.3 

44 

(4 

.0125 

28.0 


44 

.0111 

24.9 

1 . 0000 * 

44 

.0100 

22.4 


44 

.0087 

19.6 


It 

.0080 

17.9 


44 

.0067 

15.0 


44 

.0057 

12.8 


41 

.0050 

11.2 


44 

.0014 

9.77 


44 

.0011 

9.18 

44 

41 

.0033 

7.39 


11 

.0029 

6.52 


II 

.0025 

5.60 


11 

.0020 

1 18 


41 

.0017 

3.81 


44 

.0013 

2.91 


44 

.0010 

2.24 


«l 

.0003 

0.65 


4 ( 

.0000 

0.00 


For other tables of grades, see pp 176, 723, 724, 725. 

Art. 63. The following principle is one of great practical importance. When 


* Near enou 8 h for P raet ' ce 1 actually .99995, or less by one part in 20000, or about 1 II. inVtm^ 




































FORCE IN RIGID BODIES. 


355 






Pig 65 


a plane w x, Fig 64, has just that inclination, y x w, at which the fric of any given 
body is balanced; and sliding is about to commence, from 
the action of any force h o, applied to the plane, through 
the body, in any direction k o, not perp to it; if from the 
point o of application, we draw a line op, at right angles to 
the surf of the plane, then the angle hop will always be 
equal to the angle of fric y x w of the body. If the plane 
is so steep that sliding must take place, then the angle 
formed between the force ho, and a perp op to the plane, be¬ 
comes greater than the angle of fric; but if the steepness is 
so slight that the body rests firmly on the plane, then said 
angle is less than the angle of fric. 

The practical applications of this principle are very numerous; they 
extend to pressures in any directions whatever; and apply to plane 
surfs in any position whatever, whether inclined, vert, or hor ; for any 
given pres produces precisely the same amount of fric, whether we im¬ 
part it to the ceiling, the floor, or the wall of a room; provided they all 
be of the same material. The angle of fric of cut stone 
upon cut stone is about 32°; that is, one block of cut stone 
will not slide upon another at a less slope than about 32°; 
the fric then being full -jfh 0 f the pres. Therefore, if the 
floor /, Fig 65, ceiling c, and walls w, w, of a room be of 
cut stone; and p, p, p, p, lines at right angles to them ; we 
may press a piece s of cut stone against them with any 
force whatever , applied in the direction of the stone itself, 
without danger of its sliding; provided only that the di¬ 
rection of the pres along s does not form with the perp p 
an angle exceeding 32°. But sliding will take place, 
whether the pres be great or small, if, as at o, o, o, o, said 
angle exceeds 32°. The angle of fric is, by some writers, 
called, in such cases, the limiting*angle of 

resistance. 

Rem. The friction at the feet of 
rafters when highly inclined diminishes very 
much their horizontal pressure and tendency to 
split off the ends of the tie-beams. 

The angle of fric of oak endwise against hard limestone, is, according to 
Morin, 20%°; therefore, if the walls, &c, of a room consisted of such lime¬ 
stone, we could not press a piece of oak endwise against it without sliding, 
if the angle withp exceeded 20%°; and the legs of a wooden trestle, Fig 66, 
would not spread, on the level surf of such limestone, under any wt w, if 
the angle ahc be less than 20%°; but certainly would if it be greater, unless 
other preventives besides fric at the feet be depended on. In this case the 
fric amounts to very nearly of the pres; that being the proportion cor¬ 
responding to 20%°. These two illustrations show how wide is the applica¬ 
tion of this principle: for the announcement of which we are (the writer , . _... . 

believes) indebted to Moseley. 

Fi»66 

Art. 64. To find the effector an extraneous force (fa, Fig 67 J 
imparted in any direction, to a rigid body (B) on an inclined 

plane, i Ji 5 when we know the angle ot fric, and the wt of the body, lhe prin¬ 
ciple laid down in the preceding Art 
enables us to do this. 

Through the cen of grav c, of the 
body, draw avert line aw, and extend 
the direction f g of the force, to meet 
this line, as at o. Make o a by scale, to 
represent the wt of the body: and o z 
to represent the amount oi the force 
fg. Then is o a point at which we 
may assume both these forces to be im¬ 
parted to the body. (Art 29.) Complete 
the parallelogram of forces a x z o, by 
drawing a x, and 2 x, parallel and 
equal to o z, and o a. Draw the diag 
x o and extend it to meet the plane, as . 

at < Make the line t v perp to the surf of the plane. This done, we have a single 
force x o, equal in effect upon the rigid body, to its wt, and /g combined. 

This single force may (Art 18) be considered as imparted to the body at any point that lies in its 
line of direction x /; therefore/we will assume it to be imported at t where it encounters the force 
of fric acting in the direction « «, of the joint formed between the body, and the plane. Now. l \ 
atrikuc within the base s e t v being at right angles to this joint, it follows from the last Art, that 
if the angle x t v is less than the angle of fric corresponding to the nature of the materials which 



V, 


a ; - 

• / 


0!-<— 
cf B 

Z O' 
/ 53 




11 , 


W 


Fio 67 






































356 


FORCE IN RIGID BODIES, 


compose the body and plane, then the body will remain at rest on the ptane. But if the angle xtv 
be greater than said angle of fric, the body will slide up or down the plane, (according to circum- 
stances, stated in the next paragraph ;) if the augles be equal, the body will be just on the poiut of 
beginning to slide either up or down. 

When the angle x t v is on the down hill side of v t, as ia the tig, the tendency of the body will evi- 1 
dently be to move up the plane; but if, in cousequeuce of a diff direction of the force / g, (and couse- | 
queutly of the resultant x o,) the angle x t v is on the up hill side of t> t, then the tendency will be 
down the plane. 

Rem. 1. If the direction of the resultant i o, or the point t, falls outside of its base s e, the body, 
instead of sliding, will upset. It will fall up hill, if t strikes p i up hill from the base; and down hill, 
if t strikes down hill from the base. See Art 65. 

Rem. 2. In order to draw the parallelogram of forces ax z o, and its resultant diag x o, the line a o, 
which represents the wt, may sometimes have to be regarded as pulling instead of pushing down¬ 
ward at the point o, where the other force meets it. See Fig aud Fig 69. below. 



Rem. 3. It follows from the foregoing, that when at the joints p q, r s, 
Fig 68, of a mass of masonry ; or at the joints of timbers iu carpentry, ‘ 
iron work, &c, the fric alone is depended on to prevent sliding, the re¬ 
sultant as m c, c o, o n, &c, of all the forces acting at any joint, must not 
form an angle m c i, c o a, o n e, with a perp ci,o»,n«, to the joint, 
greater than the angle of fric corresponding to the nature of the ma¬ 
terials whose surfaces constitute the joint. 

Rem. 4. The extraneous force reqd to move a body up a plane, will 
be the least when its direction, t n. Fig 67, makes with surf i p, of tUc 
plane, an angle, n i p, equal to the angle of fric. 



Art. 65. To find the force required to prevent a 

body S, Fig 69, from moving; when the direction, ow, of its wt, 
strikes outside of its base tt'. Thus, suppose we wish to impart 
a pulling force at e, and iu the direction ea, to prevent the body 
from sliding either up or down the frictionless plane ip. and 
from upsetting down the plane. Through the cenof grav c, draw 
a vert line xw\ and continue the line of direction of a e to meet 
it at o. From o draw oy at right angles to the plane ip. By scale 
make ow equal to the wt of the body : and from w draw wy paral¬ 
lel to o a. Make e a equal to wy; then is e a the reqd force, which 
will resist all tendency of the body to move. For in the par¬ 
allelogram of forces o w y z, we have'the force o w tending to make 
the body fall; aud the force o z (equal to e a) tending to prevent 
it from falling; and the resultant o y, of these two forces, equal to 
their joint effect, is at right angles to the surf of the plane; and 
is consequently (Art 19) all imparted to it as pres; no part being 
left unresisted, to produce motion iu any direction. For as before 
said, when two forces, as o w, o z, are compounded into one result¬ 
ant force o y, those two forces must be considered as no longer ex¬ 
isting ; thus, in this case, so long as we regard the joint effect of o w aud o z as being concentrated 
in their resultant o y. we cannot of course, consider them as acting in other directions at the same 
time ; so that there is, as it were, no longer any wt. o w, tending to make the body fall; nor any force 
e a, tendiug to uphold it; but only the single force oy. which presses the inert body against, and at 
right angles to, the surf ip; imparting to it a tendency to move only iu the direction o y ; which ten¬ 
dency is reacted against by the inherent 
cohesive force, or strength, of the plane. 

If the body is prevented, by friction or 
by a stop at its lower toe t, from sliding 
down the plane, and we wish to know 
the least force in the direction oz which 
will just prevent the body from over¬ 
turning about t ; the line oy, instead of 
being drawn perpendicular to the plane 
ip as in Fig 69, must be drawn from o 
through the lower toe t as in Fig 69 a. 

The lines ow and zy of the parallel¬ 
ogram ozyw, representing the weight 
of the body, will of course remain the 
same as for the same body in Fig 69. but 
the lines o z and w y, representing the 
extraneous force, although the same in 
direction as before, will evidently be much shorter. 



Fi£\70 

o 



Art. 66. Stability. The stability of a structure, or of any body, is, strictly 
speaking, that resistance which its wt alone enables it to oppose against forces tend- \ 
ing to change its position. Such resistance may be assisted by extraneous wts, or 
by other forces properly applied; but such must he distinguished from the stability 
inherent in the structure, or body itself. To insure the stability of a structure, the 
disposition of its parts, as well as that of the entire mass, must he such that neither 
of them shall move, either by sliding , or by overturning , under the action of the im¬ 
parted forces. Stability is therefore a branch of Statics; or of forces at rest, or iD 
equilibrium with each other; Art 16. 





















FORCE IN RIGID BODIES. 


357 


Stability must not be confounded with strength. A structure 

may be very strong, and yet very uustable. A block of stone is quite as strong while sliding down 
a smooth plane, or rolling down a steep bank, as when resting on a firm hor base ; but it has stability 
only in the last case. A pyramid of weak chalk may have great stability ; while a globe of granite 
or cast iron, has very little. We generally have to examine into the strength, as welfas the stability 
of our structures: but it must be done by diff processes. The stability has reference to the structure 
considered as consisting of one or more rigid bodies, which may be moved as entire masses, but not 
broken, or changed in form, by the applied forces. See Remark 2, Art 29. 

Those forces which tend to impair the stability of a structure, are called acting ones ; and those 
which tend to maintain it, resisting ones. This distinction is merely a matter of convenience- for 
all the forces act, and resist. 

The forces which affect the stability or a rigid structure considered as one mass, are its wt; extra¬ 
neous wts, or strains; and the foundation, or support; which last reacts as an antiresultant 
against the others. When these three balance each other, the structure is stable. When the struc¬ 
ture is to be considered as composed of several rigid bodies, then the joints or surfaces of contact be¬ 
tween these bodies must also be regarded as so many secondary foundations, and these also must re¬ 
spectively balance the forces acting upon them; otherwise these parts may slide, or overturn, while 
other parts may remain firm. 

• In order to guard against accidents, a structure must generally be so designed as to 
be capable of resisting much greater forces than those which it sustains under ordinary circum¬ 
stances. The proportion which, with this object, we give to the resisting forces, in excess of the act¬ 
ing ones, is called the coefficient of stability; or simply the stability, or the safety, of the structure. 
Thus, if we make it capable of resisting 2, 3, or 6 times the amount of the ordinary acting forces, we 
say it has a stability, or a coeff of stability, or a safety, of 2, 3, or 6. 

Art. 68. Since the stability of a structure, considered apart from its founda¬ 
tion, consists entirely in the resistance which its several parts, as well as the entire 
mass, can present against both sliding and overturning, it follows that two precau¬ 
tions, already adverted to in previous articles, must be resorted to. Namely, 1st, 
against sliding', take care that the resultant of all the forces acting upon 
any joint, (including that between the base and the foundation,) shall act either at 
right angles to said joint; so as to be entirely imparted to it as strain, (press or pull,) 
leaving no part unresisted to tend to produce motion; or else that it shall not de¬ 
viate from a right angle, to a greater extent than the angle of fric corresponding to 
the materials which compose the joint; so that the portion of it which is not im¬ 
parted at right angles, shall be resisted by friction; and thus be prevented from 
producing a motion of sliding. 

Otherwise, instead of relying upon the position of the joints, resort must be had to the cohesive 
strength of joint-fastenings, such as bolts, spikes, cramps, joggles, mortises and tenons, mortar, 
cement, &c, to prevent sliding. As to mortar and cement, however, it is important to remember that 
frequently, and especially in very massive work, they have not time to harden, or acquire their full 
strength, before the acting forces are broueht to bear upon them : therefore, great care is necessary, 
when we use them as substitutes for position. On this account we frequently cannot consider a mass 
of masonry to be a single rigid body, but must reghrd it as composed of several detached rigid 
bodies ; the stability of each of which must be separately provided for, before we can secure that of 
the whole. Therefore, in large massive structures of importance, we should, as far as possible, omit 
all consideration of the strength of the mortar, and rely for stability chiefly upon placing the joints 
at or nearly at right angles to the forces acting upon them. 

Art. 69. Moment of stability. We have already stated (see Arts 46 
and 49) that the resistance which any rigid body as B, Fig 71, opposes against being 
overturned about any given point a, is equal to that 
which would be produced if the entire wt of the 
body were concentrated at its cen of grav g ; and 
acted at the end % of a straight lever a i, of which a 
is the fulcrum; or at the end o, s, or n, of airy straight 
lever (Art 49) ao, as. a n ; or of any bent lever a i n, 
a is, a in, a s n, provided that in every case there is 
the same leverage at, measured from the fulcrum a, 
and at right angles to the direction m n of the force 
of grav of the body. So far as regards tendency to 
resist overturning, it is immaterial (Art 18) at what 
point of the body, in tins line, of direction in n, we 
conceive the grav to act: or whether as a push at o, 
or a pull at t, as denoted by the arrows. We have 
also said that the tendency, or moment, of this force 
of grav, or wt, to produce or to resist motion about the fulcrum a, through the me¬ 
dium of any of these levers, is found bymult the force or wt in lbs, by the leverage 
ai in feet. The prod in ft-Ibs is generally sailed the moment of the force about the. 
point a ; but in cases like that before us, in which this moment becomes the measure 
of the stability of the body, it is called the moment of stability (or simply the sia- 






358 


FORCE IN RIGID BODIES. 



bility) of the body , about that point. Therefore, if bodies of the same size and shape 
and with their centers of gravity in the same position, have dill wts, or sp gravities, 
their respective stabilities will be in proportion to their wts, or sp grs. 

A body may have diff moments of stability, about diff points. Thus, it would be 
far more difficult to overturn B about the point b, than about a ; because the le\er- 
ige bi is ‘l]/ 2 times as great as at; and since the wt and the point ot the cen ol grav 
remain unchanged, the moment about b is 2% times as great as about a. 

Rem. 1. Let a b c o, Fig 72, be a squared block of stoue 6 feet long; ou a hor base ; and weighing 12 

tons ; and h, a force applied to overturn it about the toe o. Since its 
length o c, is 6 feet, its cen of grav t, will be dist o y, or 3 ft back from 
Consequently, the moment with which the block resisl 
u-turned, is 12 (tons) X 3 (ft leverage) = 36 fttons. Now, 


o. Consequently, tne moment witu »in«u iuc uiv^n resists beiug 
overturned, is 12 (tons) X 3 (ft leverage) = 36 ft tons. Now. suppose 
the upper half a b o, to be removed; the remainder o b c will weigh 
but 6 tons. But its cen of grav », is farther from o, than that of the 
whole block was. Being now triangular in shape, the dist o y will be 
% of o c; or will be 1 ft. Consequently, the resisting moment will be 
6 (tons) X 4 (ft leverage) = 24 foot-tons. So that although the block 
has but half the wt of the first one, it has % as great resisting 
liVff VP power. It is on this principle, that in order to save masonry, the 

JD 1 U / U f ace8 of retaining-walls. <fcc, are sloped, or battered back. 

Rf.m 2. In Fig T>]4 let the upper part, 
cn, of A, and the lower part, oy, of B, be 
made of lead, and the remaining part of 
each, of cork. Then the center of gravity 
? of the entire body A will be near the dot t ; 
and that of B, near the dot s. Let the 
weights of A and B be equal, and their cen¬ 
ters of gravity, t and s, at the same hori¬ 
zontal distance, ao, from the fulcrums, o 
and o, around which the bodies are to be 
overturned. Their moments of stability 
must then be equal. Consequently they will 
require, in order to begin to overturn them, 
equal forces, applied in any same given 
direction, and at any same given point; 
as, for example, the equal forces, / and g, 
applied horizontally at i and i. But, in 
order to entirely overthrow B, the overturn¬ 
ing force must act through n greater space, 
must do more work, than is required to overthrow A; for A will be overthrown 



Fiy 7Z -£ 


t e, 


when the force,/, has acted through only the small space necessary to move it into 
the position of the dotted lines J. Its center of gravity being then at e, which is 
beyond the base, A must necessarily fall. But the force, g, acting upon B, must move 
through the longer space necessary to move B into the position N ; for not until then 
will its center of gravity, s, be in a position, /, beyond the base. As either A or B 
turns, about its fulcrum o, from its original position (A or B) to that (J or N) in which 
it is about to fall, its leverage, ao, of stability plainly diminishes, until the center 
of gravity is over o. The leverage is then = 0. Since the weight remains unchanged, 
the moment of stability decreases (and ceases) with the leverage- of stability, as does 
also the overturning moment required to keep the body moving. The overturning 
leverage varies with the rising, and subsequent falling, of the corner, i, at which 
the force is applied; but in bodies shaped like A and B this variation is but slight 
and hence the horizontal force, f or g, required, decreases nearly as the leverage of 
stability decreases, ceasing when it ceases. The object of the civil engineer is gen¬ 
erally to secure his structures against beginning to overturn. Hence he is chiefly 
anxious about the amount of the force, f or g, f in pounds, tons, etc) required to start 
motion; and seldom needs to concern himself about the amount of work (in foot¬ 
pounds, foot-tons, etc) required to complete the overthrow. 

Rkm. 3. It is not alone the wt of the body itself, which contributes to its stability in all cases; for 
this may be assisted by extraneous wts or loads. Thus, the wt of a pier P, Fig 73, gives it in itself a 
certain degree of stability ; but when we add the wt of the two equal arches, 
its stability is thereby increased, supposing the foundation to be secure. And 
a passing load, when it is directly over the pier, increases it still more. It is 
true that the wt of the arches might crush the pier to fragments, if the stone 
be soft; but this is a matter of Strength of Materials ; not of stability ; and 
must be examined into by itself. If the two arches be of unequal sizes, or if 
there be but one arch, the stability of the pier may become either increased 
or diminished, according to circumstances: as will appear farther on. 

Whether the force acting for or against the stability of a structure, be 
gravity, or pushes, or pulls, produced from other sources, is, as in other cases, 
a matter of no importance: for force is simply force, no matter whence de¬ 
rived. We have, therefore, only to look at the diff forces acting upou our 
structures, as so many tendencies to produce motions in certain directions. If these tendencies are 
reacted against, or destroyed by others, they will not produce it; but if they meet no resistance, mo¬ 
tion must take place. The only peculiarity we need assign to grav. is that the direction of its action 
x* always vert downward ; while other forces may he imparted in either that, or any other direction, 
hie resultant of grav combined with other force or forces, may be in any direction. 





Fio73 






















































FORCE IN RIGID BODIES. 


359 


Art. 70. Fig 73^. In the principal cases of 
to the civil engineer, both the acting and resisting 
forces iv x, /, /, / &c, may all be considered to be 
imparted and acting in the same plane; which is a 
vertical one, eloc, passing through the center of 
gravity, v, of the’structure, mnpqrstu: and of 
course, coinciding with the’ line of direction w x, 
of its weight, or force of gravity. The plane eloc, 
and all the forces, therefore, may be considered as 
coinciding with a leaf of paper standing vertically 
on one edge. This renders the calculations much 
more simple than if the forces were in different 
planes; in which case diagrams alone would not 
suffice for determining the resultants, leverages, 
moments, &c. See Art 44. Whereas, when they 
are in the same plane, such diagrams, neatly 
drawn to a convenient scale, will usually possess 
all the accuracy required in practice. 


stability that present themselves 


III B 




Thus, In examining the strains on the diff pieces composing the 
truss of a roof, or of a bridge, &c, not only their wts, but all the 
Btraiuing forces, may be assumed to act in a vert plane passing 
lengthwise of the truss from end to end : splitting it into two equal 
parts. In the case of a structure of masonry, such as a retaining- 
wall, or a stone bridge B, Pig 74, we base our calculations upon a 
thin vert slice of it. having a length a o, i i, or y f, of only one ft; 
no matter what may be the height y s; or the thickness y c, of the 
structure. Through the center of this one ft of length, we suppose 
a vert plane to pass; splitting, as it were, the body B into two pre¬ 
cisely similar parts; and all the forces actually diffused equally 
throughout the whole one ft of length, are supposed to be concen¬ 
trated. and to act upon one another, in this plane. See Remark, 
Arts 49 and 57, 



Art. 71. In order to ascertain the effect of diff forces to 
produce either sliding, or overturning, of either un entire rigid 
body, or of one composed of several rigid bodies placed together without joint- 
fastenings; we first find the direction of the resultant of those forces. If the body 
or structure is not composed of diff parts ; or if, being so composed, these parts are 
so firmly united together by joint fastenings, as to constitute virtually but a single 
rigid mass, then we need do nothing more than (as in Art 35) find the resultant ad, 
Fig 19, of all the forces. If the direction of this resultant strikes inside of the base, 
the body will not overturn (Bemark 2, Art 72, ); and if, besides striking inside 

of the base, it forms with a line b x, at right angles to the base, and angle abx, less 
than the angle of fric between the body and its support n m, then it cannot slide. 

At Fi'e, 67, the direction of the resultant x o strikes at t. inside of the base s e ; therefore the 

body B cannot upset; whether it will slide or not, depends upon whether the angle o t v is greater or 
less’than the angle of fric corresponding to the materials composing the body, and the plane. If both 
are of ordinary dressed granite, this angle must not exceed about 32°. See other angles, under head 
Friction. 

Questions on overturning, may also be solved on the principle of leverage. See Art 49, 

Art. 72. We will now consider a case in which the structure is assumed to be 
composed of several rigid bodies, merely placed together without joint-fastenings of 
any kind, such as cramps, bolts, mortar, &c; but depending entirely upon their wt 
and positions, for securing their stability. The process in this case is the same as 
in Fig 19, except that instead of assuming the body to have but the one joint 

n m; and finding the effect of the resultant ad with reference to this joint alone; 
we consider it to have several joints, as P Z, F L, W X, &c, Fig 75; and then examine 
the effect of the diff resultants which they must respectively sustain in consequence 
of the diff wts of the several parts NMPZ, NMFL, NMWX, resting on them. 

het HK JT be one half of a stone atfch ; or rather a vert slice of 
it, 1 ft thick; and let NMWX be a similar slice of a dressed stone abut which has 
been designed to sustain the thrust of the arch; and the fitness of which for the 
purpose, we wish to ascertain. 

Suppose the thrust of the slice of the arch, (that is. the resultant of its wt, and of its hor pres,7 to 
have been previously ascertained by Ex 2, p 830, to be 30 tons; that this is concentrated at o, (the 
center of the skewback ;) and that its direction is o b. Also, suppose the wt of the part NMPZ be 
found to be 10 tons, and to be concentrated at its cen of grav G ; and, consequently, to act in the 
vert direction G a. Now (Arts 18 and 35) we may suppose the 30 tons resultant of the arch, and the 
10 tons grav of the part N M P Z, to act at the same point c, at which their directions G a and o b meet. 
Make c d by scale equal 30 tons, and ca 10 tons; complete the parallelogram of forces cdya ; and 
draw its diag cy, which by the same scale will give the resultant of all the forces acting upon the 
part NMPZ. 

Now we see that the direction cy of this resultant strikes at i, or within the base P Z; consequently 
(Art 60, &c> NMPZ cannot upset, no matter how great may be the pres cy; see Remark 2. From i 






































360 


FORCE IN RIGID BODIES. 


draw it, at right angles to the line P Z; and measure the angle c it, which the resultant cy forms 
with it. This® we find, is greater than 32°; that is, it exceeds the angle of fric between surfaces of 
dressed stone. Therefore, the part N M P Z must slide along the joint P Z. Tins might P°s slb 'y 
prevented by good mortar, if time be allowed it to solidify properly, before the centets are eased so 


H 


w 


NT 


IYl 



m t# bring the pres of the arch upon the abut; or by iron cramps, stone joggles, Ac; but these are 
expensive. The most obvious remedy, as well as the least expensive, is simply to incline the joint 
PZ into a direction somewhat like from It to Z : so as to receive the pres of the resultant cy more 
nearly at right augles ; at least so nearly as to be fairly within the limits of the angle of Iric. If this 
is done, stability is secured ; for the part N M P Z, being now safe against both sliding and overturn¬ 
ing, can move in no other way ; unless the strength of the stoue composing the masonry is insufficient 
to bear the pres, aud may therefore crush to pieces under it. But this is a question of Strength of 
Materials. The point i, of Fig 75, comes much nearer to Z than would be 

desirable in practice ; for it might cause crushing at Z. See Item 2, following. 

Having thus provided for both the sliding and the overturning stability of the abut as far down as 
the joint P Z, we will now examiue as far down as the joint F L. Taking the entire part N M F L of 
the abut, we first find its weight, say 25 tons; and this we assume to act at its cen of grav K., and in 
the vert direction v K l. The amount and direction of the thrust of the arch at o, of course, remain 
as before. Therefore, from the point v, where the two directions meet, lay off tie to represent as 
before the 30 tons thrust of the arch ; aud v l, the 25 tons wt of the abut. Complete the parallelogram 
of forces v e si, and draw its diag v s ; which, measd by the same scale, will give the resultant of all 
the forces acting upon the part N M F L. Now we see that the direction of this resultant does not 
fall within the base F L; but. on the coutrary, passes out of the body aty; outside of which it meets 
no force to resist it. Consequently, since this resultant must be considered as an 

only force acting upon an inert body or abut.'N M F L, without wt, (Art 35.) that body must upset 
around the point L ; or around the nearest joint in the masonry between L andy ; and' cannot con¬ 
tinue to stand of itself, unless its base be abovey'. It is true that bv p'acing earth behind it. espe¬ 
cially if well compacted by ramming, the abut of a small arch might be made to stand safely even 
upon the base W X ; and in the case of arches of moderate spans, this aid may be resorted to for 
strengthening the abuts, when there is no danger that the earth may be washed away by Hoods or 
rains, and thus expose them to ruin ; and this is generally and properly done. 

Rkm. 1. If in the same manner that the point i was found in the joint PZ. others between P Z and 
W X be determined also: then a curve, commencing at the skewback o, and drawn through them, 
will represent the line of pressures or of resistance, or of thrust, through the 

abut. At any point whatever in this line, say at i, the entire pres above said point may be supposed 
to be concentrated ; while the entire length, as cy, of that resultant which cuts said point, gives the 
amount of said pres at that point; and the direction, as c t, of the same resultant is also the direc¬ 
tion in which said pres acts upon said point. See Art 13 of Hydrostatics, p 231. 

Rkm. 2. The line of pres enables us to determine another very important point connected with the 
stability of a structure. It is not sufficient in practice that this line should strike merely within the 
base; it must strike at a considerable dist within. If the structure aud its foundation were abso¬ 
lutely rigid, so that no conceivable force could bend or break them, this would not be necessary ; but 
all materials are more or less weak, so that, if great pressures come ton near to their edges, there is 
danger of splitting or crushing at those points ; or if near the edge of a base, an unequal settlement 
of the soil beneath may take place. Therefore, even in structures of but small size, the dist i Z, Fig 
75, of the line ot pres, from the outer point Z. should never be less at any joint than y of the width 
•f that joint; except, perhaps, in a case like that of a small arch in which the earth filling is depos- 


























FORCE IN RIGID BODIES, 


361 


ited behind the abuts before the centers are removed. In important works, it should not be less than 
about % of the width of the joint; and it is still better, when possible, at ]4 ; or, in other words, at 
the center of each joint. When, as at Q, a footing U is added at a base, W X should be taken as the 
joint; not W Q. 

Rem. 3. Tlie line of pressure in an arch itself, as H J N T, Fig 75, 

may also be found much in the same way, thus: First divide the half arch II J N T, 
and the filling above it, by vert lines ru, wx , Ac, which need not be at equal dists 
apart. Four such lines will suffice for a flat arch, and about six for a semicircular 
one. We then consider in turn, and separately, each part, as r u H J, w x H J, 
N D T H J, Ac, which extends from these lines to the center H J of the half arch; 
the last of these being the entire half arch. The cen of grav, and the wt, of each 
of these parts, must be found; also, (Fx 6, p 341,) the hor pres at the keystone. 

Now each of these parts, like the beam in Ex I, or the half arch in Ex 2, p 330, is acted upon, and 
kept in equilibrium, bv three forces; namely, the hor pres at the keystone, (see Ex 6, p 341;) its own 
wt, acting vert; and the reacting force of the part next behind it. We proceed with each part sepa¬ 
rately, as we did with the entire beam alluded to, thus: Beginning with the part ruH J, from its 
cen of grav, m, draw a vert line m f. From the center E of the keystone, draw a hor line E n, to meet 
mf. From n lay off nf by scale, to represent the wt of the part ru H J; and from / lay off / g, hor 
by scale, to represent the hor pres at the key. Draw the diag n g ; which will give, by scale, the re¬ 
sultant of these two forces. The point 6, at which the diag n g intersects the vert ru, is a point in 
the line of pres reqd. Next go to the part wxH J; and in the same manner find another point in 
the line wx, using the cen of grav and the wt of that part. Finally, treat the entire half arch N D T 
H J in the same way. The resultant diag of this last will pass through o, the center of the skewback, 
if the archstones have the same depth throughout. A curve drawn by hand through the points thus 
found, will be the reqd line of pres. These points will not all fall equally well within the thiekness 
of the archstoties. 

Art. 73. The stability of bodies on inclined planes, as regards 
overturning, is measd in the same way as when the base is hor; namely, by mult 
their wt, by the perp dist (an, or ct, at A, B, and D, Fig 76,) from the fulcrum, or 
turning-point a or c, to the vert line of direction (g o) drawn from the cen of grav 



of the body. Hence, it is evident that the body B has less overturning stability 
about its toe a, than the similar body A has, when the force, n, tends to upset it 
down hill. But it has more than A, when the force tends to upset it up hill, or about 
the toe c; for the leverage t c of B is greater than that, o c, of A. 

The bodv C, which would overturn upon a level base, because the line g o strikes outside of the 
base: would be stable against overturning, if placed as D upon an inclination, where the vert y o 
strikes within the base. In our fig the longer sides of D are supposed to be vertical, so that the line 
of direction g o cuts the center of the base a c. The two leverages, a o and t c, are therefore equal, as 
are, consequently, the moments of stability of 1) about a and c respectively. Similarly, a given up¬ 
ward vertical force would have the same upsetting effect whether applied at a (to upset up lull! or at 
c (to upset down hill) because its leverage, ao + tc, is the same in both cases. But a horizontal 
force, applied at anv given height, as at g, has a greater leverage ( = go) when pushing down hill 
than when pushing up hill. For in order to upset down hill the body must rotate on the corner a, 
whereas, in order to upset up hill, it must rotate on the eoruer c, which gives to a horizontal force 
applied at g, only the shorter leverage = g t. 

Structures built upon slopes are, however, liable to slide; 

that is they are deficient in frictional stability. In practice this is remedied by cutting the slope 
into hor steps as at K. But works so constructed are uot as strong as if the base were a continuous 
hor line • because the vert faees of the steps break the bond of the masonry ; and because the mortar 
in the higher portious s d, being in greater quantity than that in the lower portions e y, necessarily 
allows more settlement of the masonry iu the former; and thus renders the work liable to crack, or 
SDlit oDen vertically. The case is analogous to that of a foundation, firm in some parts, and com¬ 
pressible in others. Therefore, whcu circumstances permit, the foundation should be levelled off as 
at d o; or ir the masonry has to sustain down-hiUward pressures, v should be lower thau d; and the 
courses of masonry he laid with a corresponding inclination. 

24 





















362 


GRAVITY—FALLING BODIES. 


GRAVITY, FATTING BODIES. 

Art. 1. Bodies falling vertically. A body, falling freely in vacuo 

from a state of rest, acquires, by tbe end of the first second, a velocity of about 
32.2 feet per second ; and, in each succeeding second, an addition of velocity, or 
acceleration, of about 32.2 ft per sec. This dist is called tbe acceleration of 
gravity, and is denoted by g. It increases from about 32.1 ft per sec at tbe 
equator, to about 32.5 at the poles. In the latitude of London it is 32.19. These 
are its values at sea-level; but at a height of 5 miles above that level it is dimin¬ 
ished by only about 1 part in 400. For most practical purposes it may be taken 
at 32.2. 

Caution. Owing to the resistance of the air none of the follow¬ 
ing rules give perfectly accurate results in practice, especially at great vels. 
The greater tbe sp gr of the body the better will be the result. The air resists 
both rising and falling bodies. 

If a body be thrown vertically upwards with a given vel, it will 

rise to the same height from which it must have fallen in order to acquire said 
vel; and its vel will be retarded at the rate of 32.2 ft per sec. Its average ascend¬ 
ing velocity will be half of that with which it started ; as in all other cases of 
uniformly retarded vel. In falling it will acquire the same vel that it started 
up with, and in the same time. See above Caution. 


In the following, the falls are in feet, the times in seconds, 
and the velocities anti accelerations in feet per second. 

Acceleration acquired 


in a given time . = g X time 

in a given fall from rest = j/~2 g X fall, 
in a given fall from rest 1 twice tbe fall 
and given time 


}- 


See table, p 258 


for a given acceleration = 


time 
Time required 

acceleration 


9 


for a given fall from rest = = 

\149 


fall 


for a given fall from rest! 
or otherwise / 


fall 


final velocity 
fall 


}/■i (initial vel + final vel) 


in a given time 


mean vel 

Fall 

time X the final vel 


ri 

>;=" 


time 2 x 


time X mean vel = time X 


(starting from rest) 
in a given time (starting! 

from rest or otherwise) j 
reqd for a given acceleration ) acceleration 2 
(starting from rest) ) ~ 2 g 

during any one given second (counting from rest) 


9 


initial vel + final vel 


See table, p 258. 


= ?X (number of the second (1st, 2d, &c) — A 

during any equal consecutive times (starting from rest) a 1, 3, 5, 7, 9, &c. 
Calling g = 32.2, we have 


At the end of the 



1st. 

2d. 

03 

Gu 

4th. 

5th. 6th. 7th. 
seconds 

8th. 

9th. 

10th. 

Velocity; ft per sec. 
Dist fallen since end 

32.2 

64.4 

96.6 

128.8 

161.0 

193.2 

225.4 

257.6 

289.8 

322.f 

of preceding sec; ft. 

1G.1 

48.3 

80.5 

112.7 

144.9 

177.1 

209.3 

241.5 

273.7 

305.9 

Total dist fallen; ft. 

16.1 

64.4 

144.9 

257.6 

402.5 

579.6 

788.9 

1030.4 

1304.1 

1610.C 








































DESCENT ON INCLINED PLANES. 


363 


Art. 2. I>escent on inclined planes. When a body, U, is placed 
upon an inclined plane, AC, its whole weight W is not employed in giving it 
velocity (as in the case of bodies falling vertically) 
but a portion, P, of it (= W X cosine of o = W X 
cosine of a*) is expended in perpendicular pressure 
against the plane; while only S, (= W X sine of o 
= W X sine of a,*) acts upon U in a direction parallel 
to the surface AC of the plane, and tends to slide it 
down that surf. See Art. 60, p 352. 

The acceleration, generated in a given body in a 
given time, is proportional to the force acting upon 
the body in the direction of the acceleration (Art. 

9, p 310). Hence if we make W to represent by scale 
the acceleration g (say 32.2 ft per sec) which grav 
would give to U in a sec if falling freely, then S will 
give, by the same scale, the acceleration in ft per 
sec which the actual sliding force S would give to U in one sec if there were 
no friction between U and the plane. We have therefore 

theoretical acceleration down the plane = g X sine of a. 

Therefore we have only to substitute 11 g. sin a” in place of u g and the 
sloping distance or “slide” A C in place of the corresponding vertical distance 
or “ fall ” A E in the equations, p. 362, in order to obtain the accelerations etc as 
follows: 

on an inclined plane without friction. 

In the following 1 , the slides A C are in feet, the times in 
seconds, and the velocities and accelerations in feet per 
second. 

Acceleration of sliding velocity 



._._.•_ vert accel acquired in falling) . . . „ 

in a given time — vert during the same time | X s n a 

ss g. sin a X time 

in a given slide, as A C, | _ slide 


from rest 


y time 


( vert accel acquired in falling - ) 

= ■< freely thro the corresponding >= j/ 2 g. A E 
( vert ht A E J 


— y 2 g. sin a X slide 


for a given sliding acceleration 


Time required 

sliding acceleration 


for a given slide, as AC, from _ 
rest 


g. sin a 
slide 


y final sliding velocity 


V 


slide 


X A 9 • sin a 

time reqd to fall freely thro the correspond¬ 
ing vert ht A E 


sin a 


slide 


slide 


for a given slide, from)______.___ 

rest or otherwise i mean sliding vel ^(initial + final sliding vels) 


Cosine a 


horizontal stretch, as E C, 
base EC of an y leng th, as A C 

length A C that length 


]/AC 2 — A E 2 
j'TC 


height A E _ fall , A E, in any given le ngth, AC 
Sine a = j engt h A~C ~ that length 


1 / A C2 — E C 2 
AC 


* Because o and a are equal. 
































364 


GRAVITY—PENDULUMS. 


Slide, as A C 

in a given time, starting from rest = time X final sliding vel 

= time 2 X A g- sin a. 

‘Vr other“i“e' r ™“ re8 ‘ = «'"<» X mean sliding vel ' 

= time X (initial + final, sliding vels) 

required for a given sliding accel- _ sliding acceleration 2 
eration (starting from rest) 2 g. sin a J 

But in practice tlie sliding on the plane is always op¬ 
posed by friction. To include the effect of friction, we have 
only to substitute 

“ 9 X [sin a — (cos a. coeff fric) J ” in place of “ g. sin a ” in the above equations. 
Because 

Friction = Perpendicular pressure P X coefficient of friction 
= weight W X cosine a X coefficient of friction 

and 

retardation of friction = g X cosine a X coefficient of friction. 

(For table of coefficients for various substances, see p 373.) 

Resultant sliding 1 acceleration 

= theoretical sliding accel (due to the sliding force, S) — retardation of fric 
= (g. sin a) — (g. cosine a. coeff fric) 

= 9 X £sin a — (cosine a. coeff fric)J 

If the retardation of friction (— g. cos a X coeff fric) is not less than the total 
or theoretical accel (“ g. sin a”) the body cannot slide down the plane. 




PENDULUMS. 


The numbers of vibrations which diff pendulums will make in any given 
a given time, are inversely as the square roots of their lengths; thus, if one of them 
is 4, 9, or 1(3 times as long as the other, its sq rt will be 2, 3, or 4 times as great; but 
its number of vibrations will be but %, or A as great. The times in which diff 
pendulums will make a vibration, are directly as the sq i ts of their lengths. Thus, 
if one be 4, 9, or 16 times as long as the other, its sq rt will be 2, 3, cr 4 times as 
great; and so also will be the time occupied in one of its vibrations. 

The length of a pendulum vibrating seconds at the level of the sea, in a vacuum, 
in the lat of London (51%° North) is 39.1393 ins; and in the lat of N. York (40%° 
North) 39.1013 ins. At the equator about y^ inch shorter; and at the poles, about y 1 ^ 
inch longer. Approximately enough for experiments which occupy but a few sec, 
we may at any place call the length of a seconds pendulum in the open air, 39 ins ; 
half sec, 9% ins; and may assume that long and short vibrations of the same pen¬ 
dulum are made in the same time ; which they actually are, very nearly. For meas¬ 
uring depths, or dists by sound, a sufficiently good sec pendulum may be made of a 
pebble (a small piece of metal is better) and a piece of thread, suspended from a 
common pin. The length of 39 ins should be measured from the centre of the pebble. 










PENDULUMS, ETC. 


365 


In starting the vibrations, the pebble, or bob, must not be thrown into motion, but 
merely let drop, after extending the string at the proper height. 

To find the length of a pondulum reqd to make a given number of 
vibrations in a min, divide 375 by said reqd number. The square of the quot will be 
the length in ins, near enough for such temporary purposes as the foregoing. Thus, 
for a pendulum to make 100 vibrations per min, we have ||| = 3.75; and the square 
of 3.75 = 14.06 ins, the reqd length. 

To find file number of vibrations per min for a pendulum of 
given length, in ins, take the sq rtof said length, and div 375 by said sq rt. Thus, 

lor a pendulum 14.06 ins long, the sq rt is 3.75; and = 100, the reqd number. 

Rem. 1. By practising before the sec pendulum of a clock, or one prepared as just 
stated, a person will soon learn to count 5 in a sec, for a few sec in succession ; and will 
thus be able to divide a sec into 5 equal parts; and this may at times be useful for 
very rough estimating when he has no pendulum. 

Centre of Oscillation and Percussion. 

Rem. 2. When a pendulum, or any other suspended body, is vibrating or oscillating 
backward and forward, it is plain that those particles of it which are far from the 
point of suspension move faster than those which are near it. But there is always 
a certain point in the body, such that if all the particles were concentrated at it, so 
that all should move with the same actual vel, neither the number of oscillations, 
nor their angular vel, would be changed. This point is called the center of oscilla¬ 
tion. It is not the same as the cen of grav, and is always farther than it from the 
point of suspension. It is also the centre of percussion of the suspended vibrating 
body. The dist of this point from the point of susp is found thus : Suppose the body 
to be divided into many (the more the better) small parts; the smaller the better. 
Find the weight of each part. Also find the cen of grav of each part; also the dist 
from each such cen of grav to the point of susp. Square each of these dists, and 
mult each square by the wt of the corresponding small part of the body. Add the 
products together, and call their sum p. Next mult the weight of the entire body 
by the dist of its cen of grav from the point of susp. Call the prod g. Divide p by g. 
This/> is the moment of inertia of the body, and if divided by the wt of the 
body the sq rt of the quotient will be the Itadius of Gyration. 

Angular Velocity. 

When a body revolves around any axis, the parts which are farther from that 
axis move faster than those nearer'to it. Therefore we cannot assign a stated 
linear velocity in feet per second, or miles per hour etc, that shall apply to every 
part of it. But every part of the body revolves around an entire circle, or 
through an angle of 360°, in the same time. Hence, all the parts have the same 
velocity in degrees per second, or in revolutions per second. This is called the 
angular velocity. Scientific writers measure it by the length of the arc de¬ 
scribed by any point in the body in a given time, as a second, the length of the 
arc being measured by the number of times the length of its own radius is con¬ 
tained in it. When so measured, 

Angular velocity linear velocity (in feet etc) per sec 
in radii per second - length of radius (in feet etc) 

Here, as before, the angular velocity is the same for all the points in the body; 
because the velocities of the several points are directly as their radii or dis¬ 
tances from the axis of revolution. 

In each revolution, each point describes the circumference of I he circle in 
which it revolves = 2 n r (it = 3.1416 etc; r — radius ol said circle). Conse¬ 
quently, if the body makes n revolutions per second, the length of the arc de¬ 
scribed by each point in one second is 2nrn; and the angular velocity of the 
body, or linear velocity of any point measured in its own radii, is 

a = 2n - -- = 2 nn = say 6.2832 X revs per second = say .1047 X revs per minute, 
r 

Moment of Inertia. 

Suppose a body revolving around an axis, as a grindstone; or oscillating, like 
a pendulum. Suppose that the distance from the axis of revolution (which, in 
the pendulum, is the point of suspension) to each individual particle of the 
body has been measured; and that the square of each such distance has been 
multiplied by the weight of that particle to which said distance was measured. 







366 


MOMENT OF INERTIA. 


The sum of all these products is the moment of inertia of the body. Or 


Moment 
of inertia 


-{ 


l 


the sum, 

(or all the particles f 


( weight square of dist 
of -< of X of particle from 
(particle axisof revolution 


or. 


1 = 2 d* w. 


Scientific writers frequently use the mass of each particle ; 

t e, - ; - : - --— - instead of its weight, in calculating 

acceleration (<j) of gravity, or about 32.2 

the moment of inertia. 

In practice we may suppose the body to be divided into portions measuring 
a cubic inch (or some other small size) each ; ami use these instead of the theo¬ 
retical infinitely small particles. The smaller these portions are taken, the 
more nearly correct will be the result. 

When the moment of inertia of a mere surface is wanted (instead of that of a 
body), we suppose the surface to be divided into a number of small areas, and 
use them instead of the weights of the small portions of the body. For an ex¬ 
ample, see p 486. 

weight of body, 0 __ - 

Moment of inertia = or " X rQrl ; s „ qu f re ° r .• : 

area of surface radlus of gy ratl0n 

For definition of Radius of gyration, see p 440. A body may have any number 
of radii of gyration, depending upon the position of the axis of revolution. 

Table of Radii of Gyration. 


Body 

Revolving 

around 

Radius of Gyration 

Any body or 
figure 

any given axis 

(moment of inertia around thegiven axis 
A weight of body, or area of surface 

Solid cylin¬ 
der 

ditto 

ditto, infinitely 
short (circular 
surface) 

its longitudinal 
axis 

a diam, midway 
between its ends 

a diameter 

radius of cylinder X 
= radius of cylinder X about .7071 

/length 2 radius 2 of cylinder 

A 12 ' 4 

radius of cylinder 

2 

^follow cyl¬ 
inder 

its longitudinal 
axis 

_ (inner rad 2 + outer rad 2 

ditto, infinitely 
thin 

ditto 

A 2 

radius of cylinder 

ditto, of any 
thickness 

a diam midway 
between its ends 

/inner rad 2 + outer rad 2 lentrth 2 

A 4 + 12 

ditto, infinitely 
thin 

ditto 

/radius 2 of cylinder _ length 2 

A 2 “*■ 12 

ditto, infinitely 
thin and infinitely 
short (circumfer¬ 
ence of a circle) 

a diameter 

radius of cylinder X 
= radius of cylinder X about .7071 

Solid sphere 

a diameter 

/radius 2 of sphere 

A 2.5 

= radius of sphere X j/^4~ 

= radius of sphere X about .63246 















































RADII OF GYRATION. 


367 


Table of Radii of Gyration.— Continued. 


Body 

Revolving 

around 

Radius of Gyration 

H ollow 

sphere of any 
thickness 

a diameter 

12 (outer rad 5 — inner rad 5 ) 

\ 5 (outer rad 3 — inner rad 3 ) 

ditto, thin 

ditto 

approx (outer rad + inner rad) X .4085 

ditto, infinitely 
thin (spherical 
surface) 

ditto 

any point, r, in its 
length 

• 1 2 

radius of sphere X 
= radius of sphere X about .8165 

Straight line, 

a b 

la x? x b 3 

M 3 ab 

a x c b 


1 V 


either end, a or 6 

length ab X -Vf-jp 



= length ab X about .5775 


its center, c 

oc x 



= length ab X about .2887 

Solid cone 

Circular 
plate, of rect¬ 
angular cross sec¬ 
tion 

Circular 
ring, of lectan- 
gular cross section 

its axis 

See Solid cylin¬ 
der 

See Hollow cylin¬ 
der 

radius of base of cone X j/^3 - 
= radius of base of cone X .5477 

For the thickness of plate or ring, 
. measured perpendicularly to the plane 
of the circumference, take the length of 
the cylinder. 

Square, rect¬ 
angle and 
other sur¬ 
faces 

For least radiusof gyration, or that around the longest axis, 
see p 440 and 441. 



























CENTRIFUGAL FORCE. 


368 


CENTRIFUGAL FORCE. 


When a body a, Fig 3 moves in a circular path abd, it tends, at each point, as 
a orb, to move in a tangent at or bt' to the circle at that point. But at each 




Fig. 3 













point, as a etc, in the path, it is deflected from the tangent by a force acting 
toward the center, c, of the circle. This force may be exerted by a string ca, by 
the rails on a curve etc etc. At each point, therefore, the tangential tendency 
at is resolved into two components. Oneof these, ao = bt, is its centrifug'al 
force, or resistance to the deflecting or centripetal force an. The other 
is a new tangential force ab, which carries the body to the next point b in its 
path.*f 

The centrifugal force, being directly opposed to the centripetal force, neces¬ 
sarily acts directly/row the center. Hence its name. Being merely a resist¬ 
ance to the centripetal force, it is necessarily equal to it, and ceases when it 
ceases. These two equal and opposite forces, acting against each other through 
the material of the string, rail, fly-wheel etc, produce among its particles a 
strain equal to one of them. See Art. 13, p 311. The moment this strain ex¬ 
ceeds the strength of the string etc, the latter breaks; the centrip and centrif 
forces therefore instantly cease; and the tangential force, now undisturbed,car¬ 
ries the body in a tangent, at or bt' etc, to its circular path, or at right angles 
to the direction of the centrif force. 


Centrifugal force = weight of body X 


velocity 2 in feet per second 


radius, ca Fig 3, in feet X 9 
_ weight of v No. 2 of revs per min X tt 2 X rad ca Fig 3, in ft 
body A ~ 


900 g 


7r = about 3.1416. See page 123. — about 9.8696. See caution, p. 369. 

g = 32.2. See page 362. 


Demonstration. (1st) Let a h. Fig 3, be the minute space described by the re¬ 
volving body a in an exceedingly short time t* so that we may consider the 
straight line, or chord, ab, as equal to the arc , ab. 

Now (Euclid, vi. 8) 

diameter ad: ab :: ab : an, or an — 

a d 


But a n is = tb 


= the distance through which the body a deviates from the tangent 
a t toward the center c in the time t, or while moving from a to b f 
= half the acceleration toward c produced by the centripetal force 
in the time t : 


* We of course cannot draw this correctly. In order to do so, we should have 
to make ab far shorter than we could represent it by a line. As it is, we may 
imagine the path of the body a to be, not a circle, but a polygon, of which a b 
is one of the sides; the ball flying freely in straight lines from point to point, 
as from a to b, and striking, at each point, a flat surface tangential to the circle. 
These surfaces take the place of the string c a ; each of them, as the ball strikes 
it, reacting with a centripetal force = oa = an. 

t When the body reaches the next point, as b, in its path (see foot-note *) a 
new deviation has to take place ; i e, from the neiv tangent b t! . Hence the accel¬ 
eration of the velocity of deviation does not accumulate as the body revolves, as 
does that of the velocity of a falling body during its fall. 



















CENTRIFUGAL FORCE. 


369 


whence we have 


centripetal acceleration in time t 


ab 2 
a d ‘ 


The same proportion holds good if we let 

ab = the velocity of revolution in feet per second 
an — half the centripetal acceleration in one second. 

(1 /)2 

Therefore, since an — —- or 2 an = z - 

ad y^ad 

centripetal acceleration veh city 2 of revolution in feet per second 
given in one second - Sdiusacin feet - 


But (see Art. 9, page 310) forces are proportional to the accelerations which 
they can produce in a given body in a given time: and gravity (or the weight 
of the body) would, in one second, give to it an acceleration, g, of about 32.2 leet 
per second (see page 362). Therefore, 

n . centrip accel . . n . vel 2 of rev in ft per sec .. . weight . centrip or 
•' ’ given in 1 see • ■ ” • ~ radius in feet of body ' centrif force 

or, 

, „ . , . . . velocity 2 of revolution in ft per sec 

centrifugal force = weight X ---—- - —- 

radius X 0 

as in the first formula. 

(2d) Velocity in feet per second 

revolutions per minute X circumference in feet 

60 (seconds per minute) 

_ revolutions per minute X radius in feet X 2^ 

60 

Hence, velocity 2 in feet per second 

revolutions 2 per minute X radius 2 in feet X 47r 2 

3600 

revolutions 2 per minute X radius 2 in feet X w 2 
“900 


But, by the first formula, page 368, 


centrifugal force — weight X 
Hence, 

centrifugal force = weight X 
— weight X 

as in the second formula. 


velocity 2 in feet per second 
radius X g 

revolutions 2 per minute X radius 2 X v 2 
radius X <7 X 900 

revolutions 2 per minute X n 2 X radius 
900^7 > 


The centrif force thus found will be in lbs, tons etc, according as the wt of the 
body is taken in lbs, tons etc. 

Caution. If the dimension at of the body, Figs 1 and 2, measd in the direc¬ 
tion of the rad ca ,does not exceed about one-third of the entire rad ca, we may, 
near enough for ordinary practice, measure the rad from the cen c of the circle 
to the cen of grav n of the body Fig 2, or to the cen of grav n of cross section 
of a revolving ring Figs 1 and 4. 

When at is equal to one-third of ac, the rad thus found will be but about one- 
fiftieth part too short; when one-half of ac, about one-nineteenth part too short; 
and when two-thirds of ac, about one-ninth part too short. But when, as is 
often the case, as in fly-wheels, etc, at is much less than one-third of ac, the 
error is not worth notice in practice. 

But if greater accuracy is required, or if, as in Fig A, at is greater than about 
one-third of ca, use the radius of gyration, see p 366, and the 

velocity of the center of g-yration. 

Velocity ofcenlerofgyration Velocity ofouter particles radius of gyration 

in feet per second in feet per second X - outer radius 

ca Figs 1, 2 and 4 


Revolutions per minute 
“~60 


X radius of gyration X 2 tt 


The ring Fig 1, or the body, Fig 2, must be united to the center c, either by arms, or by a string 
or wire, etc, and the weight of these arms, etc, will slightly shorten the radius of gyration. Butin 
practice this effect is usually too small to be regarded ; or a trifling allowance is made for it by guess. 





















370 


FRICTION. 


FRICTION. 

Art. 1. When one rough body rests upon another, the projections and de¬ 
pressions forming the roughnesses of their surfaces of contact interlock 
to a greater or less extent; and, in order to slide one over the other, we must 
expend a portion of the sliding force, either in separating the bodies (as by lilt¬ 
ing the upper one) sufficiently to clear the projections, or in breaking on some 

of the projections and clearing the others. 

Thus, let B Fig 6 p 314 represent oneof the minute projections (highly magni¬ 
fied) on the surf of the lower body : isg one of those of the upper body ; and Fg 
a force tending to slide the upper body hor over the lower one. To do so it must 
separate the two bodies (bv pushing the upper one up the inclined plane i/) un¬ 
til 5 clears t; but it may reduce the height of this lift by grinding or breaking 
off a part off or of s. It. is immaterial whether the surf of contact is hor, in¬ 
clined or vert. The separation of the two bodies must take place in opposition 
to whatever force, acting perp to the surf of contact, tends to hold them to- 

^ The surfs of all bodies are more or less rough. Hence this interlocking takes 
place to some extent between even the smoothest, surfs. _ Without it, the most 
powerful vise could not prevent the lightest wt front falling out of its jaws; and 
the smallest conceivable force would slide the greatest conceivable wt. 

The resistance which the slidiug force thus encounters is called friction.* 

Art. 2. Friction always tends to prevent relative motion of the two bodies 
between lohich it acts; i e , motion of one of the bodies relatively to the 
other. In doing so, however, it tends equally to cause relative motion be¬ 
tween each of those two and a third, or outside body. Thus, the fric between 
a belt and the pulley driven by it, tends to prevent slipping between them; 
but thus tends to make the belt slip on the driving pulley, and sets the 
driven pulley and its shaft in motion relatively to the bearing in which the shaft 
revolves. This motion is resisted by the fric between journal and bearing; and this 
fric, in turn, tends equally to make the beariitg revolve with the journal, and to 
make the belt slip on the driven pulley. 

Art. 3. The fric between t wo bodies at rest relatively to each other is called 
static friction, or fric of rest. That between two bodies in relative motion 
is called kinetic friction or fric of motion. 


Art. 4. (a). The ultimate or maximum static fric between two 

bodies, as U and L Fig 1, (or the greatest fric resistance which they are capable 

of opposing to any sliding force when at rest) is 
equal to a force (as that of the wt F) which is just 
upon the point of making U begin to slide upon 
L.f Thus fric, like other forces, may be expressed 
in weights, as in lbs. 

(»>) A resistce cannot exceed the force which it 
resists.}: Therefore if F is less than the ult static 
fric bet ween U and L, th e frictional resistce actually 
exerted by them is also less. When F is = the ult 
fric (and U is therefore on the point of sliding) the actual resistce is = the ult 
stat fric. If F exceeds the ult stat. fric, the excess gives motion to U. 

Art. 5. If, when a body is in motion, all extraneous forces and resistces are 
removed or kept in equilib, it moves at a uniform vel. Hence, if the force, F 
Fig 1, is just = the ult kinetic fric between U and L, their vel is uniform. If F 
exceeds this, the excess accelerates the vel. If the ult kinetic fric exceeds F, the 
excess retards the vel. Thus tlie actual fric resistce exerted by two 
bodies in relative motion is = their ult kinetic fric = that force (as F) which 
can just maintain their relative vel uniform. 

Hence, if the hor surfS upon which L rests, could be made perfectly friction¬ 
less, the pres of L against the lug m (which would then always be = the actual f ric 
resistce between U and L) would also be = their ult fric so long as U continued in 
motion over L, and might therefore be greater or less than or = F; but when 


UMl/llU 

t-r 



Fix.l 


t 

t 


t 

t 

I 

t 


! 


I 




* “Friction" (meaning rubbing) is a misnomer in so far as it implies that rubbing must take 
place in order to produce the resistance. For we meet this resistance, not only during rubbing, lyit 
also before motion (or rubbing) takes place. “ Resistance of roughness” would better express its 
nature. 

t We here neglect the fric of the string and pulley, and assume that all the force of the wt F is 
transmitted by the string to U. 

1 If a resisting force exceeds the force resisted, the excess is not resistce, but motive force. 













FRICTION. 


371 


U was at rest the pres against m would be = F, and less than (or at most just =) 
the ult trie. 

Art. 6 . Since no surface can be made absolutely smooth, some separation of 
the two bodies must in all cases take place in order to clear such projections as 
exist. Hence the trie is always more or less affected by the amount of the perp 
pres which tends to keep them together. 

The proportion which the ult fric, in a given case, bears to the perp pres, is 
called the coefficient of'friction for that case. Or, 


Coefficient of friction 


ultimate friction 


perpendicular pressure 


and 


Ultimate friction = perp pres X coeff of fric 



D 

E 


Thus, if a force F Fig 1, of 10 lbs, just balances the ult fric between U and L, 
and if the wt of U (the perp pres in this case since the surf between U and L is 

hor) is 50 lbs, then the coeff of fric between U and L is = ~ S - = .2. 

50 lbs 

The coeff* is usually expressed decimally, or by a common fraction ; 
but sometimes, as in the case of railroad cars and engines, in lbs (of fric) per ton 
(of perp pres). Or by the “ angle of fric ” in degs and mins. (Art. 61, p 358.) 

The coeff is diff for diff materials; and, in a given material, varies with the 
smoothness, cleanness, dryness etc of the surf. 

Art. 7. The coeff* of static fric may he found experimentally, 
either as in the above example, or by inclining the surf of contact, as in Art. 61, 
p 353. Expts on fric with unguenjs cannot well be made on a small scale in the 
latter way; on account of the stickiness or cohesion of the unguent. 

Art. 8. (a) To find the coeff of Kinetic fric, allow one of the bodies, 

U Fig2, to slide down an inclined plane AC 
formed of the other one and having any con¬ 
venient known steepness ACE greater than 
the angle of fric (Art. 61, p 353). Note the vert 
dist A E through which U descends in sliding 
any dist as AC, (AE = AC X sine of ACE, 
table of sines etc, pp 60 etc); also its actual 
sliding ve 1 in ft per sec on reaching C. Calcu¬ 
late the vert dist AD through which it would 
have to descend along the plane (from A to B) 
/ velocity 2 in ft per sec \ 

to acquire that vel iflheie were nofiic. ^ twice the accel g of grav*/ 

Find DE( = AE — AD),and thehordist EC corresponding to AC (EC = ACX 
cosine ACE = iXAC 2 — AE 2 ). Then 

„ DE 

Coeff 1 of the average fric in sliding from A to C = 

because if we let AE represent the total sliding force expended (in moving U 
from A to C and in overcoming the fric); then A D represents the portion of 
A E expended on vel; and DE that expended on fric, and therelore = the fnc 

AE sliding force 
itself. And (Art. 62, P 353) - -^TpreT * 

„ DE friction _ „ 

Hence - ^ =-- = coeff. 

EC perp pres 

(I>) Or, find the sine and the tangent (table pp 60 etc) of ACE; and the dist 
A C /= time 2 in secs Y l o* X sine of ACE) through which U would slide m a 
given time if there were nofric. Measure the dist A B through which it actually 
slides in that time; and find BC = AC AB. Then 

tan DCE = tan ACE X 


coeff of the average 
fric in sliding from A to B. 

because (1st) AC: AB:BC:: AE: AD 


AC 

DE :: theoretical vel due to total slid- 


* g = about 32.2 ; 2 g = about 64.4. 






















372 


FRICTION. 


ing force : actual vel : frictional retardation :: total sliding force : sliding force 
used in giving the actual vel : fric, or the sliding force reqd to balance it. And 
if A E = the total sliding force, then EC = the perp pres ; and (Art. 62, p 353) 
D E 

coeff=tan DOE. (2d) Owing to the similarity of the two triangles 


EC 

ADD and ACE, AC : B C :: A E : D E :: 


AE DE 


EC EC 


tan ACE : tan DCE. 


Art. 9. In 1831 to 1834, Clen'l Arthur Morin* experimented with i 
pressures not exceeding about 30 lbs per sq in ; and arrived at the following j 
conclusions in regard to sliding fric where the perp pres is considerably less j 
than would be necessary to abrade the surfs appreciably. These were for a long 
time generally regarded as constituting the three fundamental laws of 
fric. But see Art. 11, p 374. 

1st. The ult fric between two bodies is proportional to the total perp force 
which presses them together; i e, the coetf is independent of the perp 

pres and of its intensity (pres per unit of surf). Hence 

2d. For any given total perp pres, the coetf is independent of the | 

area of surf in contact. 

If upon a hor support we lay a brick, measuring 8X4X2 ins, first upon its j 
long edge (8X2 ins) and then upon its side (8X4 ins), we double the area of 
contact, while the total pres (the wt <Jf the brick) remains the same, and thus re¬ 
duce the pres per sq in by one-half. Consequently (thecoeff remaining practically 
the same) we have only half the fric per sq in. But we have twice as many sq ins 
of contact, and therefore the same total fric. 

But if we can increase or diminish the area of contact without affecting the pres 
per sq in, the total pres will of course vary as the area , and the total fric will vary 
in the same proportion, for the coetf remains the same. Tints, if we place two 
similar sheets of paper between the leaves of a book (taking care not to place 
both sheets between the same two leaves) and then squeeze the book in a letter- 1 
copying press, it will require about twice as much force to pull out both sheels 
as to pull out only one of them. 


3d. Although thecoeff of static fric between two bodies is often much greater 
than their coeff of kinetic fric; yet tlie coeflf of kinetic fric is inde¬ 
pendent of the vel. 

This applies also (approx) to the fric, and hence to the n orfc (in /oot-pounds etc) 
of overcoming fric t hrough a given dist; for then the work ( = resistce X dist) is 
independent of the vel. But in a given time , the dist (and consequently the 
work also) of course varies as the vel. See Art. 21, p 374/. 


Art. 10. (a) Some kinds of surfaces appear to interlock their projections 
much more perfectly when at rest relatively to each other, than when in even 
very slow motion ; and in some cases the degree of interlocking seems to in¬ 
crease with time of contact. Hence there is often a great diff in amount between 
fric of restand fric of motion. Thus, Gen’l Morin found that with oak upon 
oak, fibres of the two pieces at right angles, the resistce to sliding while still at , 
rest, and after being for “some time in contact,” was about one eighth greater 
than when the pieces had a relative vel of from 1 to 5 ft per sec. 

(1>) But experience shows that even very slight jarring suffices to remove this ! 
diff, and since all structures, even the heaviest, are subject to occasional jarring 1 
(as a bridge, or a neighboring building, or even a hill, during the passage of a 
train ; or a large factory by the motion of its machinery ; or in numberless cases 
by the action of the wind) it is expedient, in construction, not to rely on fric for 
stability any further than the coeflf for moving fric will justify. When it is to be 
regarded as a resistce, which we must provide force for overcoming, it should be ' 
taken at considerably more than our tabular statement, p. 373. 


* See his “ Fundamental 
New York, I860. 


Ideas of Mechanics", translated by Jos. Bennett; D. 


Appleton & Co., 











FRICTION 


373 


Tabic of moving friction, of perfectly smooth, clean, and 
dry, plane surfaces, chiefly from Morin. 


Materials Experimented with. 


Oak on oak ; all the fibers parallel to the motion. 

“ “ moving fibres at right angles to the others; and to the motion... 

“ “ all the fibres at right angles to the motion. 

“ “ moving fibres on end; resting fibres parallel to the motion. 

“ cast iron, fibres at right angles to motion. 

Elm on oak, fibres all parallel to motion. 

Oak on elm, “ “ “ . 

Elm on oak, moving fibres at right angles to the others, and to motion. 

Ash on oak, fibres all parallel to motion. 

Fir on oak, “ “ “ “ .. 

Beech on oak “ “ “ “ . 

Wrought iron on oak, fibres parallel to motion. 

Wrought iron on elm, “ “ “ “ . 

Wrought iron on cast iron, fibres parallel to motion. 

“ “ ou wrought iron, fibres all parallel to motion. 

Wrought iron on brass. 

Wrought iron on soft limestone, well dressed. 

*• “ “ hard “ “ “ . 

a a «« k a a a wet 

“ “ or steel on hard marble, sawed. By the writer.about.. 

“ “ “ “ “ smoothly planed, and rubbed mahogany, fibres par¬ 
allel to motion. 

“ “ “ “ “ smoothly planed wh pine. 

Cast iron on oak, fibres parallel to motion. 

“ “ “ elm, “ “ “ “ . 

“ “ “ cast iron. 

a u a brass. 

Steel on cast iron.. 

Steel on steel. By the writer. 

Steel on brass.•. 

Steel on polished glass. By the writer.about... 

“ quite smooth, but not polished; on perfectly dry planed wh pine, fibres 

parallel to motion.about.. 

“ quite smooth,but not polished; on perfectly dry planed and smoothed 

mahogany, fibres parallel to motion.about.. 

Yellow copper on cast iron.•. 

“ “ on oak.1. 

Brass on cast iron... 

“ on wrought iron, fibres parallel to motion. 

“ on brass. 

“ ou perfectly dry planed wh pine, fibres parallel to motion.about.. 

i* “ “ “ “ and smoothed mahogany, fibres parallel to mo¬ 
tion.about.. 

Polished marble on polished marble. By the writer.average. 

“ “ ou common brick. “ . 

Common brick on common brick. “ . 

Soft limestone well dressed, on the same. 

Common brick, on well-dressed soft limestone. 

• < “ »‘ “ “ hard “ . 

Oak across the grain, on soft limestone, well dressed. 

• * “ “ “ “ hard “ “ “ . 

Hard limestone on hard limestone, both “ “ . 

“ “ “ soft “ “ “ “ . 

Soft “ “ hard “ “ “ “ . 

Wood on metal, generally, .2 to .62.mean.. 

Wood, very smooth, on the same, generally, .25 to .5 . 

Wood, “ “ on metal, “ .2 to .62. J •• 

Metal on metal, very smooth, dry “ .15 to .22. “ •• 

Masonry and brickwork, dry “ .6 to .7 . •• 

a a “ with wet mortar.about.. 

a << “ “ slightly damp mortar. “ .. 

“ on dry clay. It •• 

“ “ moist“ ..... ‘ •• 

Marble, sawed ; on the same; both dry. By the writer.*.average 

.. a “ “ “ both damp. 

a a on perfect.lv dry planed wh pine. 

“ " on damp planed wh pine. 

“ polished, on perfectly dry planed wh pine 
White pine, perfectly dry ; planed; on the same; all the fibres parallel to 

motion. 

“ a damp, planed ; on the same. 


.*. 

.*. 


.about.. 


Coefif of 
Fric: or 
Propor¬ 
tion of 
Fric to the 
Pres. 

Angle of 
Fric. 

.48 

Deg. 

25 

Min. 

38 

.32 

17 

45 

.34 

18 

47 

.19 

10 

46 

.37 

20 

19 

.43 

23 

17 

.25 

14 

3 

.45 

24 

16 

.40 

21 

49 

.36 

19 

48 

.36 

19 

48 

.62 

31 

47 

.25 

14 

3 

.19 

10 

46 

.14 

7 

58 

.17 

9 

39 

.49 

26 

6 

.24 

13 

30 

.30 

16 

42 

.17 

9 

39 

.18 

10 

12 

.16 

9 

6 

.49 

26 

6 

.20 

11 

19 

.15 

8 

32 

.15 

8 

32 

.20 

11 

19 

.14 

7 

59 

.15 

8 

32 

.11 

6 

17 

.16 

9 

6 

.18 

10 

12 

.19 

10 

46 

.62 

31 

48 

.22 

12 

25 

.16 

9 

6 

.20 

11 

19 

.19 

10 

46 

.24 

13 

30 

.16 

9 

6 

.44 

23 

45 

.64 

32 

38 

.64 

32 

38 

.65 

33 

2 

.60 

31 

00 

.38 

20 

48 

.38 

20 

48 

.38 

20 

48 

.67 

33 

50 

.65 

33 

2 

.41 

22 

18 

.38 

20 

48 

.41 

22 

18 

.18 

10 

12 

.65 

33 

2 

.47 

25 

30 

.74 

36 

30 

.51 

27 

00 

.33 

18 

15 

.4 

21 

49 

.55 

28 

49 

.45 

24 

14 

.6 

31 

00 

.26 

14 

35 

.4 

21 

48 

.6 

31 

00 


* But after a few trials the surfaces become so much smoother as to reduce the angles as much as 
from 2° to 5°; the sliding blocks weighing about 110 lbs each. 


























































































374 


FRICTION. 


Art. 11. Recent experiments, with much greater variations of pres and of 
vel, and with more delicate apparatus for detecting slight changes in the coeff; 1 
although giving conflicting results,* show that tfie three laws in Art. 9 
are far from correct for surfs moving at high vels, and under great pres; and 
that they are only approximately eorreet for ordinary vels and 
pressures; for the coeff is found to vary both with the intensity of the pres and 
with the vel, as also with the temperature* But in the cases with which the 
civil engineer has mostly to deal, slight diffs iu the character of the surfs, or 
even in the dampness of the air, will often cause much greater changes of coeff' 
than those due to any probable changes of pres, vel and temp: so that, within 
the limits of abrasion, we may generally take Morin’s rules as sufficiently cor¬ 
rect for such cases. 


Art. 12. Prof. A. S. Kimball, of 

the Worcester (Mass) Inst of Industrial 
Science, has made some very delicate experi¬ 
ments upon the fric between surfs of pine 
wood.f The results are given in Fig 3, 
merely to show how the coeff varied with 
vel and pres. Our table gives a coeff' of .4 
for pine on pine. 



00 80 70 GO 50 40 30 20 

Velocity of Sliding, In Inches per second. 



It will be seen that at low vels the coeff decreased when the pres per sq in was 
almost imperceptibly increased; but this diff disappeared as the vel increased. 
At vels from 4 to 120 ins per sec, the coeff generally decreased as the vel in¬ 
creased ; rapidly at first, but more slowly as the vel became greater. This agrees 
with other recent expts. But at very low vels (.08 to 5 ins per sec) Prof. Kimbal 
found the coeff (line E) increasing very rapidly with the vel. 

We have made the scale of coeffs large in order to show their variations, which 
are so slight that they would otherwise be scarcely perceptible. Less delicate 
expts would have failed to show them at all. 


Art. 13. (a) In 1878 (’apt. Douglas Dalton and Mr. Thos Westing- 

house made careful expts in England upon fric as affectiug the action of rail¬ 
way brabes.J The fric and pres were automatically recorded by means of 


* This is not surprising in view of the extent to which the coeff is affected by the nature of the 
surf. If the shape of the minute projections is such that they fit into each other as perfectly under 
small pressures as under great ones, and if they are too strong to be broken by the pressures applied, 
the coeff, as stated in the 1st law, should be independent of the pres. But if high pres wedges the 
projections of one body more closely between those of the other, the coeff should increase under such 
pres. On the other hand, if the higher pres breaks down the projections while the lower ones are 
unable to do so. the coeff should decrease under the higher pres. The particles thus broken off may 
either act as a lubricant and thus still further reduce the fric and its coeff, or (if angular and hard) 
may increase it. Change of area of contact, under a given total pres, may, by affecting the intensity 
of the pres, make changes in the coeff similar to those just mentioned. 

At high vels the roughnesses have not time to interlock as perfectly as at low vels. Tfenco we 
should expect a less coeff at high vels. But high vel generally increases the number of projections 
broken away ; and these may either increase or diminish the’ coeff, as explained above. High vel 
often indirectly affects it by increasing the temperature. 

t Silliman's Journal (American Journal of Science) March 1876 and May 1877. 

J See Proc, Instn of Mechl Engrs, London, June and Oct 1878 and April 1879; and “ Engineer¬ 
ing," London, 1878; vol. 25, pp 432, 469, 490; vol. 26, pp 153, 386, 395. 
































































FRICTION. 


374 a 


hydraulic gauges. With cast iron brake-blocks and steel-tired wooden wheels, 
43J ins diam, they found coeffs about as shown in Fig. 4. 

The points in lines A, B and C show the average brake coeffs, or coeffs of slid¬ 
ing fric between the tread of a rolling wheel and the brake-block. 


Speed of Car, in miles per hour. 



Line A shows brake coeffs obtained immed’y after application of brake 
“ B “ “5 secs “ “ 

• u q u »i jg « « “ 

“ D shows rail coeffs or coeffs of sliding fric between the tread of a slid¬ 
ing or “ skidding ” wheel (held fast by the brake) and the rail. 

(b) From lines A B and C it appears that the brake coeff obtained at a 

given length of time after the application of the brake was generally greater 
at lew than at high vels. But where the vel was maintained uniform 
the brake coeff diminished as block and wheel remained 
longer in contact. Thus, lines A and B show that at 37J miles per hour 
the brake coeff was .154 when the brake was first applied (point g), but fell to 
.096 in 5 secs ( x ). Line A (immed’y after application) shows a higher brake coeff 
(.132 at/) at 47J miles than line B (5 secs after application) shows at 37£ miles 
(.096 at. x). , , , 

5 The diminution of the rail coeff with length of time of application of brake, 

I was scarcely noticeable. 

(c) When the brake fric (owing to the reduction of vel and consequent m- 
l ' crease of coeff) becomes = the “adhesion ” or static fric between the rail and 

the tire of the rolling wheel, the vel of rotation rapidly falls below that due to 
the vel of the car; i e, the wheel begins to “skid” or slide along the 
h rail: and in from .75 to 3 secs the rotation of the wheel ceases entirely. 

(d) The rail coeff, line D, is generally much less than the 
c brake coeff, lines A, B and C. The pres on the rail ( = the wt on a wheel) 

• was about 5000 lbs per sq in, or greatly in excess of the limit of abrasion. That 
k i at the brake was about 200 lbs per sq in. A few expts were made with brake 
• blocks having but 5 of the usual area of contact, and therefore 3 times the pres 
f per sq in under a given total pres. They failed to show conclusively that this 
caused anv marked change in the coeff. 

fe) The rail coeff, line D, like the brake coeff, increases as the vel 
' diminishes; slowly at first, but much more rapidly as the speed becomes 
, r less; until, at the moment of stopping, it is generally even greater than the 
« brake coeff just before skidding. With steel tires on iron rails at high vela.it. was 
h somewhat greater than on steel rails, but this dilFdisappeared as the vel dimin- 
8 ished. 

(f) Locomotives overcome resistces = from J to § or more of the wt on 
! all the drivers; i c , they have a coeff of .33 or more, although the expel inient.al 
coeff for steel on steel in motion at low pres, is only about .15. But the cases are 
e so diff that a similarity in their coeffs could hardly be expected. The great wt, 
! say from 2 to 6 or even 7 tons, on a driver, is concentrated on a surf (where the 
wheel touches the rail) about 2 ins long X about § inch wide, or = say 1 sq in. 
The pres per sq in thus greatly exceeds not only that upon which the tables are 
based, but also the limit of abrasion. Besides, any point in the tread, during 














































































































































































































3746 


FRICTION. 


the instant when it is acting as the fulcrum for the steam pres in the cyl, is 
stationary upon the rail, its fric (miscalled “ adhesion ”) is therefore static. 

Capt. Gal ton found that tlie coeir of “adhesion” was independent of 
the vel, and depended only on the character of the surfs in contact. With a 
four-wheeled car having about 5000 lbs load on each wheel, it was generally over 
.‘10 on dry rails: in some cases .25 or even higher. On wet or greasy rails, with¬ 
out sand*it fell as low as .15 in one case, but averaged about .18. Witli sand 
on wet rails it was over .20. Sand applied to dry rails before starting gave .85 
and even over .40 at the start, and an average of about .28 during motion; but 
sand applied to dry rails while the car was in motion was apt to be blown away 
by the movement of the car and wheels. 

(g) Owing to the constancy of the coeff of “ adhesion ” under given conditions 
of tire and rail, the brake fric necessary to “skid ” the wheels in any case was 
also practically constant for all vels. But at high vels, owing to the lower brake 
coeff, a higher brake pres was reqd to produce this fixed amount of brake fric. 
The skidding also reqd a longer time than at low speeds. 

Art. 14. If the pres is sufficient to produce abrasion (indeed, while it is 
much less) the fric often varies greatly, but no precise law has yet been discov¬ 
ered for estimating it. Rennie gives the following table of coefTs of fric 
of dry surfaces, under pressures gradually increased up to 
tlie limits of abrasion. It will be noticed that in this table the 
coeff generally' increases with the intensity of the pres: 

CoefTs of friction of dry surfaces, under pressures grad¬ 
ually increased up to the limits of abrasion. (By G. Rennie, C E ) 


Pres, in Lbs. 

Wrought Iron 

Wrought Iron 

Steel 

Brass 

per 

OQ 

on 

on 

on 

Square Inch. 

Wroughtlron. 

Cast Iron. 

Cast Iron. 

Cast Iron. 

32.5 

.140 

.174 

.166 

.157 

186 

.250 

.275 

.300 

.225 

224 

.271 

.292 

.333 

.219 

336 

.312 

.333 

.347 

.215 

448 

.376 

.365 

.354 

.208 

560 

.409 

.367 

.358 

.233 

672 


.376 

.403 

.233 

700 


.434 


*234 

784 




232 

821 




.273 


Art. 15. (a) Rolling friction, or that between the circumf of a roll¬ 
ing body and the surf upon which it rolls, is somewhat similar to that of a 
pinion rolling upon a rack. In disengaging the interlocking projections, or in 
lifting the wheel over an obstacle o, Figs 5 and 6, the motive force F, instead of 
dragging one over the other, as in Fig 6, p 814, acts at the end of a bent lever 
Fit W Figs 5 and 6, the other end W of which acts in a direction perp to the 
contact surf; and in practical cases of rolling fric proper the leverage RW of 
the resisting wt of the wheel ard its load is very much less, in proportion to 
that (FR) of the force F, than in our exaggerated*figs. Hence the force F reqd 
to roll a wheel etc is usually very much less than would be necessary to slide it. 


(b) There are usually two ways of applying tlie force in overcon 
ing rolling fric: 1st (Fig 5) at the axis of the rolling body; as the force of 

horse is applied at 


°i 

R 

w y 



mzz. 


FU 



but at both top and bottom of the wheel 

(c) When the obstacles o are very small as in 
smooth hard roads, or of car-wheels on iron 


- ... the axle of . 

wagon-wheel; or that of a man at tin 
axle of a wheel-barrow : 2d (Fig 6) al 
the circumf; as when workmen pusl 
along a heavy timber laid on top of 
two or more rollers; or as the ends of 
an iron bridge-truss play backward 
and lorwanl by contraction and ex¬ 
pansion, on top of metallic rollers oi 
balls (p614). In Fig 5 we have, in ad 
dition to the rolling fric of the cir 
cninf of the wheel on its support, tin 
sliding fric of the axle in its bearing 
In Fig 6 we have only rolling fric 


the case of cart-wheels o 
or steel rails, the leverag 


! 


I 

I 



































FRICTION. 


374 c 


(F FI) of F becomes, practically, in Fig 5 the radius , and in Fig 6 the diam, of the 
wheel; while that (RW)of the resistce is very small. Hence, neglecting axle 
trie in Fig 5, the force F reqd to overcome rolling fric in such cases is directly 
as the wt W of and on the wheel, and inversely as the diam of the wheel. 

1 he few expts ihat have been made upon the coeffs of rolling fric, apart from 
axle fric, are too incomplete to serve as a basis for practical rules. See Art 20, 
and “Traction,” p 375. 

(«1) The fric (or “ adhesion ”) between wheel and rail, which enables a 
locomotive to move itself and train, or which tends to make a car-wheel revolve 
notwithstanding the pres of the brake, is a resistce to the sliding of the wheel 
on the rail ; and is therefore not rolling but sliding fric; static when the wheels 
either stand still or roll perfectly on the rails; and kinetic when thev slip or 
“ skid ”. See Art. 13 (c, d, e and /). 

Art 16. The friction of liquids moving in contact with solid bodies 
is independent of the pressure, because the “lifting” of the particles 
of the fluid over the projections on the surf of the solid body, is aided by the 
pres of the surrounding particles of the liquid, which tend to occupy the places 
i of those lifted. Hence we have, for liquids, no coeflf of fric corresponding with 
that (= resistce -t- pres) of solids. The resistce is believed to he directly as the 
I area of surf of contact. Recent researches indicate that Resistce = a coeff'X 
area of surf X vel n , in which both n and the coeff depend upon the vel and 
upon the character of the surf: and that at low vels » = l,but that at a certain 
“critical ” vel (which varies with the circumstances) n suddenly becomes = 2, 
owing to the breaking up of the stream into marked countercurrents or eddies. 
The resistance of fluid fric arises principally from the counter currents thus set 
in motion, and which must be brought into compliance with the direction of 
the force which is urging the stream forward. 

Art. 17. Table of coefficients of moving' friction of smooth 
plane surfaces, when kept perfectly lubricated. (Morin.) 


Dry 

Soap. 

Olive 

Oil. 

Tal¬ 

low. 

Lard. 

Lard <fc 
Plum¬ 
bago. 

.164 

.... 

.075 

.067 


.... 

.... 

.083 

.072 


.136 

.... 

.073 

.066 


.... 

• • • • 

.080 



. . . . 

.... 

.098 



. . . . 

.... 

.055 



.137 

.... 

.070 

.060 


.139 





.... 

.... 

.066 



.214 


.085 



, . . . 

.055 

.078 

.076 


.... 

.066 

.103 

.076 


.... 

.070 

.082 

.081 


.... 

.078 

.103 

.075 


.189 





.... 

.075 

.078 

.075 


.... 

.061 

.077 

.... 

.091 

.197 

.064 

.100 

.070 

• 05§ 

, . . . 

.078 

.103 

.075 


, . . , 

.... 

.069 



.... 

.066 

.072 

.068 


.... 

.077 

.086 



.... 

.072 

.081 


.089 

«... 

.058 




.... 

.079 

.105 

.081 


.... 

.... 

.093 

.076 


.... 

.053 

.056 

.... 

.067 

• • • • 

.133 

.159 



• ■ • • 

.191 

.241 




Substances. 


Oak on oak, fibres parallel to motion. 

“ “ “ fibres perpendicular to motion. 

“ on elm, fibres parallel to motion. 

“ on cast iron, fibres parallel to motion. 

“ on wrought iron, “ “ “ . 

Beech on oak, fibres “ “ “ ... 

Elm on oak, “ “ *• “ .. 

“ on elm, “ “ “ “ . 

I “ cast iron, “ “ “ “ . 

Wrought iron on oak, fibres parallel, greased aud wet, .256. 

*• “ “ “ fibres parallel to motion. 

“ “ on elm, “ “ “ “ . 

** “ on cast iron, “ “ “ . 

“ “ on wrought iron, “ “ “ . 

“ “ on brass, fibres “ “ “ . 

Cast iron on oak, fibres parallel to motion. 

•< •• “ “ “ “ “ “ greased aud wet, .218 

“ “ on elm, “ “ “ “ . 

“ “ on cast iron, with water, .314 . 

“ “ on brass. 

Copper on oak. fibres parallel to motion. 

Yellow copper on cast iron. 

Brass on cast iron. 

“ on wrought iron. 

“ on brass. 

Steel on cast iron. 

“ on wrought iron. 

“ on hrass . 

Tanned oxhide on cast iron, greased and very wet, .365 . 

“ “ ou brass . 

•* “ on oak, with water, .29. 


The launching: friction of the wooden frigate Princeton was found by 
a committee of the Kranklin Institute in 1844. to average about .067 or one-fifteenth of the pressure 
during the first .75 of a sec, and .022 or one forty-fifth for the next 4 secs of her motion. The slope 
of the wavs was 1 in 13, or 1 deg, 24 mins. They were heavily coated with tallow. Pressure on them 
— 15.84 lbs per sq inch, or 2280 lbs per sq ft. In the first .75 of a sec the vessel slid 2.5 ins ; in the 4 
next secs 15 ft. 6.5 ins; total for 4.75 sec 15.75 ft. 


























































374c? 


FRICTION. 


Art. 18. The friction of lubricated surfaces varies greatly with 
the character of the surfs and wit h that of the lubricant and the manner of its 
application. If the lubricant is of poor quality, and scantily and unevenly ap¬ 
plied under great pres, it may wear away in places and leave portions of the dry 
surfs in contact. The conditions then approximate to those of unlubricated 
surfaces. But if the best lubricants for the purpose are used, and supplied reg¬ 
ularly and in proper quantity, so as to keep the surfs always perfectly separated, 
the case becomes practically one of liquid fric (Art. 16), and the resistce is very 
small. Between these two extremes there is a wide range of variations (see 
table, Art. 19 (d)), the coelf being affected by the smallest change in the condi¬ 
tions. Where any degree of accuracy is reqd, we would refer the reader to the 
experimental results given in Prof. Thurston’s very exhaustive work,* devoted 
exclusively to this intricate subject. 

Art. 19. (a) Expts by Mr. Arthur M. Wellington upon the fric of 
lubricated journals f gave a gradual and continuous increase of coelf as 
the vel of revolution diminished from 18 ft per sec ( = a car speed of 12 miles ] 
per hour) to a stop. This increase was very slight at high vels, but much more 
rapid at low ones ; as in Figs 3 and 4. At vels from 2 to 18 ft per sec the coelf 
was much less under high pressures than under low ones; but at starting there 
was little dilf in this respect. The coelf increased rapidly as the tempera¬ 
ture rose from 100° to 120° and 150° Fahr. 

(b) Prof. Thurston, also experimenting with lubricated journals,t 
found that at starting, the coelf increased with increase of pres, as it did also 
when in motion, if the pres greatly exceeded the max (say 500 to 600 lbs per sq 
in) allowable in machinery. He also found that at high vels the coelf increased 
very slowly (instead of continuing to decrease) as the vel increased. 

(c) Prof. Thurston gives the following approx formula' for journal 
friction at ordinary temperatures, pressures and speeds, with journal and 
bearing in good condition and well lubricated: 


Coetf for starting' = (.015 to .02) X 1^ pres in lbs per sq in. 


Coelf when the shaft . X/'’ veTln It per min 

is revolving = (-02 to .03) X _ : 

f/prestulbs per sq in. 


At pressures of about 200 lbs per sq in : 


Temperature of mini 
fric ; in Fahr degs 


i in 


: 15 X 1/vel in ft per min 


Caution. The leverage, with which journal fric resists motion, in¬ 
creases with the diam of the journal. 


(d) The following figures, selected from a table of experimental results given 
by Prof. Thurston, merely show the extent to which the coeflf of 
journal fric is affected by pres, vel and temperature; and 
hence the risk incurred in rig'dlv applying general rules to such cases. In 
these expts the character of journal and bearing, the lubricant and its method 
of application, remained t he same throughout. Where these vary, still further, 
and much greater, variations in the coeff may occur. 

Steel journal in bronze bearing, lubricated with standard 

sperm oil. 


u 



4, V 



130 ° 

90° 


Speed of revolution 

30 feet per minute j 100 feet per minute joOO ft per min\ 1200 ft per min 

Pressures. 


200 

100 

4 

200 

100 

4 

200 

100 

200 

100 

lbs 

per sq 

in 

lbs per sq 

in 

lbs per sq in 

lbs pe 

r sq in 

Coeff 

Coeff 

Coeff 

Coeff 

Coeff 

Coeff 

Coeff 

Coeff 

Coeff 

Coeff 

.0160 

.0044 

.125 

.0087 

.0019 

.0630 

.0053 

.0037 

.0065 

.0075 

.0056 

.0031 

.094 

.0040 

.0019 

.0630 

.0075 

.0061 

.0100 

.0150 


* Friction and Oost Work in Machinery and Mill Work. John Wiley & Sons, New York, 1885. 
t Trans Anier Soc of Civil Kncrs, New York. Dec. 1884. 







































FRICTION. 


374 e 


(e) Where the force is applied first on one side of the jour¬ 
nal and then on the opposite side, as in crank pins, the fric is less 
than where the resultant pres is always upon one side, as in fly-wheel shafts; 
because in the former case the oil has time to spread itself alternately upon both 
sides of the journal. 



(f) Friction rollers. If a journal .T, in¬ 
stead of revolving on ordinary bearings, be sup¬ 
ported on friction rollers R, R, the force required 
to make J revolve will be reduced in nearly the 
same proportion that the diam of the axle o or 
o of the rollers, is less than the diam of the 
rollers themselves. 

Mr. Wellington experimented with a patent 
hearing on this principle, invented by Mr. A. 
Higley. Diam of rollers ER, 8 ins; of their 
axles o o If ins; of the journal c, 3£ ins. Here, 
theoretically, 


fric of patent journal = fric of 3£ in journal X 


diam of axles oo _ If ins 
diam of rollers RE 8 ins 


or as 1 to 4.6. Under a load of 279 lbs per sq in, Mr. Wellington found it about 
as 1 to 4 when starting from rest; and about as 1 to 2 at a car speed of 10 miles 
per hour. 


Art. 20. (a) Resistance of railroad rolling- stock. This con¬ 
sists of rolling fric between the treads of the wheels and the rails (the treads 
j also sometimes slide on the rails, as in going around curves); of sliding fric be¬ 
tween the journals and their bearings, and between the wheel flanges and the 
rail heads; of the resistce of the air; and of oscillations and concussions, which 
consume motive power by their lateral and vert motions, and also increase the 
wheel and journal fries. 

Its amount depends greatly upon the condition of the road-bed and rails (as 
to ballast, alignment, surf, spaces at the joints, dryness etc); upon that of the 
rolling stock (as towt carried’, kind of springs used, kind and quantity of lubri¬ 
cant, condition and dimensions of wheels and axles etc); upon grades and curv¬ 
ature; upon the direction and force of the wind; and upon many minor con¬ 
siderations. Experiments give very conflicting results. 


(b) During the summer of 1878, Mr. Wellington experimented with 
loaded and empty box and flat freight cars, passenger and sleeping cars, and at 
speeds varying from 0 to 35 miles per hour. The cars were started rolling (by 
grav) down a nearly uniform grade of .7 foot per 100 feet, or 36.5 feet per mile, 
and 6400 ft long. Their resistces were calculated as in Art. 8 (a). “The rails 
were of iron, 60 lbs per yd, and the track was well ballasted and in good line and 
surf, but not strictly first, class.” The following approx figures are deduced 
from Mr. Wellington’s expts upon cars fitted with ordinary journals :* 


Car Resistance in pounds per ton (2240 lbs) of weight of 
train, on straight and level track in good condition. 


Speed of 
train in 
miles per 


Empty cars 



Loaded cars 


Axle, 

Oscilla¬ 

tion 



Axle, 

Oscilla¬ 

tion 



hour 

tire and 
flange 

and 

con- 

cuss’n 

Air 

Total 

tire and 
flange 

and 

con- 

cuss’n 

Air 

Total 

0 

14 

0 

0 

14 

18 

0 

0 

18 

10 

6 

.6 

.4 

7 

4 

.6 

.4 

5 

20 

6 

2.7 

1.3 

10 

4 

2. 

1 . 

7 

30 

6 

5.3 

2.7 

14 

4 

4.7 

2.3 

11 


(c) With the Higley patent anti-fric roller journal, the resistce to starling was 
but about 4 lbs per ton. But see Art. 19 (f). 

(d) About midway in the track experimented upon, was a curve of 1° de¬ 
flection angle (5730 ft rad) 3000 ft long, with its outer rail elevated 3 to 4 ins 


* Transactions, American Society of Civil Engineers, Feb 1879. 































FRICTION. 


374/ 


above the inner one. The rise of the outer rail was begun on the tangent, about 
500 ft before reaching the curve. In the first 500 ft of the curve the resiaice was 
greater than that encountered just before reaching the curve, by from .6 to 2.1 
(average 1.1) lbs per ton. In t lie last 500 ft of thecurve this excess had diminished 
to from .2 to .9 (average .6) lbs per ton. Owing to the continuance of the down 
grade on the curve, the vel increased as the train traversed the curve; but it 
does not clearly appear whether the decrease in curve resistce was due to the 
increase in vel*or to the fact that the oscillations caused by entering tlie curve 
gradually ceased as the train went on. 

(e) ]?Ir. P. II. Dudley, experimenting with his “ dynagrapli ”* ob¬ 
tained results from which the following are deduced: 

Train Resistance in pounds per ton (2240 lbs) of weight of 

train, including grades. 


Trip 

Average 
speed. 
Miles per 
hou r 

Average 

resist¬ 

ance. 

Toledo to Cleve- 

20 

8.34 

land. 95 miles 

Cleveland to Erie 
95.5 miles 

20 

7.67 

Erie to Buffalo. 
88 miles 

20 

8.89 


Description of train 


Loaded 

cars 

Empty 

cats 

Weight tons 
(2240 lbs) 

29 

2 

526 

37 

0 

633 

25 

2 

458 


“With the lone and heavy trains of the L. S. & M. S. Ry, of 600 to 650 tons, it 
reqd less fuel with the sameengiue to run trains at 18 to 20 miles per hour than 
it did at 10 to 12 miles per hour”, owing to the fact that at the higher speeds 
steam was used expansively to a greater extent, and hence more economically. 

Art. 21. The work, in ft-lhs. reqd to overcome fric through 
any dist, is = the fric in lbs X the dist in ft. In order that a bodv,started slid¬ 
ing or rolling freely on a hor plane and then left to itself, mav do’this work ; ie, 
may slide or roll through the given dist, its kinetic energy ( = its wt in lbs X its 
vel 2 in ft per sec -f- 2g\) must = the first-named prod. Conversely, tlie dist 
in ft through which such a body will slide or roll on a hor plane, 'is 


its kinetic energy i n ft-lbs, at start 
fric in lbs 


wt of b ody in lbs Xjnitial vel 2 in ft per sec _ initial vel 2 in ft per sec 
wt of body in lbs X coeffof fric X 2gf coelf of fric X 2gf 


The time reqd, in secs, is == 


dist in ft, so found 
mean vel, in ft per sec 


_____ dist in ft 
5 initial vel in ft per sec 


Suppose two similar locomotives, A and B, each drawing a train on a level 
straight track ; A at 10 miles, and B at 20 miles, per hour. The total resistce of 1 
each eng and train (which, for convenience, we suppose to be independent of 
vel i is 1000 lbs. Hence the force, or total steam pres in the two cyls reqd to 
balance the fric and thus maintain the vel, is the same in each eng.’ In travel¬ 
ing ten miles this force does the same amount of work (1000 lbs X 10 miles 
— 10000 pound-miles) in eacii eng, and with the same expenditure of steam in 
each; although B must supply steam to its cyls twice as fast as A, in order to 
maintain in them the same pres. In one hour the force in A does 10000 lb-miles 
as before, but that in B does (1000 lbs X 20 miles ==) 20000 lb-miles, and with 
twice A’s expenditure of steam. 

But in fact the resistce of a given train is much greater at higher vels. See 
table Art. 20 ( b) And even if we still assumed the resistce to be the same at 
both vels, B must exert more force than A in order to acquire a vel of 20 miles 
per hour while A is acquiring 10 miles per hour. 


* An inst for measuring the strain on the draw-bar of a locomotive, or the force which the 
exerts upon the train. mner 

t g— acceleration of gravity = say 32.2 ; ig= say 64.4. See p 362. 




























TRACTION 


375 


A double purchase crane, with a weight of 7000 lbs suspended from it, showed a fi ic of g the weight, 
or nearly 800 ttis. One ton suspended at each end of a chain passing over 2 cast iron sheaves of 2 teet 

| diani; with wrought-iron journals, working in brass bearings, well oiled, gave -rV of the weight; or 
i 2 tons 

——320 lbs; or 160 lbs per ton. 

Morin says the fric of a sled on dry ground is % of the pres. Babbage states that a block of stone 
of 1080 lbs was drawu aloug a rock surface by 758 lbs; or fric 70 per ct: on a wooden sled on a wooden 
floor, 60 per ceut_; with both wooden surfaces greased, only 6 per cent; aud with the block on top of 
wrooden rollers 3 ins diam, only 2.6 per cent. Rubble masonry on wet clay .2 to .35. 


TRACTION. 


Traction oil common roa«l«. ami canals; or the power reqd to draw 

vehicles and boats along them. In connection with this subject read the preceding and the following 
one. 


The following table shows tolerable approximations to the force in lbs per ton. reqd to draw a stage 
eoach and passeugers, up ascents on the Holyhead turnpike road in England, (a fine road,) by horses ; 
- as ascertained by means of a dynamometer. The entire weight was 1)4 tons; but in the table, the 
j results are giveu per siugle ton. From the nature of such cases, no great accuracy is attainable. 


t 

1 

i| 

1 


s 

t 


I 

) 

I 

I 

I 

! 

I 


Proportional 

Ascent. 

Ascent in Ft. 
per Mile. 

At 4 Miles 
per Hour. 

At 6 Miles 
per Hour. 

At 8 Miles 
per Hour. 

At 10 Miles 
per Hour. 




Lbs. 

Lbs. 

Lbs. 

Lbs. 

1 in 

15)4 

340.7 

210 

216 

225 

240 

i ** 

20 

264. 

196 

202 

212 

229 

i “ 

26 

203.1 

155 

160 

166 

175 

i “ 

30 

176. 

137 

142 

147 

154 

i “ 

40 

132. 

114 

120 

124 

130 

i “ 

64 

82.5 

109 

115 

120 

126 

i “ 

118 

44.7 

102 

107 

113 

120 

i •• 

138 

38.3 

99 

103 

109 

117 

i “ 

156 

33.9 

98 

101 

106 

112 

i “ 

245 

21.6 

93 

96 

101 

107 

i •* 

600 

8.8 

81 

85 

91 

96 

Level. 

0. 

76 

80 

85 

91 


The following results, most of them with the same instrument, are also in Rs per ton ; with a four- 
wheeled wagon, at a slow pace, on a level; and the roads in fair condition. 

On a cubical block pavement. 32 lbs per ton...........to aO. 

•• McAdam road, of small broken stone. 62^ ( u probably to 75. 

“ gravel road. •••••• . ; , 4 

“ Telford road, of small stone on a paving of spawls 46 () „ 

“ broken stone, on a bed of cement concrete. 46 • .<?• 

- common earth roads. 200 to 300. On a plank road 30, to 50 lbs. 

The tractive power of a horse diminishes as his speed in¬ 
creases; and perhaps, within certain limits, say from % to four miles per hour, 
nearly in inverse proportion to it. Thus, the average traction of a horse, on a level, and actually 
pulling for 10 hours in the day, may be assumed approximately as follows: 

Miles per hour. Lbs. Traction. Miles per hour. Lbs. Traction. 

* .B” 58:::::::::::::: SB: 11 

200 : . 

1*:::::::::::::: JK 3 «:« 

2 .125. 4 . 62.50 

If he works for a smaller number of hours, his traction may increase as the hours diminish ; down 

to about 5 hours per dav and for speeds of about from 1% to 3 miles per hour. Thus, for 5 ho “ 1 1 .® V® r 
da his traction at 2V4 miles per hour will be 200 lbs, Ac. When ascending a hill, his power dimin¬ 
ishes so rapidly, front having partially to raise his own.weight .(which‘ 


thp moo of his own weight. Assuming tnai ou a levn picuc m ’ Y .Y- 

ne a carf and load together weighing 1 ton, have to exert a traction of 60 Rs; then on ascending a 
hifi 0 f 40 inclination (or 1 in 14.3; or 369)4 ft per mile,) he would have to exert 156 lbs, against the 
gravity of the l ton :’ and 67 lbs. against that of his own weight: or 223 lbs in all. He may for a few 
^ t ininrv about twice his regular traction. This calculation shows that up a hill 

0 n f 'iO average horse is fully tasked in drawing a total load of one ton ; and should, therefore, he 
snowed in s«ch a case to choose his own gait; and to rest at short intervals. A fair load for a single 
1 Ji + h n o variable walking pace, working 10 hours per day, on a common undulating 

r'oarHiTgood order, is about half a ton, in addition to the cart which will be about half a ton more. 

™r°si:5“h“ Em; wi....»«...«... 

become more objectionable the better the road is. 

Thus, on an ascent of 2°, or 184.4 ft per mile, gravity alone requires u traction of 78 lbs pel ton , 















































376 


TRACTION, 


which is about 10 times that on a level railroad at fi miles an hour; but only about equal to that on a 
level common turnpike road, at the same speed. Therefore, (to speak somewhat at random,) it would 
require 10 locomotives instead of 1; but ouly 2 horses instead of 1. A grade ot 1 in .>o , or loO It to a 
mile- or 1° 38' is about the steepest that permits horses to be driven down a hard smooth road, in a 
fast trot, without danger. It should, therefore, not be exceeded except when absolutely necessary, 1 
especially on turnpikes. . . , . . ,. ,. .. . j 

Oil canals ami other waters, the liquid is the resisting medium that 

takes the place of friction on level roads. But unlike friction, its resistance varies as the squares ol 
the vels; (see Art 26 of page 280.) at least from the vel of 2 ft per sec, or 1.364 miles per hour; tc 
that of 1 1'A ft per sec, or 7.84 m per h. As the speed falls below 1 % m per h, the resistauce varies less I 
and less rapidly ; and this is the case whether the moved body floats partly above the surface; or is ; 
entirely immersed. In towing aloug stagnant canals, Atc, the vel is usually from 1 to 2 « m pet h ; 
for freight most frequently from 1 )4 to 2. Less force is required to tow a boat at say 2 m per h, where 
there is no curreut, than at say m per h, against a current of in per h, because iu the last, ease 
the boat has to he luted up the very gradual inclined plane or slope which produces the curreut. 

The force required’to tow a boat along a canal depeuds greatly upon the comparative transverse I 
sectional areas of the channel, and of the immersed portion of the boat. When the width of a canal 
at water-line is at least 4 times thatof the boat; and the areaof its transverse section asgreatas at least j 
6)4 times that of the immersed transverse section of the boat, the towing at usual cuual vels will be 
about as easy as in wider and deeper water. With less dimensions, it becomes more difficult. (D’Au- 
buisson.) Much also depends on the shape of the bow and other parts of the boat; and on the propor¬ 
tion of its length to its breadth and depth. Hence it is seen that the mere weight of the load is by no 
means so controlling an element as it is on land. The whole subject, however, is too intricate to be 
treated of here. Morin states that naval constructors estimate the resistance to sailing and steam 
vessels at sea, at but from about .5 to .7 of a ib for every sq ft of immersed transverse section, when 
the vel is 3 ft per sec, or 2.046 miles per hour. It is far greater on canals. 


On the Scliuylkill Navigation of Pennsylvania, of mixed canal 

and slack water, for 108* miles, the regular load for 3 horses or mules, is a boat of very full build; and no 
keel; 100 ft long, 17)4 ft beam ; and 8 ft depth of hold ; drawing 5)4 ft when loaded.* Weightof boat 
about 65 tous; load 175 tons of coal, (2240 lbs;) total weight 240 tons, or 80 tons per horse or mule. 
On the down trip with the loaded boats, for 4 days, the animals are at work, actually towing, (except 
at the locks,) for 18 hours out of the 24; thus exceeding by far the limits of time usually allowed for 
continuous effort. 

On the canal sections, (which have 60 ft water-line ; and 6 ft depth,) the speed is 1 % miles per hour; 
and on the deep wide pools, 2 miles. 

On the up trip with the empty 65-ton boats, the average speed Is about 2)4 miles per hour. The 
empty boats draw 16 to 18 ius water; and frequently keep on without stopping to rest day or night 
through the entire distance of 108 miles. The animals generally have 2 or 3 days' rest at each end of 
the trip; but are materially deteriorated at the end of the boating season. 

If our preceding assumption of 143 lbs traction of a horse at \% miles per hour, is correct, the 


143 lbs 

traction of the loaded boats on the canal sections is -= 1.83 !bs per ton. 

80 tons 

The intelligent engineer and superintendent of the Sch Nav, James F Smith, gives as the results 
of his own extensive observation, that one of these large boats loaded (240 tons in all) may, without 
distressing the animals, be drawn along the canal sections, for 10 hours per day, as follow's: By one 
average horse or mule, at the rate of 1 mile: by two animals, at 1 )4 miles; and by three, at 1% miles 
per hour. When four animals are used the gain of time is very trifling. At a time of rivalry among 
the boatmen, one of them used 8 horses ; but with these could not exceed 2)4 miles per hour in the 
canal portions. Two or more horses together cannot for hours pull as much as when working sepa¬ 
rately. 

If our preceding short table of the traction of a horse at diff vels for 10 hours is correct, then th* 
traction of the above loaded coal boats (240 tons) on the canal sections of the navigation, is as follows; 
The last column shows the traction iu lbs per sq ftof area of immersed transverse section where largest; 
viz, about 95 sq ft. 


rses. 

1. 

Miles per Hour. 

250 

Lbs. per Tod. 

Lbs. per Sq Ft. 

2 . 


3 3 3 



3. 


* -40 

4 2.8 



3 on pools 

8 . 


..240. 

3 7 5 




2 4 0 

8 00 



3 up-trip.. 

.2)4. 

* * 2 4 0 

3 00 

. 6 5. 




Lachine Canal, Canada, 120 ft wide at water-line; 80 ft at bottom ; depth 

on mitre sills 9 ft; 6 horses tow loaded schooners with ease. 

Before the enlargement of the 1-i I*i<‘ caiial.f its dimensions were 40 ft water-line ; 28 ft bottom ; 
4 ft depth of water. '1 he average weight of the boats was about 30 tons. With 75 tons of load, or 105 
tons total, they were towed by 2 horses, at the rate of about 2 miles per hour ; which bv our table gives 
a traction of nearly 2.4 lbs per ton. The boats were about 80 ft long ; 14 ft beam ; full 3)4 ft draught 
loaded ; heuce the traction by our table would be about 5.7 lbs per sq ft of immersed transverse section* 


* Cost of liosvts, 1884, (Schuylkill Canal) about $1800. Annual repairs about 
?*»• Boats last 16 to 20 years Length, 102 ft; beam, 17)4 ft; draft, 1)4 to 5)4 ft; capacity, 180 
tons j weight, about 08 tons ; speed, with 3 mules, miles per hour 

t Length 363 miles ; cost $19680 per mile. The enlarged canal lias70 ft; 42 ft; and 7 ft of water ; 
and cost $90800 per mile for the enlargement only. The cost of the several canals iu Pennsylvania 
lias ranged between $23000 and $o0000 per mile. J 



























ANIMAL POWER. 


onrr 

Oi t 


While, for 8'2-ton loaded boats on a smaller canal, (the boa ts nearly touching bottom,) the traction at 
1% miles, would be 3% fts per ton ; or about twice as great as the above 1.78 lbs. It also would be 5.7 
lbs per sq ft of immersed section. 

For traction on railroads, see Art 20, pp 374 e and 374/. 


ANIMAL POWER. 


Art. 1. So far as regards horses, this subject has been partially considered 
under the preceding head, Traction. All estimates on this subject must to a certain extent be vague, 
owing to the dilf strengths and speeds of animals of the same kind ; as well as to the extent of their 
training to any particular kind of work. Authorities on the subject differ widely; and sometimes 
express themselves in a loose manner that throws doubt on their meaning. We believe, however, 
that the following will be found to be as close approximation to practical averages as the nature of 
the case admits of with our present imperfect knowledge. We suppose a good average trained horse, 
weighing not less than about % a ton, well fed and treated. Such a one, when actually walking for 
10 hours a day, at the rate of 2% miles per hour, on a good level road, such as the tow-path of a canal, 
or a circular horse-path,* can exert a continuous pull, draught, power, 
or traction, of lOO lbs. 

Now, 2% miles per hour, is 220 ft per min. or 3% ft per sec; and Binee 10 hours contain 600 min, 
his day’s work of actual hauling on a level, at that speed, amounts to 

min ft Ihs 

600 X 220 X 100 — 13 200 000 ft-lbs per day. 

Or, 22000 ft-lbs per min, or 366% ft-lbs per sec.t Which means that he exerts force enough during the 
day to lift 13 200000 lbs 1 foot high ; or 1 320000 lbs 10 feet high; or 132000 lbs 100 ft high, &c. He may 
exert this force either in traction (hauling) or in lifting loads. If he has to raise a small load to a 
great height, the machinery through which he does it must be so geared as to gaiu speed, at the loss 
(commonly but improperly so expressed) of power. Whether he lifts the great weight through a 
small height, or the small weight through a great height, he exerts precisely the same amount of 
force or power. In connection with this subject, the student should read Arts 5, 9, 11, &e, of Force 
iu Rigid Bodies. Also, see Hauling by Horses and Carts. 

Experience shows that within the limits of 5 and 10 hours per day, (the speed remaining the same,) 

tlie draft of a horse may he increased in about the same pro¬ 
portion as the time is diminished ; so that when working from 5 to 10 hour! 
per day, it will be about as shown in the following table. Hence, the total amount of 13200 000 ft-lb# 
per day may be accomplished, whether the horse is at work 5, 6, or 8, &c, hours per day 4 This, of 
course, supposes him to be actually' lifting or hauling all the time; and makes no allowance for stop¬ 
pages for any purpose. 

Table of draft of a horse, at 2% miles per hour, on a level. 


Hours per day. 

Lbs. 

Hours per day. 

Lbs. 

10 . 

.... 100 

7 . 

... 142® 

9 . 

.... Ill| 

6 . 


8 . 

.... 125 

5 . 

... 200 


Experience also shows that at speeds between % and 4 
miles an hour, his force or draught will he inversely in pro¬ 
portion to his speed. Thus, at 2 miles an hour, for 10 hours of the day, hi# 
draught will be 

miles miles lbs lbs 
2 : 2% : : 100 : 125 draught. 

At 1% miles, it would be 166% fts; at 3 miles, 83% lbs ; and at 4 miles, 62% fts ; as per table i» 
Traction. 

Therefore, in this case also, the entire amount of his day’s work remains the same;§ and within 


* To enable a horse to work with ease in a circular horse-walk, its diam 
should not be less than 25 ft; 30 or 35 would be still better. 

t A nominal horse-power is 33000 ft-fbs per minute; this being the rata 
assumed by Boulton and Watt in selling their engines; so that purchasers wishing to substituM 
steam for horses, should not be disappointed. Their assumption can be carried out by a very strong 
horse day after day for 8 or 10 hours; but as the engine can work day and night tor months without 
stopping, which a horse cannot, it is plain that a one-horse engine can do much more work than any 
one such horse. Hence many object to the term horse-power as applied to engines; but since every¬ 
body understands its plain meaning, and such a term is convenient, it is not In fact objectionable. 
Boulton and Watt meant that a one-horse engine would at any moment perform the work of a very 
strong horse. An average horse will do but 22000 ft-lbs per min. 

} K, i s plain that although the day's labor will be the same, that of an hour, or of a min, will vary 
with the number of hours taken as a day’s work. It must be remembered that a working day of a 
given number of hours, by no means implies, in every case, that number of hours of actual work; 
but includes intermissions and rests. • 

^ 'E iiis remark about speed will not apply to loads lowed 

th ronsrli the water. Thus, if his draught at 2 miles an hour he 125 ft»s; and 
at 4 miles, 62% lbs ; he will on land draw loads in these proportions ; but in hauling a boat through 
the water at the greater speed, he has to encounter the increased resistance of the water itself; which, 
resistance at 4 miles is much more than twice as great as at 2 miles; probably 4 times as great. 
Therefore, at 4 miles on a canal, his draught of 62% fts would not suffice for a load half as great aa 
be could tow with his draft of 125 fts at 2 miles. 















378 ANIMAL POWER. 


all the foregoing limits of hours and speed, may be practically taken to be about 13 200 000 ft-fl>s per 
day • or 22000 fl-BH per miu of a day of 10 hours. But it does not follow that the horse can always 
in practice Actually lift, loads at that rate; because generally a part of his power is expeuded in ' 

overcoming the friction of the machinery which he puts in motion ; aud moreover, the nature ot the 
work may require him to stop frequently ; so that in a working day of 8 or 10 hours, the horse may ^ 
not actually be at work more than 5, 0, or 7 hours. 

As a rough approximation, to allow for the waste of force in overcoming the friction of hoisting 

machinerv and the weight of the hoisting chains, buckets, &c. we may say that the 

or paying daily net work of a horse, in hoisting by a com* 

moil gin, is about 10000000 ft-lbs. That is, he will raise equivalent to 10000000 ttis net of 
water or ore, &c. 1 foot. The load which he cau raise at once, including chains, bucket, aud an 
allowance for’friction, will be as much greater thau his own direct force, as the diam of the horse- 
walk is greater than that of the winding drum; aud it will move that much slower than he does. 

His own direct force will vary according to the number of hours per day that he may be required to 
work, as in the foregoing table. With these data, the size of the buckets can be decided on , aud of j 
these there should be at least two, so that the empty one at the bottom may be tilled while the full one 
at top is being emptied ; so as to save time. The same when the work is done by men. 

Art. 2. A practised laborer liauling along a level road, by 
a rope over his shoulders; or in a circular path, pushing before him a 
hor lever, at a speed of from 1 % to 3 miles per hour, exerts about % part as much force as a horse; 
or 2 200 000 ft-lbs per day ; or 3666% ft-fts per min of a day of 10 hours of actual hauling or pushing. 

But laborers frequently have to work under circumstances less advantageous for the exertion of 
their force than when hauling or pushingin the manner just alluded to ; and in s„eh cases they cauuot 
do as much per day. Thus in turning a winch or crank like that of a grindstone, or of a crane, the 
continual bending of the body, and motion of the arms, is more fatiguing, fhe Size Of H 
winch should not exceed 18 ins, or the rad of a circle of 3 ft diam; aud against 
it a laborer can exert a force of about 16 tbs, at a vel of 2% ft per sec, or 150 ft per min, making very 
nearly 16 turns per min; for 8 hours per day. To these 8 hours an addition must be made of about 

M part, for short rests. Or if a working day is taken at 8. or 10, &c, hours, part must generally be 
taken from it for such rests. On the foregoing data an hour’s work of 60 min of actual hoisting 
would be 

lbs ft min 

16 X 150 X 00 = 144000 ft-lbs; 

or, deducting A part for rests, 115200 ft-lbs per hour of time , including rests. In practice, however, 
a further deduction must be made for the fric of the machine, and for the wt of the hoisting chains ; 
aud in case of raising water, stone, ore, &c. from pits, for the wt of the buckets also. As a rough 
average we may assume that these will leave but 100000 ft-lbs of paying, or useful work per hour; 

that is, that si man sit a winch will actually lift equivalent to 
lOOOOO lbs of water, ore, «fcc, 1 foot high per hour’s time, in¬ 
cluding rests. This is equal to 1666% ft-lbs per min of a day of 10 hours, including rests. 
Therefore, in a day of 10 working hours he would raise 1 000000 lbs net, 1 foot high ; Or just _J_ 

part of what a horse W’Ollld do with a gin in the same time. We have 

before seen that in hauling along a level road, he can at a slow pace perforin about % of the daily 
duty of a horse. He may also work the winch with greater force, say up to 30 or even 40 lbs; but 
he will do it at a proportionately slower rate; thus, accomplishing only the same daily duty. 
With lb gin, like those for horses, but lighter, with 2 or more buckets, a prac¬ 
tised laborer will in a working day of 10 hours, raise from 1 200000 to 1 400000 ft-lbs net of water, ore, i 
&c. With a shallow well or pit, more time is lost in emptying buckets than in a deep one; but the 
deep one will require a greater wtof rope. To save time in all such operations on a large scale, there 
should be at least two buckets; the empty one to be filled while the-full one is being emptied. It is 
also best to employ 2 or more men to hoist at the same time, by winches, at both ends of the axis; 
aud the men will work with more ease if the winches are at right angles to each other. Each winch 
handle mav be long enough for 2 or 3 men. An extra man should be employed to empty the buckets. 

He may take turns with the bolsters. The same remarks apply in some of the following cases. 

Oil a tread wheel a practised laborer will do about 40 per cent more daily 
duty than at a winch ; or in a working day * of 10 hours, including rests, he will do about 1 400000 ft- 
lbs. And he can do this whether he works at the outer circnmf of the wheel, stepping upon foot¬ 
boards, or tread-boards, on a level with its axis; or walks inside of it near its bottom. In both cases 
he acts by his wt. usually about 130 to 140 lbs; and not by the muscular strength of his arms. When 
at the level of the axis, his wt acts more directly than when he walks on the bottom of the wheel; 
hut in the fir <t case he has to perform a slow and fatiguing duty resembling that of walking up a 
continuous (light of steps; while in the second he has as it were merely to ascend a very slightly in¬ 
clined plane; which he can do much more rapidly for hours, with comparatively little fatigue: and 
this rapidity compensates for the less direct action of his wt. Therefore, in either case, as experience 
lias shown, he accomplishes about the same amount of daily duty. Treadwheels may be from 5 to 25 
ft iu diam, according to the nature of the work. They are generally worked by several men at once, 
and may at times be advantageously used in pile-driving, as well as in hoisting water, stone, &c. 

By a gootl common pump. properly proportioned, a. practised laborer 

will in a day or 10 working hours, raise about 1 000 000 ft-lbs of water, net.t 

Bailing' with a light bucket or scoop, he can accomplish about 
200 000 ft-lbs net of water. By a bucket and swape. (along lever rocking vertically; 
and weighted at one end so as to balance the full bucket hung from the other; often seen at couutry 

* The working day must be understood to include necessary rests, and such intermissions as the 
nature of the work demands ; bat does not include time lost at meals. A working dag of 10 hours 
may. therefore, have but 8, 7, or 6, <fcc hours of actual labor. This will be understood when we here¬ 
after speak of a working dav, or simply a day. 

♦ Desagulier's estimates of daily work of men and horses exceed the above, but are entirely too great. 







ANIMAL POWER, 


379 


irells,) 600000 to 800000. In the last he has only to pull down the empty bucket, and thereby raise the 

counterweight. By 2 buckets at the ends of a rope suspended over 

pulley , 500 000 to 600000, Here he works the buckets by pulling the rope by hand. 

*5y a tympan, or tympanum,* * worked by a treadwheel, about 1 200000 

to 1 400 000. 

By a Persian wtieel.f a chain-pump, a chain of buckets,! or 
an Archimedes screw, all worked by a treadwheel, from 800 000 to 1000 000 
it-lbs. Of these four, the first three lose useful effect by either spilling, leaking, or the necessity for 
raising the water to a level somewhat higher than that at which it is discharged. 

U nen any of the five foregoing machines are worked by men at winches, the result will be about 
■4 less than by treadwheels. They are all frequently worked also by either steam,water, or horse-power. 

By walking- backward and forward, on a lever which rocks 
on its center, a man may, according to Robison’s Mech Philosophy, perform a 
much greater duty than by any of the preceding modes. He states that a young man weighing 135 
lbs. and loaded with 30 Bis in addition, worked in this manner for 10 hours a day wnthout fatigue; 
and raised 9% cubic feet of water, 11)4 ft high per min. This is equal to 3 984 000 ft-lbs per day of 10 
hours; or 6640 ft-Bis per min ; or nearly of the net daily work of a horse in a gin. 


A laborer standing still*, can barely sustain for a few min, a load of 100 

lbs, bv a rope over his shoulder, and thence passing off hor over a pulley. And scarcely as much, 
when (facing the load and pulley) he holds the end of a hor rope w ith his hands before him. He can¬ 
not push hor with his hands at the height of bis shoulders, w ith more than about 30 lbs force. 

Weisbach states from his own observation, that 4 practised men raised a dolly (a wooden beetle 
or rammer, of wood; with 4 hor projecting round bars for handles) weighing 120 lbs, 4 ft high, at the 
rate of 34 times per min, for 4}{ min ; and then rested for 4% min ; and so on alternately through 
the 10 hours of their working day. Therefore, 5 of these hours were lost in rests; and the duty per¬ 
formed by each man during the other 5 hours, or 300 mius, was 


120 X 4 X 34 X 300 

4 _ 


- = 1224000 ft-Bis. 


In tile old mode of driving piles, where the ram of 400 to 1200 lbs 

suspended from a pulley, was raised by 10 to 40 men pulling at separate cords, from 35 to 40 Bis of the 
ram were allotted to each man, to be lifted from 12 to 18 times per min, to a height of ‘i% to 4% feet 
each time, for about 3 min at a spell, and then 3 min rest. It was very laborious; and the gangs had 
to be changed about hourly, after performing but 34 an hour's actual labor. 


.Hauling by horses. See Traction. When working all day, say 10 working 

hours, the average rate at which a horse walks while hauling a full load, and while returning with 
the empty vehicle, is about 2 to 2}4 miles per hour; but to allow for stoppages to rest, Ac, it is safest 
to take it at but about 1.8 miles per hour, or 160 ft per min. The time lost on each trip, in loading 
and unloading, may usually be taken at about 15 min. Therefore, to find the number of loads that can 
be hauled to any given dist in a day, first find the time in min reqd in hauling one load, and return¬ 
ing empty. Thus: div twice the dist in ft to which the load is to be hauled; or in other words, div 
the length in ft. of the round trip, by 160 ft. The quot is the number of min that the horse is in mo¬ 
tion during each round trip. To this quot add 15 min lost each trip while loading and unloading ; the 
sum is the total time in min occupied by each round trip. Div the number of min in a working day 
(600 min in a day of 10 working hours) by this number of min reqd for each trip; the quot will be the 
number of trips, or of loads hauled per day. 

Ex. How many loads will a horse haul to a dist of 960 ft, in a dav of 10 working hours, or 600 min ? 

1920 

Here, 960 X 2 = 1920 ft of round trip at each load. And - -- = 12 nnn, occupied in walking. And 
„ . : „ ... ,, 600 min in 10 hours 

12 + 15 in loading, Ac) = 27 min reqd for each load. Finally, —— =- ; - - rr 22 2, or 

1 27 min per trip 


say 22 trips; or loads hauled per day. 


Table of number of loads hauled per day of 10 working 
hours. The first col is the distance to which the load is actually hauled; or half 
the length of the round trip. The cost of hauling per load, is supposed to be for one-horse carts; the 
driver doing the loading and unloading; rating the expense of horse, cart, and driver at $2 per day. 
See Cost of Earthwork, page 742. 


* The tympan revolves on a hoi- shaft: and is a kind of large wheel, the spokes, arms, or radii of 
which are gutters, troughs, or pipes, which at their outer ends terminate in scoops, which dip into 
the water. As the water is gradually raised, it flows along the arms of the wheel to its axis, where 
it is dischd. The scoop wheel is a modification of it. It is an admirable machine for raising large 
quantities of w-ater to moderate heights. We cannot go into any detail respecting this and other 
hvdraulic machines. 

*t A kind of large wheel with buckets or pots at the ends of its radiating arms ; revolves on a hor 
axis; discharges at top. The buckets are attached loosely, so as to hang vert, and thus avoid spill¬ 
ing until they arrive at the proper point, where they come into contact with a contrivance for tilting 
and emptying them. The noria is similar, except that the buckets are firmly held in place, and thus 
spill much water. It, is therefore inferior to the Persian wheel. 

I An endless revolving vert chain of buckets. D’Aubuisson and some others erroneously call thia 
the noria. It is an effective machine. 







380 


SPECIFIC GRAVITY. 


Dist. 

Feet. 

No. of 
Loads. 

Cost per 
Load. 

Dist. 

Feet. 

No. of 
Loads. 

Cost per 
Load. 

Dist. 

Miles. 

No. of 
Loads. 

Cost per 
Load. 

50 

38 

Cts. 

5.26 

1500 

18 

Cts. 

11.11 

1 

7 

Cts. 

28.57 

100 

37 

5.11 

2000 

15 

13.33 

i 'A 

6 

33.33 

2)0 

34 

5.88 

2500 

13 

15.39 


5 

40.00 

300 

32 

6.25 

3000 

11 

18.18 

2 

4 

50.00 

400 

30 

6.(»7 

3500 

10 

20.00 

3 

3 

66.67 

600 

27 

7.11 

4000 

9 

22.22 

4 

2 

100.00 

1000 

22 

9.09 

5000 

7 

28.57 

9 

1 

200.00 


If the loading and unloading is such as cannot be done by the driver alone; but requires the help 
of cranes, or other machinery, an addition of from 10 to 50 cts per load may become necessary. Haul¬ 
ing can generally be more cheaply done by using 2 or 0 horses, and one driver, to a vehicle. The neat . 
load per horse, in addition to the vehicle, will usually be from to 1 ton, depending on the condition, ; 
and grades of the road. From 13 to 15 cub ft of solid stone; or from 23 to 27 cub feet of broken stone, ; 

make i ton. In estimating' tor hauling; rough quarry stone tor 
drains, culverts, <fcc, bear in mind that each cub yard of common scabbled rubble 
masonry, requires the hauling of about 1.2 cub yds of the stone'as usually piled up for sale iu the 1 
quarry ; or about % of a cub yd of the original rock in place. A. dll) yd Of solid StOUO, 

when broken into pieces, usually occupies about 1.9 cub yds 
pertectly loose; or aboutwhen piled up. A strong cart for stone hauling, will weigh 
about % ton ; or 1500 fbs ; and will hold stone enough for a perch of rubble masonry ; or say 1.2 pers 
of the rough stone in piles. The average weight of a good working horse is about 34 a ton. 

Morin gives the following results from careful experiments made by 
him for the French Government. The draft of the same wheeled vehicle on a road, may iu practice 
be considered to be, 

1st. On hard turnpikes, and pavements: in proportion to the 

loads; in versely as the diams of the wheels ; and nearly independent of the width of tire. It increases 
to uncertain extents with the inequalities of the road ; the stiffness (want of spring) of the vehicle; 
and the speed; (considerably less than as the square roots of the last.) 

2d. On soft roads, the draft is less with wide tires than 
with narrower ones; and for farming purposes lie recommends a width of 
4 ius. With speeds from a walk to a fast trot, the draft does not vary sensibly. 




SPECIFIC GRAVITY. 

The sp grav of a body, is its weight as compared with that of an equal bulk of 
some other body, which is adopted as a standard of comparison. For other substances 
than air and gases generally, pure water is the usual standard; and since the weight 
of a given bulk of water varies somewhat with its temperature; and also with the state 
of the air, the former is assumed to be 62° Fall; and the hitter at 30 ins, at sea-level. 
On the continent of Europe, water at its greatest density, or at a temperature of 4° 
Centigrade, = 39.2° Fahrenheit, is taken as the standard. Hut where extreme 
scientific accuracy is not aimed at, all these considerations may he neglected; and 
any clear fresh water, at any ordinary temperature, sav from 60° to 80°, may he 
used. At 70°, the resulting sp gr is but 1 part in 1176 greater than at 62° F; at 75°, 
1 to 670; at 80°, 1 in 454; at 85°, 1 in 336; at 90°, 1 in 264. At 62° pure water 
weighs 62.355 lbs avoir per cub ft. 

To find file sp grav of a body, heavier than wafer. Weigh 

it first in the air; and then in water; and find the dill. The diflf is what the body 
loses in water; and is the weight of a bulk of water equal to the bulk of the body. 
Then say, Diff : wt in air : : 1 : sp grav of body. 



































SPECIFIC GRAVITY 


381 


The weight of a given bulk of a substance which is either porous, or absorbent of water, cannot be 
inferred from its sp gr. Thus pure river sand, is pure quartz ; and of course has the same sp gr ; jet, 
a solid cub ft of quartz, weighs nearly twice as much as a cub ft of sand ; on account of the interstices 
of the latter. A brick, some sandstones. &c. absorb water; so that their sp gr will not furnish the 
weight of a dry mass of the same. In such cases, the eugiueer will generally first, measure the con¬ 
tents of a piece of the substance, if a solid ; and then weigh it; thus ascertaining its weight per cub 
ft. &c. if it is in graius, or dust, he will measure, and theu weigh, a cub ft of it. Footuote p 384. 

To fllld the S|> Jfrjtv ot «l li<gusd. First carefully weigh some solid body, as a 
piece of metal, in the air. Then weigh it in water, aud note the loss, say L. Then weigh it in the 
other liquid: and note the loss, say I. Then as loss L, is to loss l. so is 1, or the sp gr of water, to the 
sp gr of the liquid. Or, if the sp gr, and weight of the solid body, are already known, merely weieh 
it in the liquid. Then as its weight in air, is to its loss in the liquid, so is its sp gr. to that of the liquid. 

Timber, when first purchased from lumber yards, even under shelter, is rarely, if ever, perfectly 
dry; but its weight, if tolerably seasoned, will be about % part greater than given in our tables, or 
rbout *4 to yi part, if green. 

Tabic of specific gravities, and weights. 

In this table, the sp gr of air, and gases also, are compared with that of water, 
instead of that of air; which last is usual. 


The specific gravity of any substance is = its weight 
in grams per cubic centimetre. 


Air, atmospheric; at 60° Fah, and under the pressure of one atmosphere or 

14.7 lbs per sq inch, weighs -g j-r part as much as water at 60°. 

Alcohol, pure. 

“ of commerce. 

“ proof spirit. 

Ash, perfectly dry. (See footnote, p 383.).average.. 

1000 ft board measure weighs 1.748 tons. 

Ash, American white, dry. “ 

1000 ft board measure weighs 1.414 tons. 

Alabaster, falsely so called; but really Marbles. 

“ real; a compact white plaster of Paris.average .. 

Aluminium. 

Antimony,'cast, 6.66 to 6.74.average .. 

“ native. “ 

Anthracite, 1.3 to 1.84. Of Penn'a, 1.3 to 1.7.usually .. 

A cubic yard solid, averages about 1.75 cub yds,when broken to any mar¬ 
ket size ; and loose. 

Anthracite, broken, of any size. Loose.average.. 

“ “ moderately shaken. “ •• 

“ heaped bushel, loose, 77 to 83. 

A ton, loose, averages from 40 to 43 cub ft. 1 

at 54 lbs per cub ft, a cub yard weighs .651 ton. 

Aspbaltum, 1 to 1.8. “ •• 

Basalt. See Limestones, quarried. “ •• 

Bath Stone, Oolite. " •• 

Bismuth, cast. Also native. “ •• 

Bitumen, solid. See Asphaltum. 

Brass, (Copper and Zinc,) cast, 7.8 to 8.4. “ •• 

“ rolled. “ •• 

Bronze. Copper 8 parts ; Tin 1. (Gun metal.) 8.4 to 8.6. •• 

Brick, best pressed. 

*• common hard. 

“ soft, inferior. 

Brickwork. See Masonry. 

Boxwood, dry. * •• 

Calcite, transparent. . ‘ •• 

Carbonic Acid Gas, is times as heavy as air. ( •• 

Charcoal, of pines and oaks. 

Chalk. 2.2 to 2.8. See Limestones, quarried. “ •• 

Clay, potter’s, dry, 1.8 to 2.1... “ 

“ dry, in lump, loose. ‘ •• 

Coke, loose, of good coal.._. ‘ •• 

14 a heaped bushel, loose, 35 to 42 lbs. “ 

“ a ton occupies 80 to 97 cub ft. “ 

In coking, coals swell from 25 to 50 per cent. 

Equal weights of coke and coal, evaporate about equal wts of 
water; and each abt twice as much as the same wt of dry wood. 

Corundum, pure, 3.8 to 4. 

Cherry, perfectly dry.average . 

1000 ft board measure weighs 1.562 tons. 

Coal, bituminous, 1.2 to 1.5. ‘ •• 

« “ broken, of any size; loose. 

“ moderately shaken. “ >• 

“ “ a heaped bushel, loose, 70 to 78 lbs. 

•* “ a ton occupies 43 to 48 cub ft. 

A cubic yard solid, averages about 1.75 yards when broken to any 
market size, and loose. 


Average 

Sp Gr. 

Average 
Wt of a 
Cub Ft. 
Lbs. 

.00123 

.0765 

.793 

49.43 

.834 

52.1 

.916 

57.2 

.752 

47. 

.61 

38. 

2.7 

168. 

2.31 

144. 

2.6 

162. 

6.70 

418. 

6.67 

416. 

1.5 

93.5 


52 to 56 


56 to 60 

1.4 

87.3 

2.9 

181. 

2.1 

131. 

9.74 

607. 

8.1 

504. 

8.4 

524. 

8.5 

529. 


150. 


125. 


100. 

.96 

60 

2.722 

169.9 

.00187 



15 to 30 

2.5 

156. 

1.9 

119. 


63. 


23 to 32 

3.9 


.672 

42. 

1.35 

84. 


47 to 52 


51 to 56 































































382 


SPECIFIC GRAVITY 


Table of specific gravities, and weights — (Continued.) 


The specific gravity of any substance is = its weight 
in grams per cubic centimetre. 


average. 


Chestnut, perfectly dry. (See footnote, p 383.). 

1000 feet board measure weighs 1,525 tons. 

Cement, hydraulic. American, Rosendale; ground, loose.average 

“ “ “ “ U S. Struck bush, 70 lbs. 

“ “ “ Louisville, “ “ 62. 

“ “ “ Copley, “ “ 67. 

“ “ English Portland, U.S. struck bush, by Gillmore, 100 to 128 

“ “ “ “ Various, weighed by writer, 95 to 102. 

“ “ “ “ a barrel 400 to 430 lbs. 

“ “ French Boulogne Portland, struck bush, 95 to 110.... 

Differences of 4 or 5 pounds either more or less than we here give per 
loose struck U.S. bush, often occur in the cement from the same 
manufactory, owing not only to the difficulty of measuring exactly, 
but to the want of uniformity in the composition of the stone, de¬ 
gree of burning, grinding, dryness, &c. Moreover, the term “loose" 
is indefinite. We mean by it the average looseness which it has 
when thrown by a scoop into a half bushel when measuring that 
quantity for sale. By shaking it may easily be compacted about % 
part, so as to weigh ^ more per bush, or cub ft. And by ramming, 
about part, so as to weigh about more. So with lime, plas¬ 
ter, <Sc. 

Copper, cast,.8.6 to 8.8. 

“ rolled.8.8 to 9.0. 

Crystal, pure Quartz. See Quartz. 

Cork... 

Diamond, 3.44 to 3.55 ; usually 3.51 to 3.55. 

Earth ; common loam, perfectly dry, loose. 

“ “ “ “ “ shaken.. 

“ “ “ “ “ moderately rammed... 

“ “ “ slightly moist, loose.. 

“ “ “ more moist, “ . 

“ “ “ “ shaken. 

“ “ “ “ moderately packed..... 

“ “ “ as a soft flowing mud. 

“ “ as a soft mud. well pressed into a box 


Ether 


Elm. perfectly dry. (See footnote, p 383.).average.. 

1000 ft board measure weighs 1.302 tons. 

Ebony, dry.... “ 

Emerald, 2.63 to 2.76. “ .. 

Fat. “ 

Flint. “ .. 

Feldspar, 2.5 to 2.8. “ 

Garuet, 3.5 to 4.3; Precious, 4'1 to 4.3. “ 

Glass, 2.5 to 3.45... “ 

“ common window. 

“ Millville, New Jersey. Thick flooring glass. “ 

Granite, 2.56 to 2.88. See Limestone, 160 to 180. “ 

Gueiss, common, 2.62 to 2.76. “ 

“ in loose piles. “ 

“ Horublendic. “ 

“ " quarried, in loose piles. “ 

Gypsum, Plaster of Paris, 2.24 to 2.30. “ 

“ in irregular lumps. “ 

“ ground, loose, per struck bushel, 70. “ 

“ “ well shaken, “ “ 80. “ 

" “ Calcined, loose, per struck bush, 65 to 75. “ 

Greenstone, trap, 2.8 to 3.2. •< 

•• “ quarried, in loose piles. “ 

Gravel, about the same as sand, which see. 

Gold, cast, pure, or 24 carat. “ 

“ native, pure, 19.3 to 19.34. “ 

“ frequently containing silver, 15.6 to 19.3. “ 

“ pure, hammered, 19.4 to 19.6. “ 

GuttaPercha. “ 

Hornblende, black, 3.1 to 3.4. « 

Hydrogen Gas, is 14M times lighter than air; and 16 times lighter than 

oxygen.average.. 

Hemlock, perfectly dry. (Footnote, p 383.).. “ 

1000 feet board measure weighs .930 ton. 

Hickory, perfectly dry. (See footnote, p 383.). “ 

1000 feet board measure weighs 1.971 tons. 

Iron, cast, 6.9 to 7.4. «i 

“ “ usually assumed at.« '' 

At 450 lbs. a cub inch weighs .2604 ft ; 8601 6 cub inches a ton ; and 
aft — 3 8t00 cub inches ; cast-iron gun inct&l. 


Average 
Sp Gr. 


.66 


8.7 

8.9 


.25 

3.53 


.716 

.56 


1.22 

2.7 

.93 

2.6 

2.65 

4.2 

2.98 

2.52 

2.53 
2.72 
2.69 


2.8 

' 2!27' 


Average 
Wt of a 
Cub Ft. 
Lbs. 


41. 

56. 


49.6 

53.6 

81 to 102 1 

76 to 81.6 


76 to 88 


542. 

555. 


15.6 


72 to 80 
82 to 92 
90 to 1U0 
70 to 76 
66 to 68 
75 to 90 
90 to 100 
104 to 112 
110 to 120 
44.6 
35. 


76.1 


58. 

162. 

166. 


186. 

157. 

158. 
170. 
168. 

96. 

175. 

100 . 

1416 

82. 

56. 

64. 



52 to 60 

3. 

187. 

107. 

19.258 

1204. 

19.32 

1206. 

19.5 

1217. 

.98 

61.1 

3.25 

203. 


.00527 

.4 

25. 

.85 

53. 

7.15 

446. 

7.21 

450. 

7.48 

467. 






























































































SPECIFIC GRAVITY 


383 


Table of specific gravities, and weights — (Continued.) 


The specific gravity of any substance is = its weight 
in grains per cubic centimetre. 


Iron, wrought, 7.6 to 7.9; the purest has the greatest sp gr.average.. 

“ large rolled bars. “ 

• usually assumed at. “ 

“ sheet. “ 

At 480 lbs, a cub inch weighs .2778 fi>; and a ffl> = 3.6000 cub ins. 
Light iron indicates impurity. 

Ivory...:.average.. 

Ice, .917 to .922. “ 

India rubber. “ 

Lignum vitae, dry. “ 

Lard. « 

Lead, of commerce, 11.30 to 11.47 ; either rolled or cast. “ .. 

Limestones and Marbles, 2.4 to 2.86, 150 to 178.8. 

“ “ “ ordinarily about. . 

“ “ “ quarried in irregular fragments, 1 cub yard solid, 

makes about 1.9 cub yds perfectly loose; or about 
1% yds piled. In this last case, .571 of the pile 
is solid; and the remaining .429 part of it is 

voids.piled.. 

Lime, quick, of ordinary limestone and marbles 92 to 98 tbs per cub ft.... 
“ “ either in small irregular lumps; or ground, loose 50 to 58.... 

In either case 1 solid measure makes about 1.8 meas loose; and then 
.555 of the mass is solid, and .445 is voids. 

To measure correctly, none of the lumps should exceed about % or 
yy of the smallest dimension of the vessel used for measuring. 

Lime, quick, ground, loose, per struck bushel 62 to 70 lbs. 

“ “ “ well shaken, “ *• ....80 “ . 

“ “ thoroughly shaken, “ ....93% “ . 

Mahogany, Spanish, dry*.average.. 

“ Honduras, dry. *‘ 

Maple, dry*. “ 

Marbles, see Limestones. 

Masonry, of granite or limestones, well dressed throughout. 

“ “ “ well-scabbled mortar rubble. About i of the mass 

will be mortar. * . 

“ “ “ well-scabbled dry rubble. 

“ " “ roughly scabbled mortar rubble. About % to % part 

will be mortar. 

“ “ “ roughly scabbled dry rubble. 

At 155 fts per cub ft, a cub yard weighs 1.868 tons; and 14.45 cub ft, 

1 ton. 

Masonry of sandstone; about % part less than the foregoing. 

“ “ brickwork, pressed brick, fine joints.average.. 

“ “ “ medium quality. “ 

“ “ “ coarse; inferior soft bricks. “ 

At 125 lbs per cub ft, a cub yard weighs 1.507 tons; and 17.92 cub 
ft. 1 ton. 

Mercury, at 32° Fah. 

“ 60° “ . 

•• 212° “ . 

Mica, 2.75 to 3.1... 

Mortar, hardened, 1.4 to 1.9. 

Mud, dry, close.. 

“ wet, moderately pressed. 

“ wet, fluid.".. 

Naphtha.••. 

Nitrogen Gas is about yy part lighter than air. 

Oak, live, perfectly dry, .88 to 1.02 *. 

“ white, “ “ .66 to .88. 

“ red, black, &c*. 

Oils, whale; olive. 

“ of turpentine. 

Oolites, or Roestones, 1.9 to 2.5. 

Oxygen Gas, a little more than yL part heavier than air.. 

Petroleum. 

Peat, dry, unpressed. 

Pine, white, perfectly dry, .35 to .45*...... 

1000 ft board measure weighs .930 ton.* 

“ yellow, Northern, .48 to .62. 

1000 ft board measure weighs 1.276 tons.* 

“ “ Southern, .64 to .80. 

1000 ft board measure weighs 1.674 tons.* 


.average. 


Average 
Sp Gr. 


7.77 

7.6 

7.69 


1.82 

.92 

.93 

1.33 

.95 

11.38 

2.6 

2.7 


1.5 


.85 

.56 

.79 


13.62 

13.58 

13.38 

2.93 

1.65 


.848 


.95 

.77 


.92 

.87 

2.2 

.00136 

.878 


.40 

.55 

.72 


Average 
Wt of a 
Cub Ft. 
Lbs 


485. 

474. 

480. 

485. 


114. 

57.4 

58. 

83. 

59.3 

709.6 

164.4 

168. 


96. 

95. 

53. 


53. 

64. 

75. 

53. 

35. 

49. 

165. 

154. 

138. 

150. 

125. 


140. 

125. 

100 . 


849. 

846. 

836. 

183. 

103. 

80 to 110 
110 to 130 
104 to 120 
52.9 
.0744 

59.3 
48. 

32 to 45 

57.3 

54.3 
137. 

.0846 

54.8 
20 to 30 
25. 

34.3 
45. 


* Oreen timbers usually weigh from one-fifth to nearly one-lialf more than 

dry ; aud ordiuary building timbers wheu tolerably seasoned about one-sixth more than perfectly diy 

















































































384 


SPECIFIC GRAVITY, 


Table of specific gravities, and weights — (Continued.) 


The specific gravity of any substance is = its weight 
in grains per cubic centimetre. 


Pine, heart of long-leafed Southern yellow, unseas. (Footnote, p 363.). 

1000 ft board measure weighs 2.418 tons. 

Pitch. 


Plaster of Paris ; see Gypsum. 

Powder, slightly shakeu. 

Porphyry, 2.66 to 2.8. . 

Platinum. 21 to 22. 

" native, in graius. 16 to 19. 

Quartz, common, pure.2.64 to 2.67. 

“ *• finely pulverized, loose. 

“ “ “ “ well shaken. 

“ “ “ “ well packed. 

“ quarried, loose. One measure solid, makes full 1% broken and 

piled. 

Ruby and Sapphire, 3.8 to 4.0. 

Rosin. 

Salt, coarse, per struck bushel; Syracuse, N. York.56 lbs .. 

“ “ “ ** Turk’s Island; Cadiz; Lisbon. 76 to 80 .. 

“ “ “ “ “ St. Barts.84 to 90.. 

“ “ “ “ “ some well-dried West India.... 90 to 96 .. 

“ “ “ “ “ Liverpool.50 to 55.. 

“ Liverpool fine, for table use.60 to 62 .. 

Sand, of pure quartz, perfectly dried, and loose, usually 112 to 133 lbs per 

struck bushel. 

At the average of 98 lbs per cub ft, a struck bushel weighs 122)4 lbs ; 
and 18.29 bushels. 1 ton ; a cub yd ~ 1.181 tons; 22.86 cub ft. 1 ton. 
Slight shaking compacts it about 2 to 3 per ct; and ramming about 
12 per ct when dry. 

“ perfectly wet, voids full of water. 

“ “ “ at the mean of 124 lbs, a cub yard weighs 1.495 tons ; 

and 18.06 cubic feet= 1 ton. 

“ sharp angular sand of pure quartz with very large and very small 

grains dry may weigh. 

If any ordinary pure natural sand be sifted into 2 or 3 or more parcels 
of differently sized grains, a measure of any of these parcels will 
weigh considerably less than an equal measure of the origiual sand. 
Thus, a sand weighing 98 lbs per cub foot.Viay give others weighing 
not more than 70 to 80 lbs. At 98 lbs per cub ft. 1 bulk of pure quartz, 
has made 1.68 bulks of sand ; of which the solid occupies .6 ; and the 
voids .4. But if this same sand be compacted to 110 lbs per cub ft, 
then 1 measure of solid quartz makes 1 )4 measures of sand ; of which 
% are solid, and % voids. Sand is very retentive of moisture ; and 
when in large hulks, is rarely as dry as that above in this table. But 
with its natural moisture, and loose, it is lighter than when dry, its 
average weight then not exceeding about 85 to 90 lbs per cub ft; or 
106)4 to 112)4 lbs per struck bushel. See Voids in Sand, p 678. 

Sandstones, fit for builditig, dry, 2.1 to 2.73.131 to 171. 

“ quarried, and piled, 1 measure solid, makes about piled... 

Serpentines, good.2.5 to 2.65. 

Snow, fresh fallen. 

u moistened, and compacted by rain. 

Sycamore, perfectly dry. (See footnote, p 383.). 

1000 ft board measure weighs 1.376 tons. 

Shales, red or black.2.4 to 2.8. average.. 

“ quarried, in piles. “ 

Slate.2.7 to 2.9. “ .. 

Silver. “ 

Soapstone, or Steatite.2.65 to 2.8. " 

Steel, 7.7 to 7.9. The heaviest contains least carbon. “ 

Steel is not heavier than the iron from which it is made; unless the 
iron had impurities which were expelled during its conversion into 
steel. 

Sulphur., average.. 

Spruce, perfectly dry. Footnote, p 383.. “ 

1000 ft board measure weighs .930 ton. 

Spelter, or Zinc.6.8 to 7.2. “ 

Sapphire; and Ruby, 3.8 to 4. “ 

Tallow. “ 

Tar. “ 

Trap, compact, 2.8 to 3.2. “ 

“ quarried; in piles. “ 

Topaz. 3.45 to 3.65. •» 


Average 
Sp Gr. 


1.04 

1.16 

1 . 

2.73 

21.5 

17.5 
2,65 


3.9 

1.1 


2.41 

2.6 " 

.59 

2.6 


.4 

7.00 

3.9 

.94 

1 . 

3. 


3.55 


Average 
Wt of a 
Cub Ft. 
Lbs. 


65. 

71.7 

62.3 

170. 

1342. 

165. 

90. 

105. 

112 . 

94. 

68.6 

45. 

62. 

70. 

74. 

42. 

49. 

90 to 106 


118 to 129 


117. 


2.8 

10.5 

2.73 

7.85 


151. 

86 . 

162. 

5 to 12 
15 to 50 
37. 

162. 

92. 

175. 

655. 

170. 

490. 


125. 

25. 

437.5 

58.6 

62.4 

187. 

107. 


* The sp gr of pure quartz sand, found as directed near foot of p 380, is of course the same as that 
of pure quartz, or 2.65. But a cub ft of dry sand weighs, as above, from 90 to 106 lbs, or only from 
1 44 to 1.7 times as much as an equal quantity of water. Most authorities give about 1.5 as the sp 
gr. See first paragraph, p 381, 




































































WEIGHTS AND MEASURES. 


385 


Table of specific gravities, and weights — (^Coutiuued.) 


The specific gravity of any substance is = its weight 
in grams per cubic centimetre. 


Average 
Average Wt of a 
Sp Gr. Cub Ft. 

Lbs. 


Tin, east, 7.2 to 7.5.average.. 

Turf, or Peat, dry, uupressed. 

Water, pure raiu, or distilled, at 22° Fab. Barorn 30 ins. 

“ “ “ •“ “ 62 ° “ “ *• “. 

<< *« «a 212° 44 41 li 44 

At 60°, a cub inch weighs .03607 lb; or .57712 oz avoir. And a lb con¬ 
tains 27.724 cub ins; equal to a cube of 3.0263 inches on each edge. 

Water, sea, 1.026 to 1.030 .average .. 

Although the wt of fresh water is generally assumed as sixty two 
and one-third lbs per cub ft, yet 62% would be nearer the truth, at 
ordinary temperatures of about 70° ; or a lb = 27.759 cub ins; and a 
cub in ~ .5761 oz avoir ; or .4323 oz troy ; or 252.175 grains. The grain 
is the same in troy, avoir, and apoth. 

Wax, bees.average.. 

Wines, .993 to 1.04. .. 

Walnut, black, perfectly dry. (See footnote, p 383.) . “ 

1000 ft board measure weighs 1.414 tons. 

Zinc, or Spelter, 6.8 to 7.2 .. “ 

Zircon, 4.0 to 4.9. “ 


7.35 


1.028 


.97 

60.5 

.998 

62.3 

.61 

38. 

7.00 

437.5 

4.45 


3 . 


459. 

20 to 30 
62.417 
62.355 
59.7 


64.08 


WEIGHTS AND MEASURES. 


United States and British measures of length and weight, 

of the same denomination, may, for all ordinary purposes, be considered as equal; 
but the liquid and dry measures of the same denomination differ widely 
in the two countries. The standard measure of length of both coun¬ 
tries is theoretically that of a pendulum vibrating seconds at the level of the 
sea, in the latitude of London, in a vacuum, with Fahrenheit’s thermometer at 
62°. The length of such a pendulum is supposed to be divided into 39.131)3 
equal parts, called inches; and 36 of these inches were adopted as the standard 
yard of both countries. But the Parliamentary standard having been destroyed 
bv fire, in 1834, it was found to be impossible to restore it by measurement of a 
pendulum; and the present British standard yard is, in consequence, shorter 
than that of the U. S. by the latest comparison, about 1 part in 40,000, or .03 inch 
in 100 feet; or 1.584 inches in a mile. But at a temperature of 62°.25Fah for 
the British standard and 59°.62 for the U.S. one, the two are of the same length, 
and ou this basis the U. S. government declares the measures of the two coun¬ 
tries to be the same; as in our tables. 


Troy Weight. 
U. 8. and British. 


24 grains. 1 pennyweight, dwt. 

20 pennyweights. 1 ounce — 480 grains. 

12 ounces. 1 pound = 240 dwts = 5760 grains. 


Troy weight Is used for gold and silver. 

A carat of the jewellers, for precious stones, is in the U. S. =3.2 grs; in London. 3.17 grs ; in 
Paris, 3.18 grs, divided into 4 jewellers’ grs. In troy, apothecaries’, and avoirdupois, the 
grain Is the same. 


Approximate Values of Foreign Coins, in U. S. Money. 

The references ( 1 , 2 , 3 and 4 ) are to foot-notes on next page. 

From Circular of U. S. Treasury Department, Bureau of the Mint, Jan. 1,1887; 
from “Question Monetaire,” by H.Costes, Paris, 1884; and from our 10th edition. 

Argentine Repub.—Peso = 100 Centavos, 96.5 cts. 23 Argentino = 5 Pesos, $4.82. 
Austria.—Florin = 100 Kreutzer,47.7 cts., 2 35.9 cts. 3 Ducat, $2.29. Maria Theresa 
Thaler, or Levantin, 1780, $1.00. 2 Iiix Thaler, 97 cts.* Souverain, $3.57. 4 
Belgium. 1 —Franc = 100 centimes, 17.9 cts., 2 19.3 cts. 3 

Bolivia.—Boliviano — 100 Centavos, 96.5 cts., 2 72.7 cts. 3 Once, $14.95. Dollar, 
9fj cts.4 

Brazil.—Mil reis = 1000 Reis, 50.2 cts., 2 54.6 cts. 3 
Canada.—English and U. S. coins. Also Pound, $4. 4 

Central America.*—Doubloon, $14.50 to $15.65. Reale, average cts. See 
Hondo ras. 

Ceylon.—Rupee, same as India. 




























386 


FOREIGN COINS. 


(Foreign Coins Continued. Small figures 0, 2 , 3 , 4 ) refer to foot notes.) 

Chili.—Peso = 10 Dineros or Deehnos = 100 Centavos, 96.5 cts., 2 91.2 cts. 3 Con¬ 
dor = 2 Doubloons — 5 Escudos = 10 Pesos. Dollar, 93 cts. 4 
Cuba.—Peso, 93.2 cts. 3 Doubloon, $5.02. 

Denmark.—Crown - 100 Ore, 25.7 cts., 2 26.8 cts. 3 Ducat. $1.81. 4 Skilling, % ct 4 
Ecuador.—Sucre, 72.7 cts. 3 Doubloon, $3.86. Condor, $9.65. Dollar, 93 cts. 4 
Reale, 9 cts. 4 

Egypt.—Pound = 100 Piastres — 4000 Paras, $4 94,3. 3 
Finland—Markka = 100 Penni, 19.1 cts. 2 10 Markkaa, $1.93. 

France. 1 —Franc=100 Centimes, 17.9 cts., 2 19.3 cts. 3 Napoleon, $3.84. 4 Livre, 
18.5 cts. 4 Sous, 1 ct. 4 

Germany.—Mark - 100 Pfennigs, 21.4 cts., 2 23.8 cts. 3 Augustus (Saxony), $3.98. 4 
Carolin (Bavaria), $4.93. 4 Crown (Baden, Bavaria, N. Germany , $1.06. 4 
Ducat (Hamburg, Hanover), $2.28. 4 Florin (Prussia, Hanover), 55 cts. 4 
Groschen, 2.4 cts. 4 Kreutzer (Prussia), .7 ct Maximilian (Bavaria), $3.30. 4 
Rix Thaler (Hamburg, Hanover), $1.10 4 (Baden, Brunswick), $1.00 4 (Prussia, 
N. Germany, Bremen, Saxony, Hanover), 69 cts. 4 
Great Britain.—Pound Sterling or Sovereign (£) = 20 Shillings = 240 Pence, 
$4.86.65. 3 Guinea = 21 Shillings Crown = 5 Shillings. Shilling («), 22.4 
cts., 2 24.3 cts. pound sterling). Penny (d), 2 cts. 

Greece. 1 —Drachma = 100 Lepta, 17 cts., 2 19.3 cts. 3 
Hayii.—Gourde of 100 cents, 96.5 cts. 23 

Honduras.—Dollar or Piastre of 100 cents, $1.01. See Central America. 

India.—Rupee = 16 Annas, 45.9 cts., 2 34.6 cts. 3 Mohur=15 Rupees, $7.10. Star 
Pagoda (Madras), $1.81. 4 

Italy, etc. 1 —Lira =100 Centesimi, 17.9 cts., 2 19.3 cts. 3 Carlin (Sardinia), $8.21. 4 
Crown (Sicily), 96 cts 4 Livre (Sardinia), 18.5 cts. 4 (Tuscany, Venice), 16 
cts. 4 Ounce (Sicily), $2.50. 4 Paolo (Rome), 10 cts. 4 PLtola (Rome). $3.37. 4 
Scudo 4 (Pbdmont), $1.36 (Genoa), $128 (Rome), $1.00 (Naples, Sicily), 95 
cts. (Sardinia), 92 cts. Teston (Rome). 30 cts. 4 Zecchino (Rome), $2.27. 4 
Japan.—Yen = 100 Sen (gold), 99.7 cts. 3 (silver), $1.04 2 , 78.4 cts. 3 
Liberia.—Dollar-, $1.00. 3 4 

Mexico.—Dollar, Peso, or Piastre — 100 Centavos (gold), 98.3 cts. (silver), $1.05, 2 
79 cts. 3 Once or Doubloon = 16 Pesos, $15.74. 

Netherlands.—Florin of 100 cents, 40.5 cts., 2 40.2 cts. 3 Ducatoon, $1.32. 4 Guilder, 
40 cts. 4 Rix Dollar, $1.05. 4 Stiver, 2 cts. 4 
New Granada.—Doubloon, $15.34. 4 

Norway.—Crown = 100 Ore = 30 Skillings, 25.7 cts., 2 26.8 cts. 3 
Paraguay.—Piastre = 8 Reals, 90 cts. 

Persia.—Thoman = 5 Sachib-Kerans - 10 Banabats = 25 Abassis = 100 Scab is, 
$2.29. 

Peru.—Sol‘= 10 Dineros = 100 Centavos, 96.5 cts., 2 72.7 cts. 3 Dollar, 93 cts. 4 
Portugal.—Milreis = 10 Testoons = 1000 Reis, $1.08. 3 Crown = 10 Milreis. 
Moidore, $6.50. 4 

Russia.—Rouble = 2 Poltinniks = 4 Tchetveitaks = 5 Abassis = 10 Griviniks = 
20 Pietaks=100 Kopecks, 77 cts , 2 58.2 cts. 3 Imperial = 10 Roubles, $7.72. 
Ducat = 3 Roubles, $2.39. 

Sandwich Islands.—Dollar, $1.00. 4 
Sicily.—See Italy. 

Spain.—Peseta or Pistareen = 100 Centimes, 17.9 cts., 2 19.3 cts 3 Doubloon (new) 
= 10 Escudos - 100 Reals. $5.02. Dnro = 2 Escudos, 4 $1.00. 2 Doubloon (old), 
$15.65. 4 Pistole = 2 Crowns, $3.90. 4 Piastie, $1.04. 4 Reale Plate, 10 cts. 4 
Reale vellon, 5 cts. 4 

Sweden —Crown -100 Ore, 25.7 cts., 2 26.8 cts. 3 Ducat, $2.20. 4 Rix Dollar, $1.05. 4 

Switzerland. 1 —Franc = 100 Centimes, 17.9 cts., 2 19.3 cts. 3 

Tripoli.—Mahbub = 20 Piastres, 65.6 cts. 3 

Tunis—Piastre = 16 Karobs, 12 cts. 2 10 Piastres, $1.16.6. 

Turkey.—Piastre = 40 Paras, 4.4 cts. 3 Zecchin, $1.40. 4 

United States of Colombia.—Peso = 10 Dineros or I)ecimos= 100 Centavos, 96.5 
cts., 2 72.7 cts. 3 Condor = 10 Pesos, $9.65. Dollar, 93 5 cts. 4 
Uruguay.—Peso = 100 Centavos or Centesinaos (gold), $1.03 (silver), 96.5 cts. 2 
Venezuela —Bolivar = 2 Decimos, 17.9 cts., 2 19.3 cts. 3 Venezolano = 5 Bolivars. 


1 France, Belgium, Italy, Switzerland, and Greece form the Latin Union. 
Their coins are alike in diameter, weight, and fineness. 

2 = 19.3 times the value ot a single coin in francs as given by Costes. 

3 Par of exchange, or equivalent value in terms of U.S. golddollar—Treasury 
Circular. 

4 From our 10th edition. 









WEIGHTS AND MEASURES, 


387 


The U. t 8. Bold dollar weighs 25.8 grs; and contalus 23.22 grs of pure gold. 

.. „ H „ ;; 258 grs; “ “ 232.20 grs “ “ 

£0 516 grs; 14 14 464.40 grs 44 “ 

Perfectly pure gold is worth $1 per 23.22 grs = $20.67183 per troy oz = $18.84151 per avoir or 
(b - C0l “j "’orth $18.6046a per troy oz = $16.95736 per avoir oz. It consists of 9 

and of 'the 1 ° f pure g ? ld > 1 part “hoy. Its value is that of the pure gold only; the cost of the alloy 

and U worth «3fi^ e,ng A h e bJ h G0V ' k A , CUb f00t of P t,rc S° ld wei ^ hs about 1201 “'oir 

and i> worth $362963. A cub inch weighs about 11.148 avoir oz; and is worth $210.04. 

. called fine, or 24 carat gold ; and when alloved, the allov is supposed to be divided 

to be iV^orV^carar^" 6 aS 10 ’ * 5 ’ ° r 2 °’ &C ’ ° f these part,s are pure 6 0ld > tlle al| oy is said 
Pure silver fluctuates in value; thus during 1878. 1879, it ranged between $1.05 and $1.18 per 
Tt. ir c 'n and $1,076 per avoir oz. A cub inch weighs about 5.528 troy, or 6.066 avoir ounces, 
x ne u. S. silver dollar weighs 412.5 grs troy; but its subdivisions weigh at the rate of about 8 
P e ct less. All consist of 9 parts silver to 1 part alloy. 

The average fineness of California native gold, by some thousands of assays at the U. 
b. mint in rhiiada, is 88.5 parts gold, 11.6 silver. Some from Georgia, 99 per ct gold. 

Apothecaries* Weight. 


U. 8. and British. 


20 grains. 1 scruple. 

3 scruples. 1 dram = 60 grs. 

8 drams... 1 ounce — 24 scruples — 480 grs. 

12 ounces.. 1 pound — 96 drams — 288 scruples = 5760 grains. 


In troy and apoth weights, the grain, ounce, and pound are the same. 


Avoirdupois, or Commercial Weight. 

U. 8. and British. 


27.34375 grains.;. 1 dram. 

16 drams. 1 ounce = 437 Hj grains. 

16 ounces. 1 pound = 256 drams = 7000 grains. 

28 pounds. 1 quarter = 448 ounces. 

4 quarters. 1 hundredweight = 112 lbs. 

20 hundredweights. 1 ton = 80 quarters = 2240 lbs. 

A stone = 14 pounds. A quintal = 100 pounds avoir. 

The standard of the avoir pound, which is the one in common commercial use, is the 
weight of 27.7015 cub ins of pure distilled water, at its maximum density at about 39°.2 Fahr, in 
latitude of London, at the level of the sea; barom at 30 ins. But this Involves an error of 
about 1 part in 1362, for the 1 ft of water = 27.68122 cub ins. 

A troy lb - .82286 avoir 11). An avoir ft = 1.21528 troy ft, or apoth. 

A troy oz = 1.09714 avoir oz. An avoir oz = .911458 troy oz. or apoth. 

Long Measure. 

C. 8. and British. 


By law, the U. 8. standards of length, as well as of weight, are made the same as the British 


12 inches. 1 foot = .3047973 metre. 

3 feet. 1 yard = 36 ins = .9143919 metre. 

yards. 1 rod, pole, or perch = 16J4 feet = 198 ins. 

40 rods.. 1 furlong = 220 yards = 660 feet. 

8 furlongs. 1 statute, or land mile = 320 rods = 1760 yds = 5280 ft = 63360 ins. 

3 miles...1 league = 24 furlongs = 960 rods - 5280 yds = 15840 ft. 


A point = j j inch. A line =6 points =r inch. A palm =3 ins. A hand = 4 ins. A 
span =9 ins. A fathom = 6 feet. A cable’s length = 120 fathoms = 720 feet. A Gunter’s 
surveying chain is 66 feet, or 4 rods long. It is divided into 100 links of 7.92 ins long. 80 Gun¬ 
ter’s chains = 1 mile. 

A nautical mile, geographical mile, sea mile, or knot, is 

variously defined as being = the length of 


1 min of longitude at the equator 
1 “ latitude “ “ 

1 “ “ “ pole 

1 “ “ at lat. 45° 

1 “a great circle of a true} 
sphere whose surface area is > 
that of the earth J 


metres feet statute miles 
= 1856.345 6087.15 1.15287 

= 1842.787 6045.95 1.14507 

= 1861.655 6107.85 1.15679 

= 1852.181 6076.76 1.15090 

( value adopted by U. S. Coast 
=< ami Geodetic Survey 
I 1853.248 6080.27 1.15157 

= 1853.169 6080.00 1.15152 


equal to that of the eart 
British Admiralty knot 

The above lengths of minutes, in metres and feet, are those published by the U. S. 
Coast and Geodetic Survey in Appendix No 12, Report for 1881, and are calculated 
from Clarke’s spheroid, which is now the standard of that Survey. 

at lat 20° = 68.78; at 40° = 

69.1 


At the equator l°of lat = 68.70 land miles; 

69.00; at 60° = 69.23 , at 80° = 69.39; at 90° - 69.41. 


26 




















388 


.WEIGHTS AND MEASURES. 


Lengths of a Degree of Longitude in different Latitudes, 
anil at tile level of the Sea. These lengths are in common laud or statute miles, 
of 5280 ft. Since the figure of the earth has never been precisely ascertained, these are but close ap¬ 
proximations. Intermediate ones maybe found correctly by simple proportion. 1° of longitude 
corresponds to 4 mins of civil or clock time; 1 min of longitude to 4 secs of lime. 


Reg of 
Lat. 

Miles. 

Deg of 
Lat. 

Miles. 

Deg of 
Lat. 

Miles. 

Deg of 
Lat. 

Miles. 

Deg of 
Lat. 

Miles. 

Deg of 
Lat. 

Miles. 

0 

69.16 

14 

67.12 

28 

61.11 

42 

51.47 

56 

38.76 

70 

23.72 

2 

69.12 

16 

66.50 

30 

59.94 

44 

49.83 

58 

36.74 

72 

21.43 

4 

6^.99 

18 

65.80 

32 

58.70 

46 

48.12 

60 

34.67 

74 

19.12 

6 

68.78 

20 

65.02 

34 

57.39 

48 

46.36 

62 

32.55 

76 

16.78 

8 

68.49 

22 

64.15 

36 

56.01 

50 

44.54 

64 

30.40 

78 

14.42 

10 

68.12 

24 

64.21 

38 

54.56 

52 

42.67 

66 

28.21 

80 

12.05 

12 

67.66 

26 

62.20 

40 

53.05 

54 

40.74 

68 

25.98 

82 

9.66 

See p 34. 

Inches reduced to Decimals of a 

Foot. 

No errors. 

Ins. 

Foot. 

Ins. 

Foot. 

Ins. 

Foot. 

Ins. 

Foot. 

Ins. 

Foot. 

Ins. 

Foot. 

0 

.0000 

2 

.1667 

4 

.3333 

6 

.5000 

H 

.6667 

10 

.8333 

1-32 

.0026 


.1643 


.3359 


.5026 


.6693 

.8359 

1-16 

.0052 


.1719 


.3385 


.5052 


.6719 


.8385 

3 32 

.0078 


.1745 


.3111 


.5078 


.6745 


.8411 

A 

.0104 

A 

.1771 

A 

.3438 

A 

.5104 

A 

.6771 

A 

.8438 

5 32 

.0130 


.1797 


.3464 


.5130 


.6797 

.8464 

3 16 

.0156 


.1823 


.3490 


.5156 


.6823 


.8490 

7-32 

.0182 


.1849 


.3516 


.5182 


.6849 


.8516 

A 

.0208 

A 

.1875 

A 

.3542 

A 

.5208 

A 

.6875 

A 

.8542 

9-32 

.0234 


.1901 


.3568 

.5234 

.6901 

.8568 

5-16 

.0260 


.1927 


.3594 


.5260 


.6927 


.8594 

11 32 

.0286 


.1953 


.3620 


.5286 


.6953 


.8620 

% 

.0313 

•y» 

.1979 

y» 

.3616 

y a 

.5313 

y» 

.6979 

A 

.8646 

13-32 

.0339 


.2005 


.3672 


.5339 

.7005 

.8672 

7-16 

.0365 


.2031 


.3698 


.5365 


.7031 


.8698 

15-32 

.0191 


.2057 


.3724 


.5391 


.7057 


.8724 


.0117 

a 

.2083 

A 

.3750 

A 

.5417 

A 

.7083 

A 

.8750 

17-32 

.0143 


.2109 


.3776 

.5443 

.7109 


9 16 

.0469 


.2135 


.3802 


.5469 


.7135 


.8802 

19-32 

.0495 


.2161 


.3828 


.5495 


.7161 


.8828 

A 

.0521 

% 

.2188 

% 

.3854 

% 

.5521 

A 

.7188 

A 

.8854 

21 32 

.0547 


.2214 


.3880 

.5547 

.7214 


.8880 

11 16 

.0573 


.2240 


.3906 


.5573 


.7240 


.8906 

23-32 

.0593 


.2266 


.3932 


.5599 


.7266 


.8932 

A 

.0625 

a 

.22.42 

A 

.3958 

A 

.5025 

A 

.7292 

A 

.8958 

25-32 

.0651 


.2318 


.3984 

.5051 

.7318 

.8984 

13 16 

.0677 


.2344 


.4010 


.5077 


.7344 


.9010 

27-32 

.0703 


.2370 


.4036 


.5703 


.7370 


.9036 

A 

.0729 

a 

.2496 

% 

.4063 

a 

.5729 

A 

.7396 

A 

.9063 

29 32 

.07.55 


.2422 


.4089 

•0 1 Di) 

.7422 

.90^9 

15-16 

.0781 


.2448 


.4115 


.5781 


.7448 


.9115 

31-32 

.0807 


.2474 


.4141 


.5807 


.7474 


.9141 

1 

.0833 

3 

.2500 

5 

.4167 

7 

.5833 

» 

.7500 

11 

.9167 


.0859 


.2526 


.4193 


.5859 

.7526 

.9193 


.0885 


.2552 


.4219 


.5885 




.92’9 


.0911 


.2578 


.4245 


.5911 


.7578 


.9245 

A 

.0938 

A 

.2804 

A 

.4271 

A 

.5938 

A 

.7604 

A 

.9271 


.0964 


.2630 


.4297 

.5904 

.7630 

.9297 


.0090 


.2656 


.4323 


.5890 


.7656 


.9323 


.1016 


.2682 


.4349 


.6010 


.7682 


.9349 

a 

.1042 

a 

.2708 

A 

.4375 

A 

.6042 

A 

.7708 

A 

.9375 


.1068 


.2734 


.4401 

.6068 

.7734 

.9401 


.1094 


.2760 


.4427 


.6094 


.7760 


.9427 

h 

.1120 

% 

.2786 


.4453 


.6120 


.7786 


.9453 

.1146 

.2813 

% 

.4479 

% 

.6146 

% 

.7813 

% 

.9479 


• 1172 


.2839 


.4505 

.6172 

.7839 

.9505 


. 1 19m 


.2865 


.4531 


.6198 


.7865 


.9531 

a 

.1224 

a 

.2891 


.4557 


.6224 


.7891 


.9557 

.1250 

.2917 

A 

.4583 

A 

.6250 

A 

.7917 

A 

.9583 


.1276 


.2943 


.4609 

.6276 

.7943 

.9609 


•1302 


.2969 


.4635 


.6302 


.7969 


.9635 

% 

.1328 

.1354 

% 

.2995 

.3021 

A 

.4661 

.4688 

A 

.6328 

.6354 

A 

.7995 

.8021 

A 

.9661 

.9688 


.1380 


.3047 


.4714 

.6380 

.8047 

.9714 


.1100 


.3073 


.4740 


.6406 


.8073 


.9740 

A 

.1432 

a 

.3099 


.4766 


.6432 


.8099 


.9766 

.1458 

.3125 

A 

.4792 

A 

.6458 

A 

.8125 

A 

.9792 


.1484 


.3151 


.4818 

.6484 

.8151 

.9818 


.1510 


.3177 


.4844 


.6510 


.8177 


.9844 


.1536 

a 

.3203 


.4870 


.6536 


.8203 


.9870 

A 

.1564 

.3229 

A 

.4896 

y» 

.6563 

A 

.8229 

A 

.9896 


.1589 


.3255 


.4922 

.6589 

.8255 

.9922 


.1615 


.3281 


.4948 


.6*115 


.8281 


.9948 


.1641 


.3307 


.4974 


.6641 


.8307 


.9974 



































































WEIGHTS AND MEASURES 


389 


Square, or Land Measure. 

U. 8. and British. 


144 square inches. 1 sq foot. 100 sq ft = 1 square. 

9 sq feet. 1 sq yard =r 1296 sq ins. 

30$*.sq yards. 1 sq rod — 272$* sq feet. 

40 sq rods. 1 rood — 1210 sq yds — 10890 sq feet. 

4 roods. 1 acre ~ 160 rods — 4840 sq yds — 43560 sq feet. 


A section of land is 1 mile sq, or 27878400 sq ft; or 3097600 sq yds ; or 640 acres. An acre 
coutaius 10 sq Guuter's chains. A sq acre is 208.710 feet; a sq half acre, 147.581 ft; and a sq 
quarter acre, 104.355 ft on each side. A circular acre is 235.504 feet: a circular half acre = 
166.527 ft; and a circular quarter acre 2 = 117.752 ft diatu. A circular inch is a circle of 1 inch 
diatn; a sq ft — 183.346 cir ins. Also 1 sq inch — 1.27324 cir ins ; and 1 cir inch — .7854 of a sq 
inch. 

Cubic, or Solid Measure. 

U. 8. and British. 

1728 cubic inches. 1 cubic, or solid foot. 

27 fcubic feet. 1 cubiq, or solid yard. 


A cord of wood = 128 cub ft; being 4 ft X 4 ft X 8 ft. A perch of masonry actually con¬ 
tains 24$* cub ft; being 16$* ft X 1$* ft X 1 ft. It is generally taken at 25 cub ft; but by some at 22. 
&c; and there is every probability that a payer will be cheated unless the number of cubic ft be dis¬ 
tinctly agreed upon in his contract. It is gradually falling into disuse among engineers; aud the cub 
yd is very properly taking its place. To reduce cub yds to perches of 25 cub ft, mult by 1.080; 
and to reduce perches to cub yds, mult by .926. The Brit rod of brickwork, of house-builders, is 
16$> feet square, by 14 inches (1$* English bricks; thick ~ 272$* sq ft of 14 inch wall. It is conven¬ 
tionally taken at 272 sq ft; which gives 317$* cub ft. In Brit engineering works the rod is 306 cub 
ft, or 11 $* cub yds. The Montreal, (Canada,) tolse = 261$* cub ft; or 9.6852 cub yds, or 10.46 
perches of 25 cub ft. The Canadian chaldron — 58.64 cub ft. A ton (2240 lbs) of Pennsylvania 
anthracite, when broken for domestic use, occupies from 41 to 43 cub ft of space , the mean of which 
is equal to 1.556 cub yds: or a cube of 3.476 ft on each edge. Bituminous coal 44 to 48 cub ft; mean 
equal to 1.704 cub yd; or a cube of 3.583 ft on each edge. Coke 80 cub ft. 


A cubic loot is equal to 

1728 cub ins, or 3300.23 spherical ins. 

.037037 cub yard, or 1.90985 spherical ft. 

.002832 myriolitre, or decastere. 

.028316 kilolitre, or cubic metre, or stere. 

.283161 hectolitre, or decistere. 

2.83161 decalitres, or centisteres. 

28.3161 litres, or cub decimetres. 

283.161 decilitres. 

2831.61 centilitres. 

28316.1 millilitres, or cub centimetres. 

.803564 U.S. struck bushel of 2150.42 cub ins, or 
1.24445 cub ft. 

.779013 Brit bushel of 2218.191 cub ins, or 1.28368 
cub ft. 

3.21426 U. S. pecks. 

A cubic inch is equal to 

16.38663 millilitres; or 1.638663 centilitres; or .1638663 decilitre; or .01638663 litre; or to .0005787 
cub ft; or to .138528 U. S. gill; or 1.90985 spherical ins. 

A eubic yard is equal to 


3.11605 Brit pecks. 

7.48052 U. S. liquid galls of 231 cub ins. 
6.42851 U. S. dry galls. 

6.23210 Brit galls of 277.274 cub ins. 
29.92208 U. S. liquid quarts. 

25.71405 U. 8. dry quarts. 

24.92842 Brit quarts. 

59.84416 U. S. liquid pints. 

51.42809 U. S. dry- pints. 

49.85684 Brit pints. 

239.37662 U. S. gills. 

199.42737 Brit gills. 

.26667 flour barrel of 3 struck bushels. 
.23748 U. 8 . liquid barrel of 31$* galls. 


27 cub feet, or to 201.974 L T . S. galls. j 

46656 cub ins. 

.0764534 myriolitre. 

.764534 kilolitre, or cub metre. 

7.64534 hec'olifres. 

7.2 flour barrels of 3 struck bushels. 

A spliere 1 foot in 

.01939 cub yard. 

.5236 cub foot. 

904.781 cub inches. 

.42075 U. S. bushel. 

1.6830 U. S. pecks. 

13.4639 U. S. dry quarts. 

26.9278 U. S. dry pints. 

3.9168 U. S. liquid gallons. 

15.6672 U. S. liquid quarts. 


76.4534 decalitres. 

764.534 litres, or cub decimetres. 
7645.34 decilitres. 

21 69623 U. S. bushels (struck). 
21.03336 Brit bushels. 

diameter, contains 

31.3344 U. S. liquid pints. 
125.3376 U. S. liquid gills. 
3.2631 Brit imp gallons. 
13.6525 Brit imp quarts. 
26.1050 Brit imp pints. 
104.4201 Brit imp gills. 
14.8263 litres. 

1.48263 decalitres. 
.148263 hectolitres. 


A sphere 1 inch in diameter, contains 


.000303 cub foot. 
.5236 cub inch. 
.07253 U. S. gill. 


.06043 Brit gill. 
8.580 millilitre. 
.8580 centilitre. 
.08580 deoil itra. 














390 


•WEIGHTS AND MEASURES 


A cylinder 1 foot in diameter, 

.02909 cub yard. 

.7854 cub foot. 

1357.1712 cub inches. 

.63112 U. S. dry bushels. 

2.5245 U. S. dry pecks. 

20.1958 II. S. dry quarts. 

40.3916 U. S. dry piots. 

5.8752 U. S. liquid gallons. 

23.5008 U. S. liquid quarts. 

A cylinder 1 inch in diameter. 

.005454 cub foot. 

9.4248 cub inches. 

.2805 U. S. dry pint. 

.3264 liquid pint. 

1.3056 U. S. gill. 


and 1 foot high, contains 

47.0016 U. S. liquid pints. 

188.0064 U. 8. liquid gills. 

4.8947 Brit imp gallons. 

19.5788 Brit imp quarts. 

39.1575 Brit imp pints. 

156.6302 Bl it imp gills. 

222.395 decilitres. 

22.2395 litres. 

2.22395 decalitres. 

.222395 hectolitre. 

and 1 foot high, contains 

.2719 Brit imp pint. 

1.0877 Brit imp gill. 

15.4441 centilitres. 

1.54441 decilitres. 

.154441 litres. 


For others, see below; also p 157. 


Liquid Measure, u. 8. only. 

The basis of this measure in the U. S. is the old Brit wine gallon of 231 cub ins; or 8.33888 lbs 
avoir of pure water, at its max density of about 39°.2 Eahr; the barom at 30 ius. A cylinder 7 ius 
diam, aud 6 ins high, contains 230.904 cub ins, or almost precisely a gallon ; as does also a cube of 
6.1358 ins on an edge. Also a gallon ~ .13368 of a cub ft ; and a cub ft contains 7.48052 galls ; nearly 
7J4 galls. This busts however Involves an error of about 1 part in 1362, for the water actu¬ 
ally weighs 8.34o008 tbs. 

cub ius. 

4 gills.. 1 pint =: 28.875. 63 gallons.1 hogshead. 

2 pints.1 quart — 57.750 — 8 gills. 2 hogsheads. 1 pipe, or butt. 

4 quarts.1 gallon - 231. = 8 pints =32 gills. 2 pipes. 1 tun. 


In the U. S. and Great Brit. 1 barrel of wine or brandy = 31 ^ galls : in Pennsylvania, a half 
barrel, 16 galls; a double barrel, 64 galls; a puncheon, *4 galls; a tierce, 42 galls. A liquid 
measure barrel of 31 % galls contains 4.211 cub ft = a cube of 1.615 ft on au edge ; or 3.384 U. S. struck 
bushels. A gill = 7.21875 cub ins. The following cylinders coutain some of these measures 
very approximately. For others, see above; also p 157. 



Diam. 

Height. 


Diam. 

Height. 

cub ius. 

Ius. 

Ius. 


Ius. 

Ius. 

Gill (7.21875)... 

... i y. . 



. 7 . 

. 6 



. 3% 


. 7 . 

. 12 


.. 3t$ . 

. 3 ” 


. 14 . 

. 12 

t^uart. 

• • • . 


10 gallons. 




To reduce IT. 

tion, divide by 1.20032; 
by 1.2. 


S. liquid measures to Brit ones of the same denomina- 

or near enough for common use, by 1.2; or to reduce Brit to U. S. multiply 

l>ry Measure. 


U. 8. only. 

The basis of this is the old British Winchester struck bushel of 2150.42 cub 

ins ; or 77.627413 pounds avoir of pure water at its max density. Its dimensions by law are 18>$ ins 
inner diam; 19>$ ius outer diam ; and 8 ins deep ; and when heaped, the cone is not to be less than 6 
ius high; which makes a heaped bushel equal to 134 struck oues; or to 1.55556 cub It. 


Edge of a cube of 

_ . _ equal capacity. 

2 pints 1 quart, — 67.2006 cub ins = 1.16365 liquid qt. 4 066 ms 

4 quarts 1 gallon, = 8 pints, = 268.8025 cub ins, = 1.16365 liq gal. 6 454 * 

2 gallous 1 peck, - 16 piuts, =r 8 quarts, = 537.6050 cub ius. 8.131 “ 

4 pecks 1 struck bushel, = 64 pints, = 32 quarts, = 8 gals, = 2150.4200 cub ins. 12.908 “ 


A struck bushel = 1.24445 cub ft. A cub ft = .80356 of a struck bushel 
The dry dour barrel = 3.75 cub ft; = 3 struck bushels. The dry barrel is 

not, however, a legalized measure; and no great attention is given to its capacity; consequently 
barrels vary considerably. A barrel of Hour contains by law, 196 Its. In ordering bv tbe barrel the 
aniouut of its coutents should be specified iu pounds or galls. ’ 

w <Ir *T measures to Brit imp ones of the same name, div 

by 1.031516; and to reduce Brit ones to U. S. mult by 1.031516; or for common purposes use 1.032. 





























WEIGHTS AND MEASURES 


391 


British Imperial measure, both liquid and dry. 

This system is established throughout Great Britain, to the exclusion of the old ones. Its basis is 
the imperial gallon of 277.274 cub ins, or 10 lbs avoir of pure water at the temp of 62° Fahr, when 

the barom is at 30 ins. This basis involves an error of about 1 part in 

1836, for 10 lbs of the water = only 277.123 cub ius. 



Avoir lbs. 
of water. 

Cub. ins. 

Cub. ft. 

Kdge of a cube of 
equal capacity. 
Inches. 

4 gills 1 pint. 

2 pints 1 quart. 

2 quarts 1 pottle. 

2 pottles 1 gallon. 

2 gallons 1 peck. 

4 pecks 1 bushel. 

4 bushelsl coomb. 

2 coombs 1 quarter. 

1.25 

2.50 

5. 

10. 

20. "1 

80. 1 Dry 
320. ( meas. 
640. J 

34.6592 

69.3185 

138.637 

277.274 

554.548 

2218.192 

8872.768 

17745.536 

1.2837 

5.1347 

10.2694 

3.2605 

4 1079 
5.1756 
6.5208 
8.2157 
13.0417 


The imp gall = .16046 cub ft; and 1 cub ft=6.23210 galls. The imp gal = 1.20032, or very nearly 
TJ. S. liquid galls. 


The weight of water affords an easy way to find the cubic contents of a vessel. First weigh the ves¬ 
sel by itself; and then full of water. The diff will be the weight of the water ; and this divided by 
62.3 or by the number in the table opp the temp of water, will be the contents in cub ft. 


To obtain the size of commercial measures by means of the 

weight of water. 

At the common temperature of from 70° to 75° Fah, a cub foot of fresh water weighs very approxi¬ 
mately 6‘2}4 lbs avoir. A cubic half foot, (6 ins on each edge,) 7.78125 lbs. A cub quarter foot, (3 ins 
on each edge,) .97266 lb. A cub yard, 1680.75 lbs; or .75034 ton. A cub half yd, (18 ins on each edge,) 
210.094 lbs; or .0938 ton. A cub inch, .036024 lb ; or .576384ounce; or9.2222drams ; or252.170grains. 
An inch square, and one foot long, .432292 ff>. Also 1 lb = 27.75903 cub ins, or a cube of 3.028 ins on an 
edge. An ounce, 1.735 cub ins ; a ton, 35.984 cub ft, all near enough for common use. 

Original. 


Liquid measures. 

IT. S. Gill. 

U. S. Pint. 

U. S. Quart. 

U. S. Gallon 8 lbs 5^ oz. 

U. S. Wine Barrel, 31J6 Gall. 


Lbs Avoir, 
of Water. 
.26005* 
1.0402 
2.0804 
8.3216 
262.1310 


Tiquid and Wry. Lbs Avoir. 

J of Water. 


British Imp Gill.31214* 

“ “ Pint. 1.24858 

“ “ Quart. 2.49715 

“ “ Gallon. 9.9886 

“ “ Peck. 19.9772 

“ “ Bushel. 79.9088 


I>ry measures. 


* 4.9942; or very nearly 5 ounces. 


U. S. Pint. 1.2104 

U. S. Quart. 2.4208 

U. S. Gallon. 9.6834 

U. S. Peck. 19.3668 

U. S. Bushel, struck. 77.4670 


* Or 4 ounces; 2 drams; 15.6625 grs. 


French measures. 


Centilitre. .02198t 

Decilitre. .2198J 

Litre. 2.1981 

Decalitre, or Centistere. 21.9808 

Metre, or Stere. 2198.0786 


t Or 5.6271 drams; or 153.866 grs. 
J 3.5169 ounces. 


- «—♦- 

METEIC WEIGHTS AND MEASUEES. 


The French metre. 

The French metre was intended to be the one ten-millionth part of the dlst from either pole of the 
earth to the equator: but after it had been introduced into use, errors were discovered in the calcu¬ 
lations employed for ascertaining that dist; so that the French metre, like the Brit standard yard, 
is not what it was intended to be. 

The U. S. Govt adopts for its length 1.093623 yds = 3.280869 ft = 39.370432 

ins U. S. or British measure. But in ordinary business transactions 

39.37 ins are a legal metre. At 3 ft 3% ins, the length is but 1 part in 8616 too great. 





















































392 


WEIGHTS AND MEASURES. 


French Measures of Length. 


By U. 8. and British Standard. 




Ins. 

Ft. 

Yds. 

Miles. 

IVf i 11 1 Tin pf rfli* . . 


039370 

.003281 



('entimftt.rpj. 

39370428 

.032809 



. 

3 9370428 

.3280869 

.1093623 


Metre +__ __ _ 

39.370428 

3.280869 

1.093623 


Decametre..) 


393.70428 

32.80869 

10.93623 


Hectometre. 


Road 

328.0869 

109.3623 

.0621375 

Kilometre.. 

' 

measures. 

3280.869 

1093.623 

.6213750 

Myriametre. 



32808.69 

10936.23 

6.213750 


* Nearly the part of an inch. t Full % inch. 

1 Very nearly 3 ft, 3% ins, which is too long by only 1 part in 8616. 


French Square Measure. 
By U. 8. and British Standard. 


Sq Millimetre. 

Sq Centimetre. 

Sq Decimetre. 

Sq Metre, or Centiare. 

Sq Decametre, or Are. 

Decare (not used). 

Sq. Ins. 

Sq. Feet. 

Sq. Yds. 

Acres. 

.001550 

.155003 

15.5003 

1550.03 

155003 

.00001076 

.00107641 

.10764101 

10.764101 

1076.4101 

10764.101 

107641.01 

10764101 

.0000012 

.0001196 

.0119601 

1.19601 

119.6011 

1196.011 

11960.11 

1196011. 

.000247 

.024711 

.247110 

2.47110 

247.110 

24711.0 

Hectare. 


Sq Kilometre. 

Sq Myriametre. 

3861090 sq miles. 
38.61090 “ 


French Cubic, or Solid Measure. 
According to U. 8. Standard. 

Only those marked “ Brit" are British. 


Millilitre, or cub 
Centimetre.... 

Cub Ins. 

.0610254 

( Liquid. 
(Dry. 

Centilitre. 

.610254 

j Liquid. 


(Dry. 

Decilitre. 

6.10254 

j Liquid. 


(Dry. 

Litre, or cubic 
Decimetre. 

61.0254 

(Liquid. 
1 Dry. 

Decalitre, or 

Centistere. 

610.254 

Cub Ft. 

.353156 

(Liquid. 

1 “ 

*-Dry. 

Hectolitre, or 
Decistere. 1. 

3.53156 

( Liquid. 
(Dry. 

Kilolitre, or 

Cubic Metre, 
or Stere. 

35.3156 

j Liquid. 
(Dry. 



Myriolitre, or 
Decastere. 

353.156 

j Liquid. 

( Dry. 


.0084537 pill. 

.0070428 Brit gill. * 

.0018162 dry pint. 

.084537 gill. 

.070428 Brit gill. 

.018162 dry pint. 

.84537 gill = .21134 pint. 

.70428 Brit gill = .17607 Brit pint. 

.18162 dry pint. 

1.05671 quart = 2.1134 pints. 

.88036 Brit quart = 1.7607 Brit pints. 
.11351 peck = .9081 dry qt = 1.8162 dry pt. 

2.64179 U. S. liquid gal. 

2.20090 Brit gal. 

.283783 bush = 1.1351 peck = 9.081 dry qts. 

26.4179 U. S. liquid gal. 

22.0090 Brit gal. 

2.83783 bush. 


264.179 U. S. liquid gal. 4 

220.090 Brit gal. VCub yds, 1.3080. 

28.3783 bush. 


2641.79 U. S. liquid gal 
283.783 bush. 


| Cub yds, 


13.080. 






































































WEIGHTS AND MEASURES 


393 


I rench Hoights, reduced to common Commercial or Avoir 
Weight, of 1 poiiiul = ltt ounces, or 7000 grains. 


Milligramme.,. 

Centigramme.. 

Decigramme. 

Gramme. 

By law a 5-cent nickel = 5 grammes 

Decagramme. 

Hectogramme. 

Kilogramme... 

Myriogramme. 

Quintal*. 

Tonneau; Millier; or Tonne. 


Grains. 


.015432 
.15432 
1.5432 
15.432 
Pounds av. 

.022046 

.22046 

2.2046 

22.046 

220.46 

2204.6 


The gramme is the basis of French weights; and is the weight of a cub centimetre of distilled 
water at its max density, at sea level, in lat of Paris ; barom 29.922 ins. 


French Measures of the “Systeme ITsuel.” 

This system was in use from about 1812 to 1840, when it was forbidden bv law to use even its names. 
This was done in order to expedite the general use of the tables which we have before given. But as 
the Systeme Usuel appears in books published during the above interval, we add a table of some of its 
values. 

Measures of Fength. 


I.igne usuel, or line. 

Yards. 

Feet. 

Inches. 



.09113 

1.09362 

13.12344 

47.245 

78.74172 

Pouoe usuel, or inch, — 12 lignes. 


.09113 

1.09362 

3.93708 

6.56181 

Pied usuel, or foot, —12 pouces. 

Aune usuel, or ell. 

Toise usuel, ~6 pieds. 

.36454 

1.31236 

2.18727 


Weights, TJsuel. 

Cubic, or Solid, Usuel. 

Grain usuel.. 
Gros usuel... 
Once usuel. 
Marc usuel.. 
Livre usuel, 
or pound, 


.8375 grains. 
60.297 “ 

1.10258 avoir oz. 
.55129 avoir lb.. 

1.10258 avoir lb. 

Litron usuel, or 1 litre 

Boisseau usuel. 

= 1.7608 British pint. 

2.7512 British gals. 

. 


Before 1812, or before the “Systeme usuel,” the Old System, “ Systeme Ancien,” was in use. 


French Measures of the “Systeme Ancien.” 


Lineal. 

Square. 

Cubic. 

Point ancien, .0148 ins. 

Sq. ins. 
.00789 

Sq. ft. 

Sq. yds. 

C. ins. 
.0007 

C. ft. 

C. yds. 


1.1359 



1.2106 




1.1359 


1.2106 


Aune ancien, 46.8939 ins = 3.90782 ft= 1.30261 yds 


40.8908 

4.5434 


261.482 

9.6845 












There is, however, much confusion about these old measures. Different measures had me same 
name in different provinces. 


* The avoirdupois quintal is 100 avoirdupois pounds. 









































































394 


WEIGHTS AND MEASURES. 


Russian. 

Foot; same as U. S. or British foot. Sachine = 7 feet.. Verst = 500 
saehine = 3500 feet = 1166% yards = .6629 mile. Pood - 36.114 lbs avoirdupois. 


Spanish. 

The Castellano of Spain and New Granada, for weighing gold, is variously 
estimated, from 71.07 to 71.04 grains. At 71.055 grains, (the mean between the 
two,) an avoirdupois, or common commercial ounce contains 6.1572 Castellano; 
and a lb avoirdupois contains 98.515. Also a troy ounce = 6.7553 Castellano ; and 
a t roy Ib = 81.064 Castellano. Three U. S. gold dollars weigli about 1.1 Castellano. 

The Spanish mark, or mareo, for precious metals, in South America, 
may he taken in practice, as .5065 of a lb avoirdupois. In Spain, .5076 lb. In 
other parts of Europe, it has a great number of values; most of them, however, 
being between .5 and .54 of a pound avoirdupois. The .5065 of a lb = 3545% 
grains; and .5076 lb = 3553.2 grains. 1 marco = 50 Castellanos = 400 tomine = 
4800 Spanish gold- grains. 

The arroba has various values in different parts of Spain. That of Cas¬ 
tile, or Madrid, is 25.4025 lbs avoirdupois: the tonelada of Castile = 2032.2 
lbs avoirdupois; the quintal = 101.61 lbs avoirdupois ; the libra = 1.0161 
lbs avoirdupois; the cantara of wine, Ac, of Castile = 4.263 U. S. gallons; 
that of Havana = 4.1 gallons. 

The vara of Castile = 32.8748 inches, or almost precisely 32% inches; or 2 
feet 8% incites. The fanegada of land since 1801 = 1.5871 acres = 69134.08 
square feet. The fancga of corn. Ac = 1.59914 U. S. struck bushels. In 
California, the vara by law = 33.372 U. S. inches; and the legua = 5000 
varas; or 2.6335 U. S. miles. 



TIME, 


39 5 


Civil, or Common Clock Time. 

60 thirds, marked ,,r 1 second, marked ,r . 

60 seconds j minute 

60 minutes 1 hour, = 3600 sec. 

24 hours 1 civil day, = 1440 min. — 86400 sec. 

] da ^f 1 week, = 168 hours = 10080 min. 

4 weeks 1 civil month, zr 28 days = 672 hours. 

For Standard Railway Time, see p 396. 

13 civil months, (or 52 weeks,) 1 day, 5 hours, 48 min, 49^ sec; or 365 days, 5 hours, 48 min. 49^ 

sec, = 1 civil year. A solar day is the time between two successive solar noons, or transits of the 
sun over the meridian of a place. These intervals are not of eoual lengths all the year round. The 
average length of all the solar days is called the mean solar day; and is the same as the common 
civil day or 24 hours of clock time. Civil noon is at 12 o'cIojk : but solar, or apparent noon, may be 
about 14% min before; or 16% min after 12 of correct clock time. A sidereal day is the interval 
between two passages of the same star past the range of two fixed objects; and is the precise time 
reqd for one complete rev of the earth on its axis. The sidereal day never varies ; but is always equal 
to 23 hours, 56 min, 4 09 sec;* so that a star will on any night appear to set, or to pass the range of 
any two fixed objects, 3 min, 55.91 sec earlier by the clock, than it did on the night before,t so that 
the number of sidereal days in a civil year is l greater than that of the civil days. 

An astronomical day begins at noon, and its hours are counted from 0 to 24. In comparing it 
with the civil day, the last is supposed to begin at the midnight before the uoon at which the first began. 

Astronomers are now (1884-5) taking measures to make their “ day ” correspond 
with the civil day. 


TABLE showing' how much earlier a star passes a given 
range, on each succeeding night. — (Original.) 


Nights. 

Min. 

Sec. 


Nights. 

H. 

Min. 

Sec. 


Nights. 

H. 

Min. 

Sec. 

1 

3 

55.91 


11 


43 

15.01 


21 

1 

22 

34.11 

2 

7 

51.82 


12 


47 

10.92 


22 

1 

26 

30.02 

3 

11 

47.73 


13 


51 

C.83 


23 

1 

30 

25.93 

4 

15 

43.64 


14 


55 

2.74 


24 

i 

34 

21.84 

5 

19 

39.55 


15 


58 

58.65 


25 

i 

38 

17.75 

6 

23 

35.46 


16 

1 

2 

54.56 


26 

i 

42 

13.66 

7 

27 

31.37 


17 

1 

6 

50.47 


27 

i 

46 

9.57 

8 

31 

27.28 


18 

1 

10 

46.38 


28 

i 

50 

5.48 

9 

35 

23.19 


19 

1 

14 

42.29 


29 

i 

54 

1.39 

10 

39 

19.10 


20 

1 

18 

38.20 


30 

i 

57 

57.30 










31 

2 

1 

53.21 


* This gives a means of regulating a watch with much accuracy and 

by a very simple process. The writer, after having regulated his chronometer watch for a year by 
this method only, differed but a few seconds from the actual time as deduced from careful solar obser¬ 
vations. Even a person not accustomed to ranging objects very accurately, need scarcely err a min¬ 
ute in a period of any number of years. It having occurred to him that the motion of a star in a 
second or two might be visible to the naked eye. he stuck a pin horizontally into a window-jamb ; and 
placing his eye close to it, sighted along one side of it, at a large star setting behind the top of a roof 
about 100 feet distaut, and found that his conjecture was correct. Those stars which are farthest 
from the poles appear to move the fastest, and are therefore the best. Those less than of the second mag¬ 
nitude are uot satisfactory. If the first observations of a given star be made as late as midnight, that 
same star will answer for about three months, until at last it will begin to pass the range in daylight. 
Before this happens, the observer must transfer the time to another star which sets later; if near 
miduight, the better, as it will serve for a longer time. A window looking west is the best. The 
longer the range, the greater will be the apparent motion of the star; and, consequently, the obser¬ 
vations will be more correct. If such a range can be secured as will strike the heavens at an angle 
of at least 40° above the horizon, the error from refraction will not appreciably affect an observation ; 
at a much less augle it may do so to the extent of three or four seconds. A candle must be so placed 
as to render the pin and the watch visible at the same time. A little practice will render the process 
very easy, and supersede the necessity for more remarks on the subject. Of course, a memorandum 
must be made and preserved of the date, hour, minute, and (approximately) second, at which the 
first passage of the star took place. Subsequent passages will occur earlier, as shown in the forego¬ 
ing table. The watch must be previously known to be right, when taking the first observation, if we 
require afterward to keep the correct time. Any person who will take the trouble thus to observe, 
and note down throughout a year, about half a dozen stars following each other at tolerably equal 
intervals of time, will on almost any clear night afterward be able, after a short calculation, to ascer¬ 
tain the correct clock time. The writer observed the passages of two or three stars behind different 
ranges, on the same nights, in order to obtain a mean of several observations; his object being to 
ascertain how pocket chronometers of the best makers would keep time under the vicissitudes of tem¬ 
perature, railroad travelling, &c, &c, to which they are ordinarily exposed. He used two of the best 
for this purpose, and the result was that their changes of rate were at times as great as from three to 
eight seconds per day. For ordinary purposes, therefore, they are of but little, if any, more service 
than a good common watch, of one-fourth the cost. 

t More accurately 3 min, 55.90944 sec. 















STANDARD RAILWAY TIME. 


396 

/ 

STANDARD RAILWAY TIME, ADOPTED 1883. 

The following arrangement of standard time was recommended by the General 
and Southern Time Conventions of the railroads of the United States and Canada, 
held respectively in St. Louis, Mo., and New York city, April, 1883, and in Chicago, 
Ill., and New York city, in October, 1883, and went into effect on most of the rail¬ 
roads of the United States and Canada, November 18th, 1S83. Most of the principal 
cities of the United States have made their respective local times to correspond with 
it. This system was proposed by Mr. W. F. Allen, Secretary of the Time Conven¬ 
tions, and its adoption was largely due to his efforts. We are indebted to Mr. Allen 
for documents from which the following has been condensed. Five standards of time 
or five “ times,” have been adopted for the United States and Canada. These are, 
respectively, the mean times of the 60th, 75th. 90th, 105th, and 120th meridians west 
of Greenwich, England. As each of these meridians, in the above order, is 15° west 
of its predecessor, its time is one hour slower. Thus, when it is noon on the 90th 
meridian, it is 1 p. M. on the 75th. and 11 a. m. on the 105th. The following gives 
the name adopted for the standard time of each meridian, and the conventional 
color adopted, and uniformly adhered to, by Mr. Allen, for the purpose of designat- ! 
ing it and its time, &c, on the maps published under his auspices: 


Longitude west 

Name of 

Conventional 

from Greenwich. 

Standard Time. 

color. 

60° 

Intercolonial. 

Brown. 

75° 

Eastern. 

Red. 

90° 

Central. 

Blue. 

105° 

Mountain. 

Green. 

120° 

Pacific. 

Yellow. 


Theoretically, each meridian may be said to give the time for a strip of country 
15° wide, running north and south, and having the meridian for its center. Thus 
the meridian on which the change of time between two standard meridians is sup¬ 
posed to take place, lies half-way between them. But it would, of course, not be 
practicable for the railroads to use an imaginary line in passing from one time 
standard to another. The changes are made at prominent stations forming the ter¬ 
mini of two or more lines; or, as in the case of the long Pacific roads, at the ends 
of divisions. As far as practicable, points at which changes of time had previously 
been made, were selected as the changing points under the new system. Detroit, 
Mich., Pittsburgh, Pa., Wheeling and Parkersburg, W. Ya., and Augusta, Ga., al¬ 
though not situated upon the same meridian, are points of change between eastern 
and central standard times. A train arriving at Pittsburgh from the east at noon, 
and leaving for the west 10 minutes after its arrival, leaves (by the figures shown 
upon its time-table, and by the watches of its train hands) not at 10 minutes after 
12, but at 10 minutes after 11. 

The necessity for making the changes of time at principal points, instead of on a 
true meridian line, necessitates also some “overlapping” of the times, or of their 
colors on the map. Thus, most of the roads between Buffalo and Detroit, on the 
north side of Lake Erie, run by “eastern,” or “red,” time; while those on the south 
side of the Lake, between Buffalo and Toledo, immediately opposite to and directly 
south of them, run by “ central ” or “ blue ” time. 

If the changes of time were made at the meridians midway between the standard 
ones, it would uot be necessary for any town to change its time more than 30 min¬ 
utes. As it is, somewhat greater changes had to be made at a few points. Thus, 
standard time at Detroit is 32 minutes ahead, and at Savannah 36 minutes back, of 
mean local time. 

In most cases the necessary change was made upon the railways by simply setting 
clocks and watches ahead or back the necessary number of minutes, and without 
making any change in time-tables. 

Halifax, and a few adjacent cities, use the time of the 60tli meridian, that being 
the nearest one to them; but the railroads in the same district have adopted the 
75th meridian, or eastern, time; so that, for railroad purposes, intercolonial time 
has never come into force. 

In 1873 there were 71 time standards in use on the railroads of the United States 
and Canada. At the time of the adoption of the present system this number had 
been reduced, by consolidation of roads, &c, to 53. By its adoption, the number be¬ 
came 5, or, practically, 4, owing to the adoption of eastern time by the intercolonial 
roads, as already explained. 













DIALS. 


397 


DIALLING. 


To make a horizontal Sun-dial, 

Draw a line a b ; and at right angles to it, draw 06. From any convenient point, as c 
in a b, draw the perp c o. Make the angle c a o equal to the lat of the place; also 
the angle c o e equal to the same ; join o e. Make e n equal to o e; and from n as a 
center, with the rad e n, describe a quadrant e s; and div it into 6 equal parts. Draw e 
y, parallel to 6, 6; and 
from n, through the 5 
points on the quadrant, 
draw lines n t, n i , &c, 
terminating in ey. From 
a draw lines a 5, a 4, &c, 
passing through t, i, &c. 

From any convenient 
point, as c, describe an 
J arc r m h, as a kind of fin- 
; ish or border to half the 
j dial. All the lines may 
now be effaced, except 
the hour lines a 6, a 5, 
a 4, &c, to a 12, or ah; 
unless, as is generally 
the case, the dial is to 
be divided to quarters 
of an hour at least. In 
this case each of the 
divisions on the quad¬ 
rant e x, must be subdivided into 4 equal parts; and lines drawn from n, through 
the points of subdivision, terminating in e y. The quarter-hour lines must be drawn 
from a, as were the hour lines. Subdivisions of o min maybe made in the same 
way; but these, as well as single min, may usually be laid oft around the border, by 
eye. About 8 or 10 times the size of our Fig will be a convenient one for an ordi¬ 
nary dial. To draw the other half of the Fig, make a d equal to the intended thick¬ 
ness of the gnomon, or style, of the dial; and drawd 12, parallel, and equal to a 12; and 
draw the arc xg w, precisely similar to the arc rm h. Between x and w, on the arc x gw, 
space off divisions equal to those on the arc rmh; and number them for the hours, 
as in the Fig. The style F, of metal or stone, (wood is too liable to warp,) will be 
triangular; its thickness must throughout fte equal to a d or hw; its base must 
cover the space adhw; its point will be at ad; and its perp height h u , over hw, 
must be such that lines vd,ua, drawn from its top, down to a and d, will make the 
angles ua h, vdw. each equal to the lat of the place. Its thickness, if of metal, may 
conveniently be from % to inch; or if of stone, an inch or two, or more, according 
to the size of the dial. Usually, for neatness of appearance, the back huv w of the 
style is hollowed inward. The upper edges, ua, vd, which cast the shadows, must 
be sharp and straight. The dial must be fixed in place hor, or perfectly level; ah 
and dw must be placed truly north and south ; ad being south, and h w north. The 
dial gives only sun or solar time; but clock time can be found by means of the “fast 
or slow T of the sun,” as given by all almanacs. If by the almanac the sun is 5 min, 
&c, fast, the dial will be the same; and the clock or watch, to be correct, must be 5 
min slower than it; and vice versa. 

To make a Vertical Snn-Dial. 

Proceed as directed above,except that the angles cao and coe on thedrawing, 
and the angle, uah or vdw of the style, must he equal to the co-latitude (= dif¬ 
ference between the latitude and 90°) of the place, and the hours must be num¬ 
bered the opposite wav from those in the above figure ; i e. from h to y number 
12, 11, 10, 9, 8, 7; and from w to g number 12,1, 2, 3, 4, 5. The dial plate must be 
placed vertically, in the position shown in the figure, facing exactly south, and 
with ah and dw vertical. 























398 


WEIGHT OF CAST-IRON 


TABLE OF WEIGHT OF CAST IRO\.* 

The weight of a pattern of perfectly dry white pine, if mult 

by ‘20, will give approximately the wt of the casting. If well seasoned, but still not 
perfectly dry, mult by 19, or by 18. 

Assuming 450 lbs to a cub ft. a pound contains 3.8400 cubic inches ; a ton 5 cub ft; 
and a cubic inch weighs .2604 lbs. 


1 Thickness 
| or Diameter 

I in Inches. 

Thick¬ 
ness or 
Diam. 
in deci¬ 
mals of 
a foot. 

Wt. of a 
Square 
Foot. 
Lbs. 

Wt. of a 
Square 
bar. 1 ft. 
long. 
Lbs. 

Wt. of a 
Round 
bar, 1 ft. 
long. 
Lbs. 

Wt. of 
Balls. 
Lbs. 

t 

1 Thickness 

1 or Diameter 
j in Inches. 

Thick¬ 
ness or 
Diam. 
in deci¬ 
mals of 
a foot. 

Wt. of a 
Square 
Foot. 
Lbs. 

Wt. of a 
Square 
bar, 1 ft. 
long. 
Lbs. 

Wt. of a 
Round 
bar, 1 ft. 
long. 
Lbs. 

Wt. of 
Balls. 
Lbs. 
t 

1-32 

.0026 

1.173 

.003 

.002 


3k 

.2604 

117.3 

30.52 

23.97 

4.162 

1-16 

.0052 

2.344 

.012 

.010 


k 

.2708 

121.8 

33.01 

25.93 

4.681 

3-32 

.0078 

3.516 

.027 

.021 

.0001 

H 

.2813 

126.5 

35.60 

27.95 

5.243 

k 

.0104 

4.687 

.048 

.038 

.0003 

k 

.2917 

131.2 

38.28 

30.07 

5.846 

5 32 

.0130 

5.861 

.076 

.060 

.0005 

% 

.3021 

135.9 

41.07 

32.25 

6.498 

3-16 

.0156 

7.032 

.110 

.086 

.0009 

H 

.3125 

140.6 

43.95 

34.51 

7.193 

7-32 

.0182 

8.203 

.150 

.118 

.0014 

Vs 

.3229 

145.3 

46.93 

36.85 

7.934 

34 

.0208 

9.375 

.195 

.154 

.0021 

4. 

.3333 

150.0 

50.01 

39.27 

8.726 

y 32 

.0234 

10.54 

.247 

.194 

.0030 

k 

.3438 

154.7 

53.18 

41.77 

9.572 

5-16 

.0260 

11.73 

.305 

.240 

.0042 

k 

.3542 

159.3 

56.46 

44.33 

10.47 

11-32 

.0287 

12.89 

.370 

.290 

.0056 

X 

.3646 

164.0 

59.82 

46.99 

11.42 

% 

.0313 

14.06 

.440 

.346 

.0072 

k 

.3750 

168.7 

63.33 

49.71 

12.43 

13-32 

.0339 

15.24 

.516 

.400 

.0092 

% 

.3854 

173.4 

66.86 

52.52 

13.49 

7-16 

.0365 

16.41 

.598 

.470 

.0114 

y* 

.3958 

178.1 

70.52 

55.39 

14.62 

15-32 

.0391 

17.56 

.687 

.540 

.0140 

k 

.4063 

182.8 

74.28 

58.34 

15.81 

k 

.0417 

18.75 

.781 

.610 

.0170 

5. 

.4167 

187.5 

78.12 

61.37 

17.05 

9-16 

.0469 

21.10 

.989 

.777 

.0243 

k 

.4271 

192.2 

82.10 

64.47 

18.35 

% 

.0521 

23.44 

1.221 

.959 

.0334 

k 

.4375 

196.9 

86.14 

67.65 

19.73 

11-16 

.0573 

25.79 

1.478 

1.161 

.0444 

% 

.4479 

201.6 

90.29 

70.52 

21.18 

H 

.0625 

28.12 

1.758 

1.381 

.0575 

k 

.4583 

206.2 

94.54 

74.26 

22.68 

13-16 

.0677 

30.47 

2.064 

1.621 

.0732 

k 

.4688 

210.9 

98.89 

77.66 

24.27 

k 

.0729 

32.81 

2.393 

1.880 

.0913 

% 

.4792 

215.6 

103.3 

81.16 

25.93 

15 16 

.0781 

35.16 

2.747 

2.158 

.1124 

k 

.4896 

220.3 

107.9 

84.72 

27.41 

1. 

.0833 

37.50 

3.125 

2.455 

.1363 

6. 

.5000 

225.0 

112.5 

88.36 

29.44 

1-16 

.0885 

39.84 

3.528 

2.771 

.1636 

k 

.5208 

234.4 

122.1 

95.89 

33.28 

k 

.0938 

42.19 

3.955 

3.107 

.1942 

k 

.5417 

243.8 

132.0 

103.7 

37.44 

3-16 

.0990 

44.53 

4.407 

3.461 

.2284 

H 

.5625 

253.1 

142.4 

111.9 

41.94 

34 

.1042 

46.87 

4.883 

3.835 

.2664 

7. 

.5833 

262.5 

153.2 

120.2 

46.77 

5-16 

.1094 

49.22 

5.384 

4.229 

.3084 

k 

.6042 

271.9 

164.2 

129.0 

51.97 

% 

1146 

51.57 

5.909 

4.640 

.3546 

k 

.6250 

281.3 

175.8 

138.1 

57.54 

7-16 

.1198 

53.91 

6.461 

5.073 

.4058 

H 

.6458 

290.7 

187.7 

147.4 

63.47 

k 

.1250 

56.26 

7.033 

5.523 

.4603 

8. 

.6667 

300.0 

200.1 

157.0 

69.82 

9-16 

.1302 

58.60 

7.632 

5.993 

.5204 

k 

.6875 

309.4 

212.7 

167.0 

76.58 

% 

.1354 

60.94 

8.253 

6.484 

.5852 

k 

.7083 

318.8 

225.8 

177.3 

83.74 

11-16 

.1406 

63.28 

8.900 

6.991 

.6555 

k 

.7292 

328.2 

239.3 

187.9 

91.35 

% 

.1458 

65.63 

9.572 

7.518 

.7310 

9. 

.7500 

337.4 

253.1 

198.8 

99.42 

13-16 

.1510 

67.97 

10.27 

8.064 

.8122 

k 

.7708 

346.8 

267.4 

210.0 

107.9 

k 

.1563 

70.32 

10.99 

8.630 

.8991 

k 

.7917 

356.2 

282.1 

221.5 

116.8 

15-16 

.1615 

72.66 

11.73 

9.215 

.9920 

% 

.8125 

365.6 

297.0 

233.3 

126.3 

2. 

.1667 

75.01 

12.50 

9.821 

1.073 

10. 

.8333 

375.0 

312.5 

245.5 

136.3 

k 

.1771 

79.70 

14.11 

11.09 

1.308 

k 

.8542 

384.4 

328.4 

257.8 

146.8 

34 

.1875 

84.40 

15.83 

12.43 

1.554 

k 

.8750 

393.7 

344.5 

270.6 

157.9 

% 

.1979 

89.07 

17.63 

13.85 

1.827 

% 

.8958 

403.1 

361.2 

283.7 

169.3 

34 

.2083 

93.75 

19.54 

15.34 

2.131 

n. 

.9167 

412.5 

378.2 

297.0 

181.5 

% 

.2188 

98.44 

21.54 

16.56 

2.467 

k 

.9375 

421.9 

395.5 

310.6 

194.2 

H 

.2292 

103.2 

23.64 

18.56 

2.835 

k 

.9583 

431.2 

413.3 

324.6 

207.3 

k 

.2396 

107.8 

25.84 

20.29 

3.241 

% 

.9792 

440.6 

431.4 

338.8 

219.2 

3. 

.2500 

112.6 

28.13 

22.10 

3.682 

12. 

1 Foot. 

450. 

450. 

353.4 

235.6 


t W r ts of balls are as the cubes of their diams. See table, p 416. 

To find the weight of a spherical shell. From the weight of a ball 
which has the outer diam of the shell, take the wt of one which has its inner diam. 


* For Copper, mult by 1.2; Lead, mult by 1.6; Brass, add l-7tli; 

by .97. All approximate. See table, 415. 


Zinc, mult 



































WEIGHT OF CAST-IRON PIPES 


399 


WEIGHT OF CAST-IRON PIPES per running foot. 

Assuming the weight of cast-iron at 450 ft»s per cub ft. or .2604 ft) per cub inch. No 
allowance is here made for the spigot and faucet-joints used in water-pipes. As 
these are now commonly made, (see Fig 38, page 295,) they add to the weight of 
each length or section of pipe of any size, about as much as that of 8 inches jji 
length of the plain pipe as given in the table. 

For load-pipe mult by 1.6; copper, mult by 1.2; brass, add 1-7th ; 
welded iron, mult by 1.0667, or add one fifteenth part. 


§ - » 

« i, V 

fc. -s a 
$ * -* 

§ °- s 

X 

% 

r 

hi 

[■HICKNESl 

% 1 X 

: 

3 OF 

y» 

PIFI 

1 

3 IN 

IX 

INCI 

IX 

IES. 

IX 

IX 


2 


Wt in 

Wt in 

Wt in 

Wt in 

Wt in 

Wt in 

Wt in 

Wt in 

Wt in 

Wt in 

Wt in 

Wt in 

Wt in 


Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

i. 

3.07 

5.07 

7.38 

9.99 

12.9 

16.2 

19.7 

23.5 

27.7 

32.1 

36.9 

47.4 

59.1 

X 

3.69 

6.00 

8.61 

11.5 

14.8 

18.3 

22.2 

26.3 

30.8 

35.5 

40.6 

51.7 

64.0 

X 

4.30 

6.92 

9.84 

13.1 

16.6 

20.5 

24.6 

29.1 

33.8 

38.9 

44.3 

56.0 

68.9 

% 

4.92 

7.84 

11.1 

14.6 

18.5 

22.6 

27.1 

31.8 

36.9 

42.3 

48.0 

60.3 

73.8 

2. 

5.53 

8.76 

12.3 

16.2 

20.3 

24.8 

29.5 

34.6 

40.0 

45.7 

51.7 

64.6 

78.7 

X 

6.15 

9.69 

13.5 

17.7 

22.2 

26.9 

32.0 

37.4 

43.1 

49.0 

55.4 

68.9 

86.7 

X 

6.76 

10.6 

14.8 

19.2 

24.0 

29.1 

34.5 

40.1 

46.1 

52.4 

59.1 

73.2 

88.6 

X 

7.37 

11.5 

16.0 

20.8 

25.9 

31.2 

36.9 

42.9 

49.2 

55.8 

62 7 

77.5 

93.5 

3. 

7.98 

12.5 

17.2 

22.3 

27.7 

33.4 

39.4 

45.7 

52.3 

59.2 

66 4 

81.8 

98.4 

% 

8.60 

13.4 

18.5 

23.8 

29.5 

35.5 

41.8 

48.4 

55.4 

62.3 

70.1 

86.1 

103. 

X 

9.21 

14.3 

19.7 

25.4 

31.4 

37.7 

44.3 

51.2 

58.4 

65.9 

73.8 

90.4 

108. 

X 

9.83 

15.2 

20.9 

26.9 

33.2 

39 8 

46.8 

54.0 

61.5 

69.3 

77.5 

94.7 

113. 

4. 

10.3 

16.1 

22.2 

28.5 

35.1 

42.0 

49.2 

56.7 

64.6 

72.7 

81.2 

99.0 

118. 

X 

11.1 

17.1 

23.4 

30.0 

36 9 

44.1 

51.7 

59.5 

67.7 

76.1 

84.9 

103. 

123. 

X 

11.7 

18.0 

24.6 

31.5 

38.8 

46.3 

54.1 

62.3 

70.7 

79.5 

88.6 

108. 

128. 

% 

12.3 

18.9 

25.8 

33.1 

40.6 

48.5 

56.6 

65.0 

73.8 

83.9 

92.3 

112. 

133. 

5. 

12.9 

19.8 

27.1 

34.6 

42.5 

50.6 

59.1 

67.8 

76.9 

87.2 

96.0 

116. 

138. 

X 

13.5 

20.8 

28.3 

36.1 

44.3 

52.8 

61.5 

70.6 

80.0 

90.6 

99.6 

121. 

143. 

X 

14.2 

21.7 

29.5 

37.7 

46.1 

54.9 

64.0 

73.3 

83.0 

94.0 

103. 

125. 

148. 

% 

14.8 

22.6 

30.8 

39.2 

48.0 

57.1 

66.4 

76.1 

86.1 

97.4 

107. 

129. 

153. 

6. 

15.4 

23.5 

32.0 

40.8 

49.8 

59.2 

68.9 

78.9 

89.2 

99.8 

111. 

134. 

158. 

X 

16.6 

25.4 

34.5 

43.8 

53.5 

63.5 

73.8 

84.4 

95.3 

107. 

118. 

142. 

167. 

7. 

17.8 

27.2 

36.9 

46.9 

57.2 

67.8 

78.7 

89.4 

102. 

113. 

126. 

151. 

177. 

X 

19.1 

29.1 

39.4 

50.0 

60.9 

72.1 

83.7 

95 5 

108. 

120. 

133. 

159. 

187. 

8. 

20.3 

30.9 

41.8 

53.1 

64.6 

76.4 

88.6 

101. 

114. 

127. 

140. 

168. 

197. 

X 

21.5 

32.8 

44.3 

56.1 

68.3 

80.7 

93.5 

107. 

120. 

134. 

148. 

177. 

207. 

9. 

22.8 

34.6 

46.8 

59.2 

72.0 

85.1 

98.4 

112. 

126. 

140. 

155. 

185. 

217. 

X 

24 0 

36.4 

49.2 

62.3 

75.7 

89.3 

103. 

118. 

132. 

147- 

163. 

194. 

226. 

10 . 

25.1 

38.3 

51.7 

65.3 

79.4 

93.6 

108. 

123. 

138. 

154. 

170. 

202. 

235. 

X 

26.4 

40 1 

54.1 

68.4 

83.0 

97.9 

113.2 

129. 

145. 

161. 

177. 

211. 

245. 

ii. 

27.6 

42.0 

56 6 

71.5 

86.7 

102. 

118. 

134. 

151. 

168. 

185. 

220. 

255. 

X 

28.8 

43.8 

59.1 

74 6 

90.4 

107. 

123. 

140. 

157. 

174. 

192. 

228. 

265. 

12. 

30.0 

45.7 

61.5 

77.7 

94.1 

111. 

128. 

145. 

163. 

181. 

199. 

937. 

275. 

13. 

32.5 

49.4 

66.4 

83.8 

102. 

120. 

138. 

156. 

175. 

195. 

214. 

254. 

294. 

14. 

35.0 

53.1 

71.4 

89.4 

109. 

128. 

148. 

168. 

188. 

208. 

229. 

271. 

314. 

15. 

37.4 

56.7 

76.3 

96.1 

116. 

137. 

158. 

179. 

200. 

222. 

244. 

289. 

334. 

16. 

39.1 

60.4 

81.2 

102. 

124. 

145. 

167. 

190. 

212. 

235. 

258. 

306. 

353. 

17. 

42.3 

64.1 

86.1 

108. 

131. 

154. 

177. 

201. 

225. 

249. 

273. 

323. 

373. 

18. 

44.8 

67.8 

91.0 

115. 

139. 

163. 

187. 

212. 

237. 

262. 

288. 

340. 

393. 

19. 

47.3 

71.5 

96.0 

121. 

146. 

171. 

197. 

223. 

249. 

276. 

303. 

357. 

412. 

20. 

49.7 

75.2 

101. 

127. 

153. 

180. 

207. 

234. 

261. 

289. 

317. 

375. 

432. 

21. 

52.2 

78.9 

106. 

133. 

161. 

188. 

217. 

245. 

274. 

303. 

332. 

392. 

452. 

22. 

51.6 

82.6 

111. 

139. 

168. 

196. 

227. 

256. 

286. 

316. 

347. 

409. 

471, 

23. 

57.1 

86.3 

116. 

145. 

175. 

206. 

236. 

267. 

298. 

330. 

362. 

426. 

491. 

24. 

59.6 

89.9 

121. 

152. 

183. 

214. 

246. 

278. 

311. 

343. 

375. 

444. 

511. 

25. 

62.0 

93.6 

126. 

158. 

190. 

223. 

256. 

289. 

323. 

357. 

391. 

461. 

531. 

26. 

64.5 

97.3 

131. 

164. 

198. 

231. 

266. 

300. 

335. 

370. 

406. 

478. 

550. 

27. 

66.9 

101. 

135. 

170. 

205. 

240. 

276. 

311. 

348. 

384. 

421. 

495. 

570. 

28. 

69.4 

105. 

140. 

176. 

212. 

249. 

286. 

323. 

360. 

397. 

436. 

512. 

590. 

29. 

71.8 

109. 

145. 

182. 

220. 

257. 

295. 

334. 

372. 

411. 

450. 

530. 

609. 

30. 

74.2 

112. 

150. 

188. 

227. 

266. 

305. 

345. 

384. 

424. 

465. 

547. 

629. 

31. 

76.7 

116. 

155. 

195. 

234. 

275. 

315. 

356. 

397. 

438. 

480. 

564. 

649. 

32. 

79.1 

120. 

160. 

201. 

242. 

283. 

325. 

367. 

409. 

451. 

495. 

581. 

668. 

33. 

81.6 

123. 

165. 

207. 

249. 

292. 

335. 

378. 

421. 

465. 

509. 

598. 

688. 

34. 

84.1 

127. 

170. 

213. 

257. 

300. 

345. 

389. 

434. 

479. 

524. 

616. 

708. 

35. 

86.5 

131. 

175. 

219. 

264. 

309. 

354. 

400. 

446. 

492. 

539. 

633. 

726. 

36. 

89.0 

134. 

180. 

225. 

271. 

318. 

364. 

411. 

458. 

506. 

554. 

650. 

746. 

42. 

104. 

156. 

210. 

262. 

315. 

370. 

423. 

478. 

532. 

588. 

644. 

753. 

864. 

48. 

119. 

178. 

239. 

298. 

359. 

4‘22. 

482. 

544. 

605. 

669. 

733. 

856. 

982. 


For wanning buildings by steam it usually suffices to allow 1 sq ft 
of cast or wrought pipe surface for each 120 cub ft of space to be warmed ; and 1 cub 
ft of boiler for each 2000 cub ft of such space. 





































400 WEIGHT OP WROUGHT IRON AND STEEL, 


Table of Weight of WROUGHT IRON* and STEEL. 

Wrought iron is here taken at 485 lbs per cub ft; or a sp gr of 7.77. Very pure 
soft wrought iron weighs from 488 to 492 lbs per cubic foot. Light weight indicates 
impurities, and weakness. Steel weighs about 490 lbs per cubic toot; therefore, for 

steel, an addition must be made to the tabular amounts, of 
about 1 pound in IOO lbs. 

At 485 lbs per cub ft, a cubic inch weighs .28067 lb; a lb contains 3.5629 cub ins; 
and a ton, 4.6186 cub ft; and this is about the average of hammered iron The usual 
assumption is 480 lbs per cub ft; which is nearer the average of ordinary rulhd iron. 
At 480 lbs, a cubic inch weighs .2778 of a lb; a lb contains 3.6 cub ins: a ton, 4.6667 
cub ft; a rod of 1 sq inch area, weighs 10 lbs per yard; or 3^ lbs per foot, exactly. 


Thickness 

or Diameter 
in Inches. 

Thick¬ 
ness or 
Diam. 
in deci¬ 
mals of 
a foot. 

Wt. of a 
Square 
Foot. 

Lbs. 

Wt. of a 
Square 
bar, 1 ft. 
long. 

Lbs. 

Wt. of a 
Round 
bar, 1 ft. 
loug. 

Lbs. 

Wt. of 
Balls. 

Lbs. 

U 

!« 0> • 
w SR 
o 

5 5" 
y .2 *- 

o 

Thick¬ 
ness or 
Diam. 
indeci¬ 
mals of 
a foot. 

Wt. of a 
Square 
Foot. 

Lbs. 

Wt. of a 
Square 
bar, 1 ft. 
long. 

Lbs. 

Wt. of a 
Round 
bar. 1 ft. 
long. 

Lbs. 

Wt. of 
Balls. 

Lbs. 

1-32 

.0026 

1.263 

.0033 

.0026 


'AX 

.2604 

126.3 

32.89 

25.83 

4.484 

1-16 

.0052 

2.526 

.0132 

.0104 


X 

.2708 

131.4 

35.57 

27.94 

5.045 

3-32 

.0078 

3.789 

.0296 

.0233 

.0001 

h 

.2813 

1556.4 

38.37 

30.13 

5.649 

X 

.0104 

5.052 

.0526 

.0414 

.0003 

X 

.2917 

141.5 

41.26 

32.41 

6.5401 

5-32 

.0130 

6.315 

.0823 

.0646 

.0005 

X 

.3021 

146.5 

44.26 

34.76 

7.000 

3-16 

0156 

7.578 

.1184 

.0930 

.0009 

X 

.3125 

151.6 

47.37 

37.20 

7.750 

7-32 

.0182 

8.8-41 

.1612 

.1266 

.0015 

X 

.8229 

156.6 

50.57 

39.72 

8.550 

X 

.0208 

10.10 

.2105 

.1653 

.0023 

4. 

.3333 

161.7 

555.89 

42.33 

9.405 

9-32 

.0234 

11.37 

.2665 

.2093 

.00543 

X 

.3438 

166.7 

57.541 

45.01 

10.32 

5-16 

.0260 

12.651 

.3290 

.2583 

.0045 

X 

.3542 

171.8 

60.84 

47.78 

11.28 

11-32 

.0287 

13.89 

.3980 

.3126 

.0060 

X 

.54646 

176.8 

64.47 

50.654 

12.31 

% 

.0313 

15.16 

.47556 

.3720 

.0078 

X 

.3750 

181.9 

68.20 

53.57 

154.549 

13-32 

.0339 

16.42 

.5558 

.45565 

.0098 

X 

.3854 

186.9 

72.05 

56.59 

14.54 

7-16 

.0365 

17.68 

.6446 

.5063 

.0123 

X 

.3958 

192.0 

75.99 

59.69 

15.75 

15-32 

.0391 

18.95 

.7400 

.58155 

.0151 

X 

.4063 

197.0 

80.05 

62.87 

17.03 

X 

.0417 

20.21 

.8420 

.6613 

.0184 

5. 

.4167 

202.1 

84.20 

66.13 

18.37 

9-16 

.0469 

22.73 

1.066 

.8370 

.0262 

X 

.4271 

207.1 

88.47 

69.48 

19.78 

% 

.0521 

25.26 

1.316 

1.033 

.0359 

X 

.45575 

212.2 

92.83 

72.91 

21.26 

11-16 

.0573 

27.79 

1.592 

1.250 

.0478 

X 

.4479 

217.2 

97.551 

76.43 

22.82 

X 

.0625 

30 31 

1.895 

1.488 

.0(520 

X 

.4583 

222.3 

101.9 

80.02 

24.45 

13-16 

.0677 

32.84 

2.223 

1.746 

.0788 

X 

.4688 

227.3 

106.6 

83.70 

26.16 

x 

.0729 

35.37 

2.579 

2.025 

.0985 

X 

.4792 

2542.4 

111.4 

87.46 

27.94 

15-16 

.0781 

37.89 

2.960 

2.325 

.1211 

X 

.4896 

2547.5 

116.3 

91.541 

29.80 

1. 

.0833 

40.42 

3.5568 

2.645 

.1470 

6. 

.5000 

242.5 

121.3 

95.23 

31.74 

1-16 

.0885 

42.94 

3.8055 

2.986 

.1763 

X 

.5208 

252.6 

131.6 

1054.3 

545.88 

X 

.0938 

45.47 

4.263 

3.348 

.20955 

X 

.5417 

262.7 

142.3 

111.8 

40.36 

3-16 

.05490 

48.00 

4.750 

3.730 

.2461 

X 

.5625 

272.8 

153.5 

120.5 

45.19 

X 

.1042 

50.52 

5.263 

4.133 

.2870 

7. 

.5833 

282.9 

165.0 

129.6 

50.40 

5-16 

.105)4 

53.05 

5.802 

4.557 

.3323 

X 

.6042 

2954.0 

177.0 

1549.0 

56.00 

% 

.1146 

55.57 

6.368 

5.001 

.3820 

X 

.6250 

3054.1 

189.5 

148.8 

62.00 

7-16 

.115)8 

58.10 

6.960 

5.406 

.4365 

X 

.6458 

313.2 

202.3 

158.9 

66.40 

X 

.1250 

60.63 

7.578 

5.952 

.4960 

8. 

.6(567 

823.3 

215.6 

169.3 

75.24 

9-16 

.1302 

63.15 

8.223 

6.458 

.5606 

X 

.6875 

333.4 

229.3 

180.1 

82.52 

% 

.1354 

65.68 

8.893 

6.985 

.6306 

X 

.7083 

5443.5 

243.4 

191.1 

90.25 

11-16 

.1406 

68.20 

9.591 

7.533 

.7062 

X 

.7292 

353.6 

247.9 

202.5 

98.45 

X 

.1458 

70.73 

10.31 

8.101 

.7876 

9. 

.7500 

363.8 

272.8 

214.3 

107.1 

13-16 

.1510 

73.26 

11.07 

8.690 

.8750 

X 

.7708 

373.9 

288.2 

226.3 

116.54 

V* 

.1563 

75.78 

11.84 

9.300 

.9688 

X 

.7917 

384.0 

304.0 

238.7 

126.0 

15-16 

.1615 

78.31 

12.64 

9.9550 

1.069 

X 

.8125 

5494.1 

320.2 

251.5 

136.2 

2. 

.1667 

80.83 

13.47 

10.58 

1.176 

10. 

.8333 

404.2 

5436.8 

264.5 

146.9 

X 

.1771 

85.89 

15.21 

11.95 

1.410 

X 

.8542 

414.3 

353.9 

277.9 

158.2 

X 

.1875 

90.94 

17.05 

13.39 

1.674 

X 

.8750 

424.4 

5471 .3 

291.6 

170.1 

% 

.1979 

95.99 

19.00 

14.92 

1.969 

X 

.8958 

4544.5 

389.2 

305.7 

182.6 

u 

.2083 

101.0 

21.05 

16.53 

2.296 

11. 

.9167 

444.6 

407.5 

320.1 

195.6 

% 

.2188 

106.1 

23.21 

18.23 

2.658 

X 

.95575 

454.7 

426.3 

3514.8 

209.2 

X 

.2292 

111.2 

25.47 

20.01 

3.056 

X 

.9583 

464.8 

445.4 

349.8 

223.5 

% 

.2396 

116.2 

27.84 

21.87 

3.492 

X 

.9792 

474.9 

465.0 

365.2 

238.4 

3. 

.2500 

121.3 

550.31 

23.81 

3.968 

12. 

1 Foot. 

485. 

485. 

380.9 

253.9 


To find the weight of a spherical shell. From the weight of a ball 
which has the outer diam of the shell, take the wt of one which has its inner diam. 


* For Copper, add l-7th part. Lead, mult by 1.47. Brass, mult by 1.06. 

Ziuc, mult by .9. Tin, mult by .95. All approximate. See table, p 415 . 






































WEIGHT OF FLAT IRON, 


401 


Weight of 1 ft in length of FLAT ROLLED IRON, at 480 lbs per 
cubic foot. For cast iron, deduct yL- part; for steel, add ig ; for copper, add 
j- for cast brass, add y 1 ^; for lead, add ]/%; for zinc, deduct yy. 



THICKNESS IN INCHES. 


cS.2 

1-16 

X 

3-16 


5-16 

% 

7-16 

X 

| % 

X 

% 

i 

1. 

.2083 

.4166 

.6250 

.8333 

1.042 

1.250 

1.458 

1.666 

2.083 

2.500 

2.916 

3.333 

X 

.2344 

.4688 

.7033 

.9375 

1.172 

1.406 

1.640 

1.875 

2.344 

2.812 

3.280 

3.75 


.2605 

.5210 

.7810 

1.042 

1.303 

1.563 

1.823 

2.083 

2.605 

3.125 

3.646 

4.166 

% 

.2865 

.5730 

.8595 

1.146 

1.432 

1.719 

2.006 

2.292 

2.864 

3.438 

4.012 

4.583 


.3125 

.6250 

.9375 

1.250 

1.562 

1.875 

2.188 

2.500 

3.125 

3.750 

4.375 

5.000 

% 

.3385 

.6771 

1.015 

1.354 

1.692 

2.031 

2.370 

2.708 

3.384 

4.062 

4.740 

5.416 

X 

.3616 

.7292 

1.09 4 

1.458 

1.823 

2.188 

2.550 

2.916 

3.646 

4.375 

5.105 

5.833 

v» 

3906 

.7812 

1.172 

1.562 

1.953 

2.344 

2.735 

3.125 

3.906 

4.688 

5.470 

6.25 

2. 

.4166 

.8333 

1.25 

1.666 

2.083 

2.500 

2.916 

3.333 

4.166 

5.000 

5.833 

6.666 

X 

.4427 

.8855 

1.328 

1.771 

2.214 

2.656 

3.098 

3.542 

4.428 

5.312 

6.196 

7.083 

X 

.4688 

.9375 

1.406 

1.875 

2.3 4 4 

2.812 

3.281 

3.750 

4.688 

5.624 

6.562 

7.500 

% 

.4948 

.9895 

1.484 

1.979 

2.474 

2.968 

3.463 

3.958 

4.948 

5.936 

6.926 

7.916 

X 

.5210 

1.012 

1.562 

2.083 

2.605 

3.125 

3.646 

4.166 

5.210 

6.250 

7.291 

8.333 

% 

.5470 

1.094 

1 641 

2.187 

2.735 

3.282 

3.829 

4.375 

5.470 

6.564 

7 658 

8.750 

H 

.5730 

1.146 

1.719 

2.292 

2.865 

3.438 

4.011 

4.583 

5.730 

6.876 

8.022 

9.166 

Vs 

.5990 

1.198 

1.797 

2.396 

2.995 

3.594 

4.193 

4.792 

5.990 

7.188 

8.386 

9.583 

3. 

.625 

1.250 

1.875 

2 500 

3.125 

3.750 

4.375 

5.000 

6.250 

7.500 

8.750 

10.00 

X 

.6515 

1.303 

1 954 

2.605 

3.257 

3.908 

4.560 

5.210 

6.514 

7.816 

9.120 

10.42 

% 

.6770 

1.354 

2.031 

2.708 

3.385 

4.062 

4.739 

5.416 

6.770 

8.124 

9.478 

10.83 

Vs 

.7031 

1.406 

2.109 

2.812 

3.516 

4.218 

4.921 

5.625 

7.032 

8.436 

9.842 

11.25 

X 

.7291 

1.458 

2.188 

2.916 

3.646 

4.375 

5.105 

5.833 

7.291 

8.750 

10.21 

11.66 

% 

.7555 

1.511 

2.266 

3.021 

3.777 

4.533 

5.288 

6.042 

7.554 

9.066 

10.58 

12.08 

X 

.7812 

1.562 

2.3 43 

.3.125 

3.906 

4.686 

5.468 

6.25 

7.812 

9.372 

10.94 

12.50 

x 

.8070 

1.614 

2.421 

3.229 

4.035 

4.842 

5.65 

6.458 

8.070 

9.684 

11.30 

12.92 


.8333 

1.666 

2.500 

3.333 

4.166 

5.000 

5.833 

6.666 

8.333 

10.00 

11.66 

13.33 

X 

.8595 

1.719 

2.578 

3.438 

4.297 

5.156 

6.016 

6.875 

8.594 

10.31 

12.03 

13.75 

X 

.8855 

1.771 

2.656 

3.542 

4.427 

5.312 

6.198 

7.083 

8.854 

10.62 

12.40 

14.16 

H 

.9115 

1.823 

2.734 

3.646 

4.557 

5.468 

6.380 

7.291 

9.114 

10.94 

12.76 

14.58 

X 

.9375 

1.875 

2.812 

3.750 

4.687 

5.624 

6.562 

7.500 

9.374 

11.25 

13.12 

15.90 

% 

.9636 

1.927 

2.891 

3.854 

4.818 

5.782 

6.745 

7.708 

9.636 

11.56 

13.49 

15.42 

X 

.9895 

1.979 

2.968 

3.958 

4.947 

5.936 

6.926 

7.917 

9.894 

11.87 

13.85 

15.83 

X 

1.016 

2.031 

.3.048 

4 062 

5.080 

6.096 

7.112 

8.125 

10.16 

12.19 

14.22 

16.25 

6. 

1.042 

2.083 

3.125 

4.166 

5.210 

6.25 

7.291 

8 333 

10.42 

12.50 

14.58 

16.66 

X 

1.068 

2.136 

3.204 

4.271 

5.340 

6 408 

7.476 

8.542 

10.68 

12.81 

14.95 

17.08 

X 

1.094 

2.188 

3.282 

4.375 

5 470 

6.564 

7.658 

8.750 

10.94 

13.13 

15.31 

17.50 


1.120 

2.240 

3.360 

4.479 

5.600 

6.720 

7.840 

8.958 

11.20 

13.44 

15.68 

17.92 


1.146 

2.292 

3.438 

4.584 

5.730 

6.876 

8.022 

9.167 

11.46 

13.75 

16.04 

18.33 

X 

1.172 

2.344 

3.516 

4.687 

5.860 

7.032 

8.204 

9.375 

11.72 

14.06 

16.40 

18.75 

X 

1.198 

2.396 

3.594 

4.791 

5.990 

7.188 

8.386 

9.583 

11.98 

14.37 

16.77 

19.16 

% 

1.224 

2.448 

3.672 

4.896 

6.120 

7.344 

8.568 

9.792 

12.24 

14.69 

17.13 

19.58 

6. 

1.250 

2.500 

3.750 

5.000 

6.250 

7.500 

8.750 

]0.00 

12.50 

15.00 

17.50 

20.00 

X 

1.276 

2 552 

3.828 

5.104 

6.380 

7.656 

8.932 

10.21 

12.76 

15.31 

17.86 

20.42 

X 

1.302 

2.604 

3.906 

5.208 

6.510 

7.812 

9.114 

10.42 

13.02 

15.62 

18.23 

20.83 

% 

1.328 

2.657 

3.984 

5.313 

6 640 

7.968 

9.297 

10.63 

13.28 

15.93 

18.59 

21.25 

H 

1.354 

2.708 

4.063 

5.417 

6.770 

8.126 

9.480 

10.83 

13.54 

16.25 

18.96 

21.66 

% 

1.381 

2.761 

4.143 

5 521 

6.906 

8.286 

9.668 

11.04 

13.81 

16.57 

19.33 

22.08 

X 

1.406 

2.813 

4.218 

5.625 

7.030 

8.436 

9.843 

11.25 

14.06 

16.87 

19.69 

22.50 

y» 

1.432 

2.864 

4.296 

5.729 

7.160 

8.592 

10.02 

11.46 

14.32 

17.18 

20.04 

22.92 

7. 

1.458 

2.916 

4.375 

5.833 

7.291 

8.750 

10.20 

11.66 

14.58 

17.50 

20.42 

23.33 

X 

1.484 

2.969 

4.452 

5.938 

7.420 

8.904 

10.39 

11.87 

14.84 

17.81 

20.78 

23.75 

X 

1.511 

3.021 

4.533 

6.042 

7.555 

9.066 

10.58 

12.08 

15.11 

18.13 

21.16 

24.16 

% 

1.536 

3.073 

4.608 

6.146 

7.680 

9.216 

10.75 

12.29 

15.36 

18.43 

21.50 

24.58 

X 

1.562 

3.125 

4.686 

6.250 

7.810 

9.372 

10.93 

12.50 

15.62 

18.74 

21.86 

25.00 

% 

1.588 

3.177 

4.764 

6.354 

7.940 

9.528 

11.12 

12.71 

15.88 

19.05 

22.24 

25.42 

X 

1.615 

3.229 

4.845 

6.458 

8.075 

9.690 

11.31 

12.92 

16.15 

19.38 

22.62 

25.83 

y» 

1.641 

3.281 

4.923 

6.562 

8.205 

9.846 

11.48 

13.13 

16.41 

19.69 

22.96 

26.25 

8. ' 

1.666 

3.3.33 

5.000 

6.666 

8.333 

10.00 

11.66 

13.33 

16.66 

20.00 

23.33 

26.66 

X 

1.69.3 

3.386 

5.079 

6.771 

8.455 

10.15 

11.85 

13.54 

16.91 

20.30 

23.70 

27.08 


1.719 

3.438 

5.157 

6.875 

8.595 

10.31 

12.03 

13.75 

17.19 

20.61 

24.06 

27.50 

% 

1.745 

3.489 

5.2.35 

6.979 

8.725 

10.47 

12.21 

13.96 

17.45 

20.94 

24.42 

27.92 

K 

1.771 

3.542 

5.313 

7.083 

8.855 

10.63 

12.40 

14 17 

17.71 

21.26 

24.80 

28.33 

X 

1.797 

.3.5.94 

5.391 

7.188 

8.985 

10.78 

12.58 

14.37 

17.97 

21.56 

25.16 

28.75 

X 

1.823 

3.646 

5.469 

7.292 

9.115 

10.94 

12.76 

14.58 

18.23 

21.88 

25.52 

29.17 

X 

1.849 

3.698 

5.547 

7.396 

9.245 

11.09 

12.94 

1 4.79 

18.49 

22.18 

25.88 

29.58 

9. 

1.875 

3.750 

5.625 

7.500 

9.375 

11.25 

13.12 

15.00 

18.75 

22.50 

26.24 

30.00 

X 

1.901 

3.802 

5.70.3 

7.604 

9.505 

11.41 

13.31 

15.21 

19.00 

22.81 

26.62 

30.42 

X 

1.927 

3.854 

5.781 

7.708 

9.635 

11.56 

13.49 

15.42 

19.27 

23.12 

26.98 

■iO.83 

y» 

1.953 

3.906 

5.859 

7.812 

9.765 

11.72 

13.67 

15.62 

19.53 

23.44 

2 7. *3 4 

31.25 

y* 

r < 

1.979 

3.958 

5.937 

7.916 

9.895 

11.87 

13.85 

15.84 

19.79 

23.74 

'Z l . i 9 

31.67 

2.005 

4.010 

6.015 

8.021 

10.02 

12.03 

14.04 

16.04 

20.04 

24.06 

28.08 

32.08 

X 

2.031 

4.0 62 

6.093 

8.125 

10.16 

12.18 

14.21 

16.25 

20.32 

24.36 

28.42 

.V2.50 

% 

2.057 

4.114 

6.171 

8.229 

10.29 

12.34 

14.40 

16.46 

20.58 

24.68 

28.80 

32.92 

[). 

2 os:; 

4.166 

6.250 

8.333 

10.41 

12.50 

14.58 

16.66 

20.82 

25.00 

29.1 6 

X;.33 


2.109 

4.219 

6.327 

8.438 

10.55 

12.65 

14.76 

16.87 

21.10 

25.30 

29.52 

33.75 

/o 

X 

2.135 

4.270 

6.405 

8.541 

10.67 

12.81 

14.94 

17.08 

21.34 

25.62 

29.88 

34.17 


















































402 


IRON AND STEEL. 


Weight of 1 ft in length of FLAT ROLLED IRON, at 480 lb?- 

per cubic foot — (Continued.) 


X! V) 

•S.3 




THICKNESS 

IN INCHES 





£.5 

1-16 

M 

3-16 

H 

5-16 

% 

7-16 

X 

% 

X 

% 

1 

10% 

2.162 

4.323 

6.486 

8.646 

10.81 

12.97 

15.13 

17.29 

21.62 

25.94 

30.26 

34.58 

% 

2.188 

4.375 

6.564 

8.750 

10.94 

13.13 

15.31 

17.50 

21.88 

26.26 

30.62 

35.00 

% 

2.214 

4.427 

6.642 

8.854 

11.07 

13.28 

15.50 

17.71 

22.14 

26.56 

31.00 

35.42 

% 

2.239 

4.479 

6.717 

8.958 

11.20 

13.43 

15.67 

17.92 

22.40 

26.86 

31.34 

35.83 

% 

2.266 

4.531 

6.798 

9.062 

11.33 

13.59 

15.86 

18.12 

22.66 

27.18 

31.72 

36.25 

11. 

2.291 

4.583 

6.873 

9.166 

11.46 

13.75 

16.04 

18.33 

22.90 

27.50 

32.08 

36.66 

H 

2.318 

4.636 

6.954 

9.271 

11.59 

13.91 

16.22 

18.54 

23.18 

27.82 

32.44 

37.08 

34 

2.344 

4.688 

7.032 

9.375 

11.72 

14.06 

16.40 

18.75 

23.44 

28.12 

32.80 

37.50 

y» 

2.370 

4.740 

7.110 

9.479 

11.85 

14.22 

16.59 

18.96 

23.70 

28.44 

33.18 

37.92 

X 

2.395 

4.791 

7.185 

9.582 

11.97 

14.37 

16.76 

19.16 

23.94 

28.74 

33.52 

38.33 

% 

2.422 

4.844 

7.266 

9.688 

12.11 

14.53 

16.95 

19.37 

24.22 

29.06 

33.90 

38.75 

% 

2.448 

4.896 

7.344 

9.792 

12.24 

14.68 

17.13 

19.58 

24.48 

29.36 

34.26 

39.16 

% 

2.474 

4.948 

7.422 

9.896 

12.37 

14.84 

17.32 

19.79 

24.74 

29.68 

34.64 

39.58 

12. 

2.500 

5.000 

7.500 

10.00 

12.50 

15.00 

17.50 

20.00 

25.00 

30.00 

35.00 

40.00 


ROLLED IRON AND STEEL.* AVERAGE PRICES. Phila, 1886. 
Iron bars. Base price, or price for “ordinary sizes”; i e, from % inch to 2 ins 
diameter, round and square; and from 1 X inch to 6 X 1 inch, flat: Ordinary 
merchant quality, called “ refined ”, 2 cts per fb; “ Extra refined ” and rivet iron, 3 
cts per lb. Sizes larger or smaller than “ordinary ” bring higher prices per lb. The 
“ extra ”, or charge in addition to the above base price, increases gradually up to .7 
ct per lb for (^ inch round and square; 2.5 cts for 7 inches round and square ; 1.2 cts 
for X ns inch flat; and 1.1 cts for 12 X 2 ins flat. Swedish ami Norway 
iron : (4 X l /i inch and % inch square, and larger; in the original bar as imported 
(called “Swedish ”), 4 cts per lb; re-rolled after importation (called “Norway”), 4^ 
cts per lb. Hoop iron : from 1% inch X No 17, 2(4 cts per lb; to % inch X No 
22, 5 cts per lb. Sheet iron; see page 403. Plate iron. Rectangular 
plates, t 3 3 inch thick and heavier; and say from 2 to 5 feet wide, and from 4 to 12 feet 
long. “Common or “puddled”, for bridge plates, sheathing of ships etc, where little 
or no bending is required, 2 cts perTh; “Shell”; for shells of boilers etc, to stand i 
bending cold with the grain to cylinders with radii of say one or two feet, but not 
flanging, 2(4 cts per lb; “ Best flange ”, 3(4 cts per lb. Fire-box plates of shell or 
flange iron, 1 ct per lb extra. Extra charges for plates of unusually great or small 
widths or lengths. Angle ami T iron; see pages 525 etc. I beams, 
channel beams, deck beams ; see pages 520 etc. Pig iron. American 
foundry. Si8 per ton of 2240 lbs; Forge, for conversion into wrought iron by pud¬ 
dling etc, S16 per ton of 2240 lbs; Charcoal foundry and forge, $20 per ton of 2240 lbs. 
Oast Steel for tools. Best American. Ordinary sizes, 8(4 cts per lb; Ma¬ 
chinery steel ; for shafting etc, ordinary sizes, 3 cts per lb. The range of “or¬ 
dinary ” sizes is nearly the same as given above for iron. For larger and smaller 
sizes, extras running up to from 50 to 100 per cent. Steel plates for boilers and 
fire-boxes, 4 cts per lb. Tire and spring steel. Bessemer, 3 cts per lb I 
Sheet steel. American.! Not lighter than No 17. Cast, 7 cts per lb; Bessemer, 
4(4 cts per lb. For each number lighter than No 17, (4 ct per lb extra. 

Rolled Star Iron. Standard sizes. Carnegie Bros & Co, Limited, Pittsburgh. 
The thicknesses are those at the end and at the root of one of the four arms, in ins. Rolled in 
lengths of 20 to 25 ft. Area in sq ins. Wt in tbs per foot run. 


Ins. 

Thick. 

Area. 

Wt. 

Ins. 

Thick. 

Area. 

Wt. 

4 X 4 

% — 9-16 

3.6 

12 

2.5 X 2.5 

5-lf, _ 13-32 

1.65 

5.5 

3.5 X 3.5 

X- X 

2.85 

9.5 

2X2 

]4 — 13-32 

1.13 

3.75 

3X3 

5-16 — 15-32 

2.18 

7.25 

1.5 X 1.5 

3 16— 5-16 

0.69 

2.3 


* Morris, Wheeler & Co, 16th and Market Sts, Philadelphia, 
t William & Harvey Rowland, Frankford, Philadelphia. 























































SHEET-IRON 


403 


Sheet iron.* Prices. Phila. 1886; in sheets 24 to 82 inches wide, 6 to 8 feet 
long; No 14 to No 28 by second table on p 411; common (puddled) iron, annealed, 2]/ y 
to 3(^ cts per pound, being least for the lower (thicker) numbers; best charcoal 
bloom iron, 3% to 5 cts per lb. 

Galvanized sheet iron.* Common (puddled) iron) 4^ to 6 cts per 
pound; best charcoal bloom iron, 5 to G}/ 2 cts per pound. The lower (thick) nos are 
the cheapest. 

Weights per square foot of galvanized sheet iron. Standard list 
adopted by the American Galvanized Iron Ass’n, at Pittsburgh, April, 1884. 


No. 

Ounces 
avoir 
per sq ft 

Sq ft 
per 

2240 Bs. 

No. 

Ounces 
avoir 
per sq ft. 

Sq ft 
per 

2240 Bs. 

No. 

Ounces 
avoir 
per sq ft. 

Sq ft 
per 

2240 Bs. 

29 

12 

2987 

24 

17 

2108 

19 

33 

1086 

28 

13 

2757 

23 

19 

1886 

18 

38 

943 

27 

14 

2560 

22 

21 

1706 

17 

43 

833 

26 

15 

2389 

21 

24 

1493 

16 

48 

746 

25 

16 

2240 

20 

28 

1280 

14 

60 

597 


The galvanizing* is simply a thin film of zinc on both sides of the 

sheet, as iu what is known as “ tinned plates,” or “ tin ; ” which are in reality sheet iron similarly 
coated with tin. Zinc, like tin, resists corrosion from ordinary atmospheric influences, much better 
than iron ; and hence the use of these metals as a protection to the iron. A well galvanized roof, 
of a good pitch, will suffer but little from 5 to 6 years’ exposure without being painted. It will then 
take paint readily, and should be painted. It is better, however, always to paint tin ones at once. 

Paint does not adhere well to new zinc, and this is the principal 

reason why new galvanized roofs are not paiuted ; but this may he remedied by first brushing the 
zinc over with the following: One part of chloride of copper, 1 part nitrate of copper, 1 part of sal- 
ammoniac. Dissolve in 61 parts of water. Then add 1 part of commercial hydrochloric acid. When 
brushed with this solution, the zinc turns black ; dries within 12 to 24 hours, and may then be painted. 

Paint of some mineral oxide of a brown color is generally used; one coat being applied to both 
sides in the shop ; and the other after being put on the roof. Repainting every 3 or 4 years will suffice 
afterward. Ungalvanized irou (called black iron, for distinction) is also very enduriug for roofs, if 
well painted every 1 or 2 years. The chief advantageof galvanized roofing is that it does not require 
painting so often as the black. The galvanizing adds about % of a B> per square foot of surface, or 
about % B per sq ft of sheet as coated on both sides; without regard to the thickness of the sheet. 
Paint for roofs should not have much dryer. See Painting, p 429. 

The sulphurous fumes from coal are very corrosive of 

either galvanized or black iron ; as may be seen in shops, railroad bridges, or engine houses, 
roofed with either; if efficient means are not provided for carrying oil the smoke ; and the same with 
other metals. The acid of oak timber is said to destroy the zinc of galvanized iron. See pp 418, 
419. Flat iron is usually nailed upon a sheeting of boards; but the strength of corrugated iron 
obviates the necessity for this, and enables it to stretch 5 or 6 ft from purlin to purlin, without inter¬ 
mediate support. The corrugated sheets are riveted together ou the roof, by rivets of galvanized 
wire about one-eighth inch thick, 300 to a pound, well driven (so as to exclude rain) 3 or 4 inches 
apart, all around the edges. The rivet-holes are first punched by machinery, so as to insure coinci¬ 
dence in the several sheets; and the rivets are driven by two men, one above, and one beneath the 
roof. For black iron, ungalvanize4 nails, boiled in linseed oil as a partial preservative from rust, are 
commonly used; as also in shingling or slating. Galvanized ones, however, would be better in all 
these cases; or even copper ones for slating because good slate endures much longer than either 
shingles or iron, and therefore it becomes true economy to use durable metals for fastening it. la 
none of these cases, however, are the nails fully exposed to the weather. 

Tlie sheets of flat iron are put together by overlapping- an«t 

folding the edgks, much the same as shown by the fig page 418, head Tin ; the joints which run 
up and down the roof being the same as at s a, and the horizontal ones as at t (; 
except that inasmuch as these are not soldered in the iron sheets, the joint is made 
about % to 1 inch wide, instead of % inch, the better to provide against leaking. 
Cleats are used as in tin, with 2 nails to a cleat. The iron plates are best laid on 
sheeting boards ; but in sheds. &c, are sometimes laid directly on rafters, not more 
than about 18 ins apart in the clear; the plates being allowed to sag a little between 
the rafters, so as to form shallow gutters. Iu such cases it is well to bevel off the tops of the rafters 
slightly, as in this fig. 

A serious objection to iron as a roof covering’, is its rapid con¬ 
densation of atmospheric moisture; which falls from the iron in drops like rain, and may do injury 
to ceilings, floors, or articles in the apartments immediately beneath the roof. Painting does not 
appreciably diminish this: it may. however, be obviated by plastering, as shown at 11, of Figs 21%, 
of Trusses, p 583. Corrugated iron makes au excellent permanent street or other awning, 

Corrugated sheet Iron. The size of sheets generally used for corrugating, 
is 30 inches wide bv 96 inches long. Corrugation reduces the width to 27% inches. When the cor¬ 
rugated sheets are laid upon the roof, the overlapping of about 2% inches along the sides, and of 4 
inches along their ends, diminishes the area of roof covered by asheet, to about seven-eighthsof that 
of the entire corrugated sheet itself; or, the weight per square foot of roof covered, will be about 
one-seventh greater than that per square foot of the corrugated sheet; or. the weight of corrugated 
iron per square foot of roof covered is about one-fifth greater than that of the flat sheets from which 
it is made. 

About 6 inches are usually allowed for the extension over the eaves. 

The weights per square foot corresponding to the different numbers of the Birmingham wire gauge, 
varv somewhat with the different makers. The two styles of corrugation given in the table below, 
5 x l% and 2% x %, are those most frequently used. 

* Marshall, Bros & Co, Front St and Girard Ave, Phila. McDaniel & Hatvey Co, 16th St and 
Washington Ave, Phila. 




























404 


CORRUGATED SHEET IRON 


No. 

Bmghm 
wire ga. 

Tlilek- 

ness 

in ins. 

Wt in lbs per 
sq ft of sheets. 

Wt in lbs per 
sq ft of roof. 

Approx prices in ets 
per lb, Phila, 1886. 

Black 

Black 

Black 

Lead coatedt 
or galv’d 

Black 

Lead coatedt 
or galv’d 

Black 

Lead coatedt 
or galv’d 

20 

.035 

1.84 

2. 

2.12 

2.3 


5% 

22 

.028 

1.50 

1.6 

1.73 

1.84 

3/4 

5 /2 

24 

.022 

1.20 

1.25 

1.38 

1.44 

3 Y % 
3M 


26 

.018 

1.00 

1.12 

1.15 

1.29 

5% 


Strength of Corrugated Iron. Experiments by the author. 


First. A sheet <1 d, of No. 16 iron, 

(about y 1 - inch thick,) 27 ins wide, by 4 ft long, 
with live complete corrugations of 5 ins by 1 inch, 
was laid on supports 3 ft 9 ins apart. A block of 
wood c, 9 ins wide, by 7 ins thick, and 30 ins long, 
was placed across the center, and gradually loaded 
with castings weighing 1600 lbs. 

This caused a deflection at the center of precisely an 
inch. On the removal of the load after an hour, no perma¬ 
nent set was appreciable. The severity of the test was pur¬ 
posely increased by applying the several castings very 
roughly, jolting the whole as much as possible.* The sus¬ 
pended area of the sheet was 8.41 sq ft; and since the actual center load of 1600 lbs is about cqulva- 

3000 

lent to 3000 lbs equally distributed, it amounts to - — 355 lbs per sq ft distributed. But 3000 lbs 

distributed would produce a deflection of but about full % of an inch. Again, 355 tbs per sq ft 
is about 4 times the weight of the greatest crowd that could well congregate upon a floor. Conse¬ 
quently this iron, at 8' 9" span, is safe in practice for any ordinary crowd. Moreover, such a crowd 
would produce a center deflection of only the %th part of % of an inch ; or yU of an inch ; or y yy 

of the clear span ; which is but two-thirds of Tredgold's limit of yyyy of the span. 

In one experiment the ends of the sheets rested upon supports dressed so as to present undulations 
corresponding tolerably closely with the shape of the corrugations; but in the other the supports 
were flat, and each end of the’sheet rested only upon the lower points of the corrugations, bio ap¬ 
preciable difference was observed in the results 

Second. An arcli of No. IS (yj 
inch) iron, corrugated like the foregoing, 
but the depth of corrugation increased to 
1J4 ins by the process of arching the sheet; 
clear span 0 ft 1 inch ; rise 10 ins; breadth 27 
ins, (of which, however, only 25 ins bore 
against the abutments) 

Each foot o of the arch abutted upon a casting j, 
the inner portion t of which was undulated on top, to 
correspond with the corrugations of the arch, which 
rested upon it. At y. (one-fourth of the span.) two 
wooden blocks were placed, occupying a width of 9 
inches, and extending across the arch ; on them was 
piled a load. I, of castings, to the extent of 4480 lbs, 
or 2 tons. Under this load the arch descended about 
half ati inch at y, becoming flatter on that side, and 
slightly more curved upward along the unloaded side n. Two similar blocks were then placed at n, 
aud two tons of load, s, were piled upon them, in addition to the 2 tons at l ; making a total of 8960 
lbs. or 4 tons. This brought the arch more nearly hack to its original shape; but still slightly 
straightened at both n and y. and a little more curved in the center. The load was then increased to 
10000 lbs. and left standing for several davs. Two iron ties, each H by 1%, which were used for pre¬ 
venting the abutment castings j from spreading, were found to have stretched nearly of an inch. 
Additional ones were inserted, and the load increased to a total of 6 tons, or 13440 lbs; parts of it on 
* and l, and part in the shape of long broad bars of iron at the center of the arch, beiow the loads * 
and l. and between n and y. So far as could be judged by eye. the shape of the arch was now almost 
perfect. The loads s and 1 did not touch each other. After standing more than a week, the load 
was accidentally overturned, crippling the arch. The load was equal to about 1000 lbs per sq ft, of 

the arch. Such arches have since come into common use instead of brick, lor 

fireproof floors. 

Curved roofs of 25 to 30 ft span, rising about % span, may lie made 
of ordinary corrugated iron of Nos 16 to 13. riveted as usual; and having no acces¬ 
sories except tie-rods a few feet apart; continuous angle-iron skewbacks; and thin 
vertical rods to prevent the ties from sagging. 




* Without letting the deflection exceed % inch ; which was prevented by a stop under the sheet, 
t Marshall, Bros & Co, Flout St aud Girard Ave, Phila. 
















































WEIGHT OF METALS 


405 


Dimensions, weights, and Zisf-prices of standard sizes of welded wrouiglit-iron 
l>i|»e,* for steam, gas, and water; (for boiler tubes, see lower table) usually 
in lengths of about 18 ft. Other sizes and lengths made to order, at extra prices. 
Pipes from ]/^ inch to 1(4 ins inner diani (“nominal ’*) are usually “ 6«ZZ-wolde(i 
larger sizes, “Zap-welded.” Discounts, 1886; on butt-welded,about 25 to !30 per 
cent; on lap-welded, about 45 to 50 per eeut. 


Inuer Diam. 

A 

| Thickness. 

L 

Wt per foot. 

Threads j>er 
| iuch of screw. | 

Price per ft run. 1 

Adopted June 11, 
1884. 

Inner Diam. 

Thickness. 

i 

i 

Wt per foot. | 

Threads per 
iuch of screw. | 

Price p« 

r ft run. 

June 11, 
84. 


'3 

Nominal.* 

1 

j Actual. 

Adopted 

18 

a 

c 

55 

o 

< 

Plain. 

Galv'd. 

Plain. 

Galv’d. 

Ius. 

Ins. 

las. 

I.bs 


$ 

$ 

Ins. 

Ins. 

Ins. 

l.bs. 


$ 

$ 

% 

.270 

.068 

.24 

27 

• 03 X 


4 

4.026 

.237 

10.66 

8 

.83 

1. 00 

hi 

.361 

•0h8 

.42 

18 

.03 X 

.05 

IX 

4.508 

.246 

12.34 

8 

1.00 

1.25 

% 

.494 

.091 

.56 

18 

.03% 

.05% 

5 

5.045 

.259 

14.50 

8 

1.20 

1.50 

X 

.623 

.109 

.84 

14 

.04% 

.06 

6 

6 065 

.280 

18.76 

8 

1.50 

2.00 

% 

.824 

.113 

1.12 

14 

.06 

.08 

7 

7.023 

.301 

23.27 

8 

2.00 


1 

1.048 

.134 

1.67 

U'A 

.08 

.10% 

8 

7.982 

.322 

28.18 

8 

2.75 


l'4 

1.380 

.140 

2.24 

nx 

.11 

.14 

9 

9.001 

.344 

33.70 

8 

3.70 


IX 

1.611 

.145 

2.68 

\\X 

.21 

.24 

10 

10 019 

.366 

40.06 

8 

4.75 


2 

2.067 

.154 

3.61 

liX 

.26 

.30 

11 

11. 

.375 

45.02 

8 

5.75 


•IX 

2.468 

.204 

5 74 

8 

.42 

.47 

12 

12. 

.375 

49.00 

8 

6.50 


3 

3.067 

.217 

7.54 

8 

.55 

.62 

13 

13.25 

.375 

54.00 

8 

7.75 


3% 

3.548 

.226 

9.00 

8 

.67 

.83 

14 

14.25 

.375 

58.00 

8 

9 00 









15 

15.25 

.375 

61.00 

8 

10.00 



Fitting* for Wrought-iron Pipes. 1, Elbow. 2, Service Elbow. 
3, Elbow with side outlet. 4, Reducing T. 5. T. 6, Reducing Cross. 7, Reducing 
i Coupling or Socket. 8, Return Rend with side outlet. 9, Return Rend with back 
outlet. 10, Cross. 11, Flange Union. 12, Oval Flange. 13, Plug. 



Dimensions, weights, and ZZsZ-p rices of standard sizes of lap-welded wrouglit- 
iron boiler tubes,* in lengths up to 20 ft. Other sizes and lengths made to 
' order, at extra prices. Discount, 1886, about 40 to 45 per cent. 


Outer 

Dia4 

Ths 

Ths. 

Wt 
per ft. 

Priced 
per ft. 

Outer 

Dia.J 

Ths 

Ths. 

Wt 
per ft. 

Price § 
per ft. 

Outer 

Dia.J 

Ths 

Ths. 

Wt 
per ft. 

Pricey 
per ft. 

Ins. 

Ins. 

B’gm 
w. ga. 

Lbs. 

$ 

Ins. 

Ins. 

B’gm 
w. ga. 

Lbs. 

$ 

Ins. 

Ins. 

B’gm 
w. ga. 

Lbs. 

S 

1 

.072 

15 

.70 

.23 

.3 

.109 

12 

3.33 

.34 

8 

.165 

S 

13.65 

1.85 

1 % 

.072 

15 

.90 

.23 

3% 

. 120 

11 

3.96 

.38 

9 

.180 

7 

16.76 

2.25 

1% 

.083 

14 

1.24 

.23 

3% 

.120 

11 

4.28 

.43 

10 

.203 

6 

21.00 

2.75 

l% 

.095 

13 

1.66 

.22 

3% 

120 

11 

4.60 

.45 

11 

.220 

5 

25.00 

3.25 

2 

.095 

13 

1 91 

.22 

4 

134 

10 

5.17 

.52 

12 

.229 

4% 

28.50 

3.55 

2 % 

.095 

13 

2.16 

.25 

4% 

.134 

10 

6.17 

.60 

13 

.238 

4 

32.06 

4.20 

2% 

.109 

12 

2.75 

.28 

5 

.118 

9 

7.58 

.72 

14 

.248 

3% 

36.00 

4.75 

2% 

.109 

12 

3.04 

.31 

6 ' 

.165 

8 

10.16 

1.00 

15 

.259 

3 

40.60 

5.75 





7 

.165 

8 

11.90 

1.45 

16 

.270 

2% 

45.20 

6 75 




* The Allison Manufacturing Co, 32i1 and Walnut Sts, Plain, manufacture pipes, tubes, 
fittings, railroad cars and furnish appliances for roofs, buildings, and bridges, railroad 

supplies, &c. dice. 

Morris, Tasker «fe Co, Limited* Pascal Iron Works, Phila, manufacture pipes and tubes, 
and their fittings, iron and brass valves and cocks, machines and tools for manipulation of pipes &o. 

f In ordering pipes, give the •• nominal ” inner diameter. It is merely an arbitrary name for the 
pipe, and, in some cases, tends to mislead. Thus, the pipe whose "nominal ’ inner diaui is oue- 
eighth iuch, has an “ actual ” inner diani of full quarter iueb. 

J In ordering boiler tubes, give the outer diameter. 

§ Adopted December 17, 1883. _ 






































































































































































406 


BOLTS, NUTS, WASHERS. 


Bolts, nuts, and washers. The following dimen¬ 
sions for bolts and nuts were proposed by Mr. Wm. Sellers 
and adopted by the Franklin Institute, of Philadelphia, as a 
standard, in 1864, and by the U. S. Navy Dept * in 1868. They 
have been adopted by the principal machinists of the country, 
and are known as “ Franklin Institute Standard,” 
or “ Sellers,” dimensions. The angle a, Fig 1, between 
the two sides of a thread, is 60°. w is the width, (measured 
lengthwise of the bolt) of the flat top and bottom of each 
thread. N is the number of threads per inch of length of 
screw. 



D 

ins 

d 

ins 

w 

ins 

N 

D 

ins 

d 

ins 

w 

ins 

N 

D 

ins 

d 

ins 

w 

ins 

N 

D 

ins 

d 

ins 

w 

ins 

IV 

X 

.185 

.0062 

20 

1 

.837 

.0156 

8 

2 

1.712 

.0277 

4X 

4 

3.567 

.0413 

3 

5-16 

.240 

.0074 

18 

IX 

.940 

.0178 

7 

2 X 

1.962 

.0277 

4 X 

4X 

3.798 

.0435 

2% 

% 

.294 

.0078 

16 

i X 

1.065 

.0178 

7 

2 X 

2.176 

.0312 

4 

4 X 

4.028 

.0454 

2 : 4 

7-16 

.344 

.0089 

14 

i% 

1.160 

.0208 

6 

m 

2.426 

.0312 

4 

4 4 

4.256 

.0476 

2% 

X 

.400 

.0096 

13 

i u 

1.284 

.0208 

6 

3 

2.629 

0357 

3 X 

5 

4.480 

.0500 

2 X 

9-16 

.454 

.0104 

12 


1.389 

.0227 

5% 

3X 

2.879 

.0357 


5>4 

4.730 

.0500 

2 X 

% 

.507 

.0113 

11 

IX 

1.491 

.0250 

5 

3X 

3.100 

.0384 


•5X 

4.953 

.0526 

2 % 

X 

.620 

.0125 

10 

1 % 

1.616 

.0250 

5 

3% 

3.317 

.0413 

3 

5 % 

5.203 

.0526 

2% 

% 

.731 

.0138 

9 









6 

5.423 

.0555 24 


Dimensions of Heads and Nuts. 


X 

H (in head) 
H (in nut) 


Rous-li. 

IX D + l A i* 1011 - 


Finished. 

1]4 B + 1-16 inch. 
D — 1-16 inch. 

it <( 



In the. Whitworth (English') standard thread, the angle a, Fig 1, is 55°. 
The tops and bottoms of the threads are rounded, instead of flat as in the American 
standards. The number (N) of threads per inch is the same as above for diams of 
bolt up to three ins, except for D = % inch; where N = 12. 


Plate-iron washers. Standard sizes. Diams of washers and bolt-holes 

in ins. Approx thickness by Birmingham wire gauge, p 410. Approx number in one ft. 


Diams. 

Ths. 

No. 


Diams. 

Ths. 

No. 


Diams. 

Ths. 

No. 

X 

X 

18 

450 


IX 

x 

.14 

43 


•2X i 15-16 

9 

8.6 

% 

5 16 

16 

210 


m 

9-16 

12 

26 


•IX 1 1-16 

9 

6.2 

% 

5-16 

16 

139 


ix 

% 

12 

22.5 


m 1 h 

9 

5.2 

% 

% 

16 

112 


m 

11-16 

10 

13.1 


3 1 % 

9 

4. 

] 

7-16 

U 

68 


2 

13-16 

10 

10.1 


■iX 1 1 X 

9 

2.8 


Price in Philadelphia, 1886, about 5 to 6 cts per ft for outer diams from 1 X to 3 X ins. 


* Except that the Navy Dept uses, for both rough and finished heads and nuts , the 
dimensions given above for rough heads ami nuts. The standard of the Navy Dept 
is known as “ United States Standard.” 



































































BOI.TS, NUTS, WASHERS 


407 


A square heal and nut together, weigh about as much as a length of the bolt equal to 7 or 8 times 
D. Hexagon, 6 or 7. 

With the above dimensions a bolt will generally fail by breaking 
off between the head and the nut, where the diameter is decreased 
by cutting the thread, rather than by stripping off its threads. 

The diam ]> of the thread must of course be greater 

than that required to bear safely the proposed tensile strain, by an amount 
equal to twice the depth of the thread. The waste of iron, which would 
result from making the entire bolt of this greater diam, is frequently 
avoided by making the bolt from a bar of only sufficient dimensions to bear 

the strain safely, and upsetting- its ends as in Fig 3, 
thus increasing their diam sufficiently to allow for the cutting of the 
threads. But see Rein, p 408. 

In carpentry, as well as in ties for masonry, washers, w w, of either cast 
or wrought iron, are placed between the timber, or stone, and the head 
and nut: in order to distribute the pressure over a greater surface, and 
thus prevent crushing; especially in timber. 

When much strained against wood, the side 

j of a square wrought-iron washer; or the diam ww of a circular one, should uot be less than 4 diams 
| of the screw, as in the fig; and its thickness, tw, X diam at least. 

Two such square washers will together weigh as much as 18 diams in 
I length of a round rod of the same diam as the screw. Two round 
| washers will weigh together as much as 14 diams of rod of same diam 

( as screw. In either case, a square head and nut will weigh as much 
as 6 diameters. Cast-iron washers, being more apt to split under 
heavy strains, may be made about twice as thick as wrought oues. 

When the strain is very great, the diam of the washer may be 5 or 
6 times that of the screw; and its thickness equal to diam; but 4 
diams will suffice for most practical purposes, or even 2.5 when there 
is but little strain, and the thickness may then be but .1 or .2 diam of 
bolt. 

Table of machine and car bolts, with 

square and hexagon heads and nuts, Figs 4 and 5 ; made by Hoopes 
& Townsend, 1330 Buttonwood St, Phila. All their bolts 
are cut with IT. S. Standard threads, as Fig-. 4. Fig. 5. 
per table on p 406, unless otherwise ordered. For washers, see p 406. 




Diam D, Fig 1, of 
bolt, ins. 

Length 
ins ex¬ 
clusive 
of head. 

Weight of 
100 bolts 
with nuts, 
in fits. 

List price* 
in dollars 
per 100. 

Size of sq 
head, ins. 

Size of hex 
head, ins. 

Size of 
nut, ins. 

Chamfered 

Square 

Nuts. 

Chamfered 

Hexagon 

Nuts. 

Min. 

Max. 

Min. 

Max. 

Min. 

Max. 

Width. 

Thickn's. 

•qiPLii 1 

1 

Thickn’s. 

Width. 

m 

.2 

2 

H 

No. per 

100 B>s. 

List pricet 
in cts per 
lb. 

No. per 

100 lbs. 

List pricet 
in cts per 
lb. 

X 

\x 

8 

4 

13.75 

2.80 

4.10 

7-16 

3-16 

7-16 

X 

X 

X 

7300 

25 

8200 

35 

5-16 

i< 

a 

7 

20.75 

3.20 

5.15 

X 

X 

X 

5 16 

19-32 

5-16 

4700 

23 

5400 

33 

% 

u 

12 

10.5 

44.5 

3.60 

7.80 

% 

5-16 

% 

% 

11-16 

S /8 

2550 

19 

3000 

26 

7-16 

n 

a 

15.2 

61.4 

4.60 

10.90 

11-16 

% 

11-16 

7-16 

25-32 

7-16 

1700 

19 

2030 

26 

X 

a 

20 

22.5 

123.2 

5.00 

16.10 

13-16 

7-16 

13-16 

X 

X 

X 

1180 

15 X 

1400 

20 

9-16 

11 

1 t 

30 

159.5 

7.20 

25.75 

Vs 

X 

Vs 

9-16 

31-32 

9-16 

890 

MX 

1080 

20 

% 

u 

1 i 

39.5 

196.5 

7 20 

25.75 

1 

X 

1 

% 

1 1-16 

% 

645 

1*X 

780 

18K 

H 

a 

it 

63 

286.8 

10.50 

32.70 

1 3-16 

% 

1 3-16 

3 4 

IK 

% 

380 

MX 

470 

16K 

X 

a 

ti 

100 

415.3 

14.90 

46.40 

Ws 

% 

1% 

Ys 

1 7-16 

Vs 

260 

MX 

310 

16H 

1 

a 

it 

153 

558. 

22.00 

62.70 

IX 

Vs 

1 9-16 

l 

Ws 

1 

172 

13 

212 

16 


* For bolts with square heads and nuts. Bolts with hex heads and nuts are about 20 per cent 
higher. Discount, 1886, about 75 per cent. Bolts are also made (at extra prices) 
with button-sliaped and countersunk heads. The price per 100 varies 
with the length. We give the extremes. The bolts have finished points, and chamfered heads and 

nuts, as shown. From inch to 8 ins the lengths increase by ins; from 

8 ins to 20 ins, by ins. 

f Not tapped. Discount, 1886, about 9% cts per lb. 

% Not tapped. Discount, 1886, about 10 cts per lb. 
































































































408 


BOLTS, NUTS, WASHERS, 


Lock-nut washers. When bolts are subjected to much rough 

jolting, as at rail-joints. Ac, the nuts are liable to wear loose, and unscrew 
themselves. On railroads this is a source of great annoyance, and innumerable 
devices for preventing it have been tried. The Verona lock-nut 
washer is a simple circular washer made of steel; with a slit s s cut through it, leaving 
sharp edges. On one side, a, of the slit, the metal is pressed upward about % inch; 
aud that on the other side, c, downward, the same distance; so that a perspective 
view would be somewhat as at t . Now, when the nut is screwed down over the 

washer, in the direction of the arrow, the slit offers no obstruction; but if the nut 

afterward tends to unscrew itself, the sharp upper edge of the slit, along a, presents 
friction against the bottom of the nut, which tends to hold it in place. Besides, the 
washer, by its elasticity, tends to resume its original shape, and thus presses the 
threads of the nut against those of the bolt; aud the additional friction thus produced, also aids in 
holding the uut. The same principles are employed in a nut-lock recently (1884) introduced upon 
the Houston and Texas Central, where the lock-nut washer is a long 
strip of steel, with two holes, each of which has its edges formed like those of a Verona washer, 
and through each of which passes one of the bolts of the rail joint. Anotlier 
device is to cut at the end of the screw a few threads of a screw of less diam 

than the main one, and in the opposite direction. The nut is then screwed upon the larger diam; 

and after it the lock-nut is screwed in the other direction upon the smaller diam, until it comes 
into contact with the main nut. In the Smith lock-nut bolt, this second nut is 
only about inch thick ; and aftei being driven home, one of its corners is bent over the edge of 
the main nut. These bolts cost, in 1884, about 5 cts each. 

See the Cambria nut-lock, p 765, 

The Atwood lock-nuts take advantage of elasticity in the nut itself, which is 
obtained either by slitting the nut, or hy reducing its thickness near the bolt hole. 

It is claimed that if the threads of an ordinary bolt and nut are carefully cut, 
so as to be in contact with each other throughout, no lock-nut 
contrivance is necessary, because the friction between the two threads is distributed over a larger 
surf, and abrasion does' not take place so readily as if the threads touched each other at only a few 
points. The nuts are therefore less apt to wear loose under repeated jarring. 

Owing to the difficulty of obtaining such perfect fitting bolts and nuts, due to the wear of the cut¬ 
ting tools used in their mfr, the Harvey Screw and Rolt i'o, 52 Wall St, 
New York, furnish bolts and nuts in which the thread on the bolt differs slightly in shape from that 
in the nut. They also furnish nuts in which the thread, instead of being of uuiforni shape through¬ 
out, gradually becomes deeper and thicker, by having its side angle a, Fig 1, p 406, made more 
acute, and its top truncated. These nuts are used with bolts having the usual uniform thread. 
The bolt enters the nut upon the side where the thread is of the same shape as its own; but its 
thread encounters, and is forced into, the gradually narrowing and deepening path between the 
threads of the nut. In both devices, the enforced conformity between the two threads, is relied 
upon to give the desired completeness of contact between them. The greater force required in screw¬ 
ing on the nut, also increases the friction between the threads. 

Table of diameters, weights, and approximate breaking: 
strains, for round rolled iron bolts, ties, or bars; assuming the 

breaking strain per square inch of average quality of rolled iron to be as follows: Up to 1 inch 
square, or 1 inch diam, ‘20 tous, or 44800 lbs ; from 1 to 2 ins sq or diam. 19 tons : 2 to 3 ins, 18 tons ; 
3 to 4 ins, 17 tons ; 4 to 5 ins, 16 tons ; 5 to 6 ins, 15 tons. The first 4 columns of the table are to 
be used when the screw end of the bolt is enlarged or upset, so that the shank or body of the bolt 
shall not be weakened by the cutting of the screw threads. But when the shank is so weakened, the 
diam and weight of the bolt must be taken from the last 2 cols. 

Rem. But it is very important to know that a long upset rod is no 
stronger than one not upset, against slowly applied loads or strains. Both will 
then break at about midlength, under equal pulls. Therefore in such cases the 
col of greatest diams in the table should be used. 

Example 1. To find the diam of a bolt, that shall just break under a strain, or a load of 52.5 tons, 
we see by the table and opposite 52.5 tons, that it will be 1 % ins if the screw end is enlarged, and 2.3 
ins if it is not. In the first case, the weight of the bolt will be 9.3 lbs per foot run ; aud in the second, 
13.8 lbs. 

Example 2. What must be the diam in order to sustain a strain of 52.5 tons, with a safety of 3 7 
Here 52.5 X 3 = 157.5 tons. In the table, the nearest we find to 157.5 tons, is 1613.6; opposite w hich 
we find the diam 3^ ius. A diam a trifle less than this will break under a straiu of 157.5 tons; and 
consequently will have a safety of 3 for 52.5 tons. 

Tliebreakgr strains in this table will also answer for square 

Saks, by merely increasing them % putt; for a round bar has very approximately ^ of the strength 
of a square one whose side is equal to the diam of the round one; or the square one has 1H times the 
Strength of the round one; or, more correctly, as 1 to .7854. For the Strength of 
COFFFiR bolts, multiply the tabular ones by the decimal .8; and for their weight, increase, 
that of iron ones ^ part. Heads, nuts, and washers, are not included in the table. 



WEIGHT OF METALS 


409 


WEIGHT AND STRENGTH OF IRON ROETS. (Original.) 
For square ones or for copper see preceding paragraph. 


Ends enlarged, or upset. 

Ends not 
enlarged. 

Ends enlarged, or upset. 

Ends not 
enlarged. 

Diam. 

Weight 

Break- 

Break- 

Diam. 

Weight 

Diam. 

Weight 

Break- 

Break- 

Diam. 

Weight 

of 

per foot 

ing 

ing 

of 

per foot 

of 

per foot 

ing 

ing 

of 

per foot 

shall k 

run. 

strain. 

strain. 

shank 

run. 

shank 

run. 

strain. 

strain. 

shank 

ruu. 

Ins. 

Pds. 

Tons. 

Pds. 

Ins. 

Pds. 

Ins. 

Pds. 

Tons. 

Pds. 

Ins, 

Pds. 

R 

.0414 

.245 

549 



If 

8.10 

45.7 

102368 

2.14 

12.0 

t 3 s 

.093 

.553 

1239 



HI 

8.69 

49.0 

109760 

2.22 

12.9 


.165 

.983 

2202 

.35 

.321 

11 

9.30 

52.5 

117600 

2.30 

13.8 

TS 

.258 

1.53 

3427 

.43 

.452 

i*§ 

9.93 

56.0 

125440 

2.38 

14.7 

3 

S 

.372 

2.21 

4950 

.50 

.654 

2. 

10.6 

59.7 

133728 

2.45 

15.7 

7 

T5 

.506 

3.00 

6720 

.58 

.897 


12.0 

63.8 

142912 

2.59 

17.5 


.661 

3.93 

8803 

.66 

1.14 

24 

13.4 

71.6 

100384 

2 73 

19.5 

T®5 

.837 

4.97 

11133 

.73 

1.41 

2f 

14.9 

79.7 

178528 

2.88 

21.6 

5 

S 

1.03 

6.14 

13754 

.80 

1.67 

2§ 

16.5 

88.4 

198016 

3.02 

23.9 

H 

1.25 

7.42 

16621 

.88 

2.03 

2f 

18.2 

97.4 

218170 

3.16 

26.1 


1.49 

8.83 

19779 

.96 

2.41 

2? 

20.0 

106.9 

239156 

3.30 

28.5 

U 

1.75 

10.4 

23296 

1.04 

2.81 

2| 

21.9 

116.8 

261632 

3.45 

31.1 

7 

n 

2.03 

12.0 

268S0 

1.12 

3.26 

3. 

23.8 

127.2 

284928 

3.60 

33.9 

1 5 
IS 

2.33 

13.8 

30912 

1.20 

3.77 

H 

27.9 

141.0 

315840 

3.86 

39.1 

1 in. 

2 65 

15.7 

35108 

1.27 

4.27 

H 

32.4 

163.6 

366464 

4.12 

44.4 

^ TS 

2.99 

16.8 

37632 

1.35 

4.77 

H 

37.2 

1S7.7 

420448 

4.41 

51.0 

H 

3.35 

18.9 

42336 

1.42 

5.28 

4. 

42.3 

213.6 

478464 

4.70 

57.8 

h% 

3.73 

21.1 

47261 

1.49 

5.81 

H 

47.8 

227.0 

508480 

4.98 

65.2 

li 

4.13 

23.3 

52192 

1.55 

6.39 

H 

53.6 

251.5 

570080 

5.25 

72.9 

h 5 5 

4.56 

25.7 

57568 

1.64 

7.04 


59.7 

283.5 

635040 

5.53 

80.5 

if 

5.00 

28.2 

63108 

1.72 

7.74 

5. 

66.1 

314.2 

703808 

5.80 

88.1 

1 7 

5.47 

30.8 

68992 

1.80 

8.48 

H 

72.9 

324.7 

727328 

6.08 

97.0 

l* 

5.95 

33.6 

7526* 

1.87 

9.20 


80.0 

356.4 

798336 

6.36 

106. 

1 T 9 n 

6.46 

36.4 

81536 

1.94 

9 88 

5f 

757.5 

3S9.5 

872480 

6.63 

116. 

1 f 

6.99 

39.4 

88256 

2.00 

10.6 

6. 

95.2 

424.1 

949984 

6.90 

126. 

in 

7.53 

42.5 

9520() 

2.07 

11.3 


See Rem, p 408. 



BUCKLED PLATES 


of iron or steel are usually 3 or 4 feet square, from l-20th to3-8th inch thick, with 
a flat rim about 2 inches wide all around, with rivet or bolt holes for holding the 
plate firmly down to its intended place. The rest of the plate is stamped into the 
form of a kind of groined arch rising from 1 to 3 inches in the center. They are 
very strong, and are used for the floors of fire-proof buildings, and of city iron 
bridges, covered with asphalt or stone paving, &c. One of 3 feet square, .25 
inch thick, curved 1.75 inches, and with a 2-inch rim well bolted down on all 
sides, required a quiet, equally distributed load of 18 tons to crush it. When 
unbolted the strength is only half as great. Ruckled plates of soft 
puddled steel bear nearly twice as much. 

Table of safe, quiet, uniformly distributed loads for buckled 
iron plates 3 feet square, arched 1.75 inches, and well bolted down on all sides. 
Keystone Rridge Co, Pittsburgh, Pa. Prices, iron 3% cents per lb at 
mill; steel, 4. 


Thickness. 

Weight of 
one plate. 

Safe load on one plate 
(= one-fourth of ultimate load). 

Birmingham 
wire gauge. 

Inch. 

Pounds. 

Tons of 

2240 lbs. 

Pounds. 

18 

.048 

17 

.27 

604 

16 

.066 

24 

.43 

963 

12 

.107 

39 

.64 

1433 


1-8 th 

45 

1.00 

2240 


3-16th 

68 

2.5 

5600 


l-4th 

90 

4.5 

10080 


5-16th 

113 

6.2 

13888 


3-8 th 

135 

9.0 

20160 

























































410 


WIRE GAUGES 


The Birmingham wire gauge is the one in most general use for iron. The 
new British w g went into effect March 1st 1884. In the “ American ” w g of Dar¬ 
ling, Brown & Sharpe, Providence R. I., each diam or thick is = the next smaller 
one X 1.122932. We take the wt of wrot iron per cub ft at 485 lbs in the first two; 
and at 486 in the last. For the wt of steel, mult that of iron by 1.01. For 
load, mult iron by 1.46. For zine, mult iron by .9. For brass (approx), mult 
iron by 1.06. For copper, mult iron by 1.134. Seep411 and Trenton Gauge p412. 


Birmingham W. Ga. 


New British W. Ga. 


No. 

Diam of 
wire, or 
thickness 
of sheet, 
ins. 

Wt of 
iron wire, 
in lbs per 
lin ft. 

Wt of 
iron 
sheets, 
in lbs per 
sq ft. 

Diam of 
wire, or 
thickness 
of sheet, 
ins. 

Wt of 
iron wire, 
in lbs per 
lin ft. 

Wt of 
iron 
sheets, 
in lbs per 
sq ft. 

Diam of 
wire, or 
thickness 
of sheet, 
ins. 

Wt of 
iron wire, 
in lbs per 
lin ft. 

Wt of 
iron 
sheets, 
in lbs 
persqft 

7-0 




.500 

.661 

20.21 




6—0 




.464 

.569 

18.75 




5-0 




.432 

.494 

17.46 




4-0 

.454 

.546 

18.35 

.400 

.423 

16.17 

.400000 

.561 

18.63 

3-0 

.425 

.479 

17.18 

.372 

.366 

15.03 

.409642 

.445 

16.58 

2-0 

.380 

.383 

15.36 

.348 

.320 

14.06 

.304790 

.353 

14.77 

0 

.340 

.306 

13.74 

.324 

.278 

13.09 

.324*61 

.280 

13.15 

1 

.300 

.238 

12.13 

.300 

.238 

12.13 

.289297 

.222 

11.70 

2 

.284 

.214 

11.48 

.276 

.202 

11.15 

.257027 

.176 

10.43 

3 

.259 

.178 

10.47 

.252 

.168 

10.19 

.229423 

.139 

9.291 

4 

.238 

.150 

9.619 

.232 

.142 

9.377 

.204307 

.111 

8.273 

5 

.220 

.128 

8.892 

.212 

.119 

8.568 

.181940 

.0877 

7.366 

6 

.203 

.109 

8.205 

.192 

.0976 

7.760 

.162023 

.0696 

6.561 

i 

.180 

.0859 

7.275 

.176 

.0820 

7.113 

.144285 

.0552 

5 842 

8 

.165 

.0721 

6.669 

.160 

.0677 

6.466 

.128490 

.0438 

5.203 

9 

.148 

.0580 

5.9S1 

.144 

.0548 

5.820 

.114423 

.0347 

4.633 

10 

.134 

.0476 

5.416 

.128 

.0434 

5.173 

.101897 

.0275 

4.125 

11 

.120 

.0382 

4.850 

.116 

.0357 

4.688 

.090742 

.0218 

3.674 

12 

.109 

.0315 

4.405 

.104 

.02S6 

4.203 

.080808 

.0173 

3.272 

13 

.095 

.0239 

3.840 

.092 

.0224 

3.718 

.071962 

.0137 

2.914 

14 

.083 

.0183 

3.355 

.080 

.0169 

3.233 

.064084 

.0109 

2.595 

15 

.072 

.0137 

2.910 

.072 

.0137 

2.910 

.057008 

.00863 

2.310 

16 

.065 

.0112 

2.627 

.064 

.0108 

2.587 

.050821 

.006S4 

2.053 

17 

.058 

.00891 

2.344 

.056 

.00832 

2.263 

.045257 

.00543 

1.832 

18 

.049 

.00636 

1.9S0 

.048 

.00610 

1.940 

.040303 

.00430 

1.631 

19 

.042 

.00467 

1.697 

.040 

.00423 

1.617 

.035890 

.00341 

1.452 

20 

.033 

.00325 

1.415 

.036 

.00344 

1.455 

.031961 

.00271 

1.293 

21 

.032 

.00271 

1.293 

.032 

.00269 

1.293 

.028462 

.00215 

1.152 

22 

.028 

.00208 

1.132 

.028 

.00207 

1.132 

.025346 

.00170 

1.026 

23 

.025 

.00166 

1.010 

.024 

.00152 

.9700 

.022572 

.00135 

.913 

24 

.022 

.00128 

.8892 

.022 

.00128 

.8891 

.020101 

.00107 

.814 

25 

.020 

.00106 

.8083 

.020 

.00106 

.8083 

.017900 

.000849 

.724 

26 

.018 

.000859 

.7225 

.018 

.000857 

.7275 

.015941 

.000673 

.644 

27 

.016 

.000678 

.6467 

.0164 

.000712 

.6628 

.014195 

.000534 

.574 

28 

.014 

.000519 

.5658 

.0148 

.000579 

.5982 

.012641 

.000423 

.511 

29 

.013 

.000448 

.5254 

.0136 

.000489 

.5497 

.011257 

.000336 

.455 

30 

.012 

.000382 

.4850 

.0124 

.000408 

.5012 

.010025 

.000266 

.405 

31 

.010 

.000265 

.4042 

.0116 

.000357 

.4688 

.008928 

.000211 

.360 

32 

.009 

.000215 

.3638 

.0108 

.000309 

.4365 

.007950 

.000167 

.321 

33 

.008 

.000170 

.3233 

.0100 

.000265 

.4042 

.007080 

.000133 

.286, 

34 

.007 

.000130 

.2829 

.0092 

.000224 

.3718 

.006305 

.000105 

.254 

35 

.005 

.0000662 

.2021 

.0084 

.000187 

.3395 

.005615 

.0000837 

.226 

36 

.004 

.0000424 

.1617 

.0076 

.000153 

.3072 

.005000 

.0000062 

.202 

37 




.0068 

.000122 

.2748 

.004453 

.0000525 

.180 

38 




.0060 

.0000952 

.2425 

.003965 

.0000417 

.159 

39 




.0052 

.0000714 

.2102 

.003531 

.0000330 

.142 

40 




.0048 

.0000608 

.1940 

.003144 

.0000262 

.127 

41 




.0044 

0000513 

177ft 




42 




.0040 

0000423 

1617 




43 




.0036 

0000344 





44 




.0032 

0000271 

1902 




45 




.0028 

0000207 

11 :*9 




46 




.0024 

0000152 

0070 




47 




.0020 

0000106 

OftOft 




48 




.0016 

0000068 

0617 




49 




.0012 

0000038 

oift*; 




50 




.0010 

.0000026 

.0404 





American W. Ga. 











































WIRE GAUGES. 


411 


No tratio Stupidity is more thoroughly senseless than the adherence to 
the various Birmingham, Lancashire, &c, gauges; instead of at once denoting the 
thickness and diameter of sheets, wire, &c, by the parts of an inch; as has long 
been suggested. Thus, No. or No. ^ wire, or sheet-metal of any kind, should 
be understood to mean }/% or ifo of au inch diam, or thickness. To avoid mistakes, 
which are very apt to occur from the number of gauges in use; and from the absurd 
practice of applying the same No. to different thicknesses of different metals, in dif¬ 
ferent towns, it is best to ignore them all; and in giving orders, to define the diam¬ 
eter of wire, and the thickness of sheet-metal, by parts of an inch. Or the weight 
per hundred ft for wire; or per sq ft for sheets, may be employed. We believe that 
the foregoing Birmingham gauge applies to zinc, copper, brass, and lead; although 
it is generally stated to be for iron and steel only. Another Birmingham gauge it 
used for sheet-brass, gold, silver, and some other metals; but we have never seen it 
stated what those others are. There are different gauges even for wire to he used 
for different purposes; and various firms have gauges of their own ; not even accord¬ 
ing among themselves. 

As Mr. Stubs makes various English gauges, the term “ Stubs gauge ” by 
itself means nothing. Generally, however, in our machine shops, it applies to the 
Birmingham gauge of the preceding table. 

Birmingham gauge for sheet Brass. Silver, Gold, and all metals 

except iron and steel ? 


No. 

Thickn’s. 

No. 

Thickn’s. 

No. 

Thickn's. 

No. 

Thickn’s. 

No. 

Thickn's. 

No. 

Thickn’s. 


Inch 


Inch 


Inch 


Inch 


Inch 


Inch 

1 

.004 

7 

.015 

13 

.036 

19 

.064 

25 

.095 

31 

.133 

2 

.005 

8 

.016 

14 

.041 

20 

.067 

26 

.103 

32 

.143 

3 

.008 

9 

.019 

15 

.047 

21 

.072 

27 

.113 

33 

.145 

4 

.010 

10 

.024 

16 

.051 

22 

.074 

28 

.120 

34 

.143 

5 

.012 

11 

.029 

17 

.057 

23 

.077 

29 

.124 

35 

.1:8 

6 

.013 

12 

.034 

|l8 

.061 

24 

.082 

30 

.126 

36 

.167 


The mills rolling sheet iron in the United States generally 
use the following, which varies slightly from the Birmingham gauge: 


No. 

lbs per 
sq ft. 

No. 

1"S per 
sq ft 

No. 

lbs per 
sq ft 

No. 

lbs per 
sq ft 

1 

12 50 

8 

6.86 

15 

2.81 

22 

1.25 

2 

12.00 

9 

6.24 

16 

2.50 

23 

1.12 

3 

11.00 

10 

5.62 

17 

2.18 

24 

1.00 

4 

10.00 

11 

5.00 

18 

1.86 

25 

.90 

5 

8.75 

12 

4.38 

19 

1.70 

26 

.80 

6 

8.12 

13 

3.75 

20 

1.54 

27 

.72 

7 

7.50 

14 

3.12 

21 

1.40 

28 

.64 


Price of brass and copper wire, approximate for 1886. Ansonia 
Brass and Copper €!o. Nos. 19 and *21 Cliff St, New York. For 100 lbs or 
more: 

Nos. 0 to 25 Copper.30 to 40 cts per lb. 

“ “ High brass.22 to 32 “ “ “ 

“ “ Low brass.26 to 36 “ “ “ 


Unannealed or hard brass wire has about ^ths the strengths of the table 
p412, and about i more weight. If annealed, only full half the strength. 

Hard copper wire may be taken at %of the tabular strengths, and full 
^ more weight. 







































412 


IRON WIRE. 


Table of Charcoal Iron Wire made by Trenton Iron Co., 

Trenton, N. J. The numbers in the first column are those of the Trenton Iron 
Co's gan^e. The corresponding diameters in the second column will be seen to 
be somewhat less than those of the Birmingham, gauge, p 410. 


No. 

Diam. 

ins. 

Lineal 
feet to the 
Pound. 

Tensile 

Str’gtli 

Approx 

lbs. 

No. 

Diam. 

ius. 

Lineal 
feet to the 
Pound. 

Tensile 

Str’gth 

Approx 

lbs. 

No. 

Diam. 

ins. 

Lineal 
feet to the 
Pouud. 

00000 

.450 

1.863 

12598 

11 

.1175 

27.340 

1010 

26 

.018 

1164.689 

0000 

.400 

2.358 

9955 

12 

.105 

34.219 

810 

27 

.017 

1305.670 

000 

.360 

2.911 

8124 

13 

.0925 

44.092 

631 

28 

.016 

1476.869 

00 

.330 

3.465 

6880 

14 

.080 

58.916 

474 

29 

.015 

1676.989 

0 

.305 

4.057 

5926 

15 

.070 

76.984 

372 

30 

.014 

1925.321 

1 

.285 

4.645 

5226 

16 

.061 

101.488 

292 

31 

.013 

2232.653 

2 

.265 

5.374 

4570 

17 

.0525 

137.174 

222 

32 

.012 

2620.607 

3 

.245 

6.286 

3948 

18 

.045 

186.335 

169 

33 

.011 

3119.092 

4 

225 

7.454 

3374 

19 

.040 

235.084 

137 

34 

.010 

3773.584 

5 

.205 

8.976 

2839 

20 

.035 

308.079 

107 

35 

.0095 

4182.508 

6 

.190 

10.453 

2476 

21 

.031 

392.772 


36 

.009 

4657.728 

7 

.175 

12.322 

2136 

22 

.028 

481.234 


37 

.0085 

5222.035 

8 

.160 

14.736 

1813 

23 

.025 

603.863 


38 

.008 

5896.147 

9 

.145 

17.950 

1507 

24 

.0225 

745.710 


39 

.0075 

6724.291 

10 

.130 

22.333 

1233 

25 

.020 

943.396 


40 

.007 

7698.253 


The wire in this table is supposed to be hard, bright, or unannealed. 
Annealing renders wire more pliable, but less elastic; and reduces its strength per¬ 
haps 20 or 25 per cent. The strengths in the last column are at the 
rate of 81000 lbs per sq inch for No. 0; 98000 for No. 10; and 111000 for No. 20; and 
are say from 15 to 25 per ct greater than those of ordinary market wire. The wires 
in the table are all made of the same specially prepared iron; and the increased 
strength per sq in of the smaller ones is due to their more frequent passage through 
the draw-plates. The choice Swedish and Norway irons would not yield wire of 
much greater strength; while the cost would be about 60 per ct more. 

Hard steel W'ire averages about twice the strength of iron wire. 


To find approximately the number of straight wires that 
can be got into a cable of given diameter. 

Divide the diameter of the cable in inches, by the diameter of a wire in inches. 
Square the quotient. Multiply said square by the decimal .77. The result will be 
correct within about 4 or 5 per cent at most, in a cylindrical cable. 

The solidity, or metal area of all the wires in a cable, will be 

to the area of the cable itself, about as 1 to 1.3. In other words, the area of the 
voids is nearly ^ that of the cable; while that of the wires is fully % that of the 
cable. All approximate. 


Price-list of Charcoal Iron and Bessemer Steel Wire.* Bright 
and annealed. Trenton Iron Co. 


Nos. 0 to 9, 10 & 11, 12, 13 A 14, 

Cts per lb. 10 11, 11% 12% 

Nos. 23, 24, 25, 26, 27, 28, 

Cts per lb. 23, 24, 25, 26, 28, 29, 


15 

& 16, 

17, 

18, 

19, 

20, 

21, 

22, 


14, 

15, 

16, 

19, 

20, 

21, 

22, 

29, 

30, 

31, 

32, 

33, 

34, 

35, 

36, 

30, 

32, 

33, 

35, 

37, 

40, 

45, 

55, 


Price-list of Cast (crucible) Steel Wire.f Trenton Iron Co. 


Nos. 

Cts per H>. 



The Co make other wire of specified qualitv to order; also Wire 
Ropes, and fittings for same. See p 413. 


* Discounts on iron and Bessemer steel win; (bright or annealed) Nos 00000 to 
36, about 60 to 75 per cent. Nos. 37 to 45, about 50 per cent. 1886. 
t Discount on crucible steel wire, about 50 to 60 per cent. 1886 









































WEIGHT AND STRENGTH OF WIRE ROPES. 


413 


ol " lr<1 Rope manufactured by Joint A. RoeblingN Sons 

1Iren ton. N. J. Prices, 1886, net. except on large orders. The prices 
and weights given are for ropes with hemp centers. When made with wire centers 
the prices are one-tenth higher, and the weights one-tenth greater. 


Rope of U3 Wires (19 wires in a strand). 


Trade 

No. 

Diam. 

Ins. 

Cir- 

cumf. 

Ins. 

Pounds 
per foot 
run. 

Breaking 

load, lbs. 

Minimum diam of 
drum, in ft. 

Price per ft run, 
in cents. 



Iron.# 

Cast steel* 

Iron. 

Cast st'l* 

Iron.* 

Ca’tst# 

i 

2K 

6K 

8.00 

148000 

260000 

8 

9 

90 

152 

2 

2 

6 

6.30 

130000 

200000 

7 

8 

67 

120 

3 



5.25 

108000 

156000 

6.5 

7.5 

59 

100 

4 


5 

4.10 

88000 

128000 

5 

6 

50 

80 

5 



3.65 

78000 

110000 

4.75 

5.5 

45 

71 

&'A 

i ; K 

*K 

3.00 

66000 


4.5 


37 


6 

IK 

4 

2.50 

54000 

78000 

4 

5 

31 

50 

7 

iK 

3K 

2.00 

40000 

60000 

3.5 

‘ 4.5 

25 

41 

8 

l 

3 K 

‘ 2 H 

1.58 

32000 

48000 

3 

4 

22 

34 

9 

K 

1.20 

23000 

40000 

2.75 

3.75 

17 

27 

10 

% 

2 K 

0.88 

17280 

26000 

2.5 

3.5 

14 

21 

™K 

1<>K 

% 

2 

0.70 

10260 

18000 

2 

3 

12 

18 

1 9 B 

IK 

0.44 

8540 

13000 

1.75 

2.75 

10 

17 

vM 

K 

IK 

0.35 

6960 

11000 

1.5 

2 

8 

15 

% 

IK 

0.26 

500J 


1 


6.5 



Rope of 49 Wires (7 wires to the strand). 








Price per foot run, 

Trade 

Diam. 

Circumf. 

Pounds per 

Breaking load, lbs. 

in cents. 

No. 

Ins. 

Ins. 

foot run. 





Iron.* 

Cast steel.* 

Iron.* 

Cast st’l* 

ii 

ns 


3.37 

72000 

134000 

41 

70 

12 

i% 

2.77 

60000 

110000 

33 

60 

13 

IK 

$ 

°/8 

2.28 

50000 

90000 

29 

50 

14 

IK 

1.82 

40000 

72000 

■ 23 

40 

15 

l 

3 

1.50 

32000 

60000 

20 

32 

16 

Vs 

2% 

1.12 

246H0 

44000 

16 

25 

17 

y 4 

2% 

0.88 

17600 

34000 

12 

19 

18 

II 

2K 

0.70 

15200 

27000 

10 

16 

19 

1% 

0.57 

11600 

20000 

9 

14 

20 

% 

iK 

0.41 

8200 

16000 

7 

11 

21 

iK 

0.31 

5660 

12000 

6 

8 

22 

S 

i K 

0.23 

4260 


4.5 

... 

23 

iK 

0.19 

3300 

8000 

4 

7 

24 


l 

0.16 

2760 

6000 

3.5 

5 

25 

25 

K 

0.125 

2060 


3 

... 


Xotes on the Use of Wire Rope, by the Roeblings Co. 

The ropes with 19 wires per strand are the most pliable, and therefore best adapted for hoisting 
and running rope. The others are stiffer and better adapted for guys. &c. Ropes of iron or steel, 
up to 8 ins diam, made to order. For the safe working load take one-fifth to one-seventh of 
ttie breaking load, according to speed. Hemp center rope is more pliable than wire center. Wire 
rope must not be coiled or uncoiled like hemp rope. When on a reel, the latter should be mounted 
on a spindle or Hat turn-table to pay off the rope. When forwarded in a small coil without a reel, 
roll it on the ground like a wheel, and thus run off the rope. Avoid untwisting and short bends. To 
preserve wire rope, apply raw linseed oil (which may be mixed with an equal quantity of Span¬ 
ish brown or lamp-black.) with a piece of sheepskin, keeping the wool against the rope. If for use 
in water or under ground, add 1 bushel of fresh slacked lime, and some sawdust, to 1 barrel of 
tar. >ioi 1 the mixture well, and saturate the rope with it while hot. Never use galvanized rope 
for running rope. The grooves of east-iron pulleys should be filled with blocks of well-seasoned 
hard wood.’ set on end. Leather or india-rubber is better where the pulleys are large and run very 
fast. Galvanized wire rope for rigging is cheaper and more durable than hemp rope; and does 
not stretch permanently under great strains. Its bulk is one-sixth, and its weight one-half, that of 
hemp rope. Roebling’s wire rope has been made the standard by the U. States Navy Department. 
Shackles, sockets, swivel-hooks, and fastenings. <fec. furnished aud put on. and splices 
made. Pulley wheels furnished. Also, galvanized steel cables for suspension bridges. 
Crucible cast-steel wire ropes are much more durable than iron ones. They should be kept well lu¬ 
bricated. The foregoing is condensed from the circular of the Roeblings Co. 

* Ropes of Bessemer steel, and of Siemens-Martin steel, are sold at the same prices as iron ropes. 
They have stood higher strains ; but. in view of the lack of uniformity in those steels, it is not advis¬ 
able to reduce their diam below that of iron rope for the same work. 






























































414 WEIGHT AND STRENGTH OF IRON CHAINS. 


On planes in Schuylkill Co. &c, a wire rope generally lasts long enough to raise «ne mil¬ 
lion of tons of coal up a plane half a mile long, and rising 1 in 10. The ordinary duration on inclined 
planes throughout the country is from 1% to 4 years, according to the amount of service; and also 
greatly to the care taken of them, and of the sheaves and rollers upon which they move. 

On the Mt Pisgah plane, 2500 ft loug, rising 6G0 ft, for raising empty coal cars, and lowering loaded 
ones, thin iron bands, 7 inches wide, and ab mt % inch thick, have been used instead of ropes. They 
scarcely exhibit any sign of wear in several years. They should be riveted; beiug apt to break if 
welded. Steel would probably be the best material in many cases. 


Table of Manilla rope. 


Diam. 

Ins. 

Circ. 

Ins. 

Wt per 
foot, 
lbs. 

Breaki 

ng load. 

Diam. 

Ins. 

Circ. 

Ins. 

Wt per 
foot, 
lbs. 

Break 

ing load. 

Tons. 

lbs. 

Tons. 

lbs, 

.239 

% 

.019 

.25 

560 

1.91 

6 

1.19 

11.4 

25536 

.318 

1 

.033 

.35 

784 

2.07 

6K 

1.39 

13.0 

29120 

.477 

IK 

.074 

.70 

1568 

2.23 

4 

1.62 

14.6 

32704 

.636 

2 

.132 

1.21 

2733 

2.39 

7K 

1.86 

16.2 

36288 

.795 

2 K 

.206 

1.91 

4278 

2.55 

8 

2.11 

17.8 

39872 

.955 

3 

.297 

2.73 

6115 

2.86 

9 

2.67 

21.0 

47040 

1.11 

z'A 

.404 

3.81 

8534 

3.18 

10 

3.30 

24.2 

54208 

1.27 

4 

.528 

5.16 

11558 

3.50 

11 

3.99 

27.4 

61376 

1.43 

4K 

.668 

6.60 

147S4 

3.82 

12 

4.75 

30.6 

68544 

1.59 

5 

.825 

8.20 

18368 

4.14 

13 

5.58 

33.8 

75712 

1.75 

&K 

.998 

9.80 

21952 

4.45 

14 

6.47 

37.0 

82880 


The strength of Manilla ropes, like that of bar iron, is very variable ; 
and so with hemp ones. The above table supposes an average quality. Ropes ot 
good Italian hemp are considerably stronger than Manilla; but their cost excludes 
them from general use. The tarring 1 of ropes is said to lessen their strength ; 
and, when exposed to the weather, their durability also. We believe that the use of 
it in standing rigging is partly to diminish contraction and expansion by alternate 
wet and dry weather. The common rules for finding the strength of rope 
by multiplying the square of the diam or circumf by a given coefficient are entirely 
erroneous. Prices in Philada, 1886, Manilla, 18 to 14 cts per lb; Italian hemp, 
20 cts; American hemp, 12 cts; Sisal hemp, 10 cts; jute, (E. Indies,) 7 cts. E. H. 
Fitler & Co., 28 N. Water St, Phila. 

The strengths of pieces from the same coil may vary 25 per ct. 

A few moil ills of exposed work weakens ropes 20 to 50 per ct. 

WEIGHT AND STRENGTH OF IRON CHAINS. 


Table of strength of chains. 

Chains of superior iron will require K to x / z more to break them. (Original.) 


1 

Diam of rod 
of which 
the links 
are m ide. 

Weight 
of chain 
per ft run. 

Breaking strain 
of the chain. 

j 

Diam of rod 
of which 
the links 
are made. 

Weight 
of chain, 
per ft run. 

Breaking strain 
of the chain. 

Ins. 

Pds. 

Pds. 

Tons. 

Ins. 

Pds. 

Pds. 

Tons. 

3-16 

.5 

1731 

.773 

1 

10.7 

49280 

22.00 

K 

.8 

3069 

1.37 

ik 

12.5 

59226 

26.44 

5-16 

1. 

4794 

2.14 


16. 

73114 

32.64 


1.7 

6922 

3.09 

i 8 J 

18.3 

88301 

39.42 

7-16 

2. 

9408 

4.20 

IK 

21.7 

1052S0 

47.00 

K 

2.5 

12320 

5.50 

v A 

26. 

123514 

55.14 

9-16 

3.2 

15590 

6.96 

IK 

28. 

143293 

63 97 

% 

4.3 

19219 

8.58 

iK 

32. 

164505 

73 44 

11-16 

5. 

23274 

10.39 

2 

38. 

187152 

83.55 

'H 

5.8 

27687 

12.36 

2K 

54. 

224448 

100.2 

13-16 

6.7 

32301 

14.42 

2 K 

71. 

277088 

123.7 

K 

8. 

37 632 

16.80 


88. 

335328 

149.7 

15-.6 

9. 

43277 

19.32 

3 

105. 

398944 

178.1 





























































415 


LEAD, COPPER, ETC. 


The links of ordinary iron chains are usually made as short as is 
consistent with easy play, in order that they may not become bent when wound 
around drums, sheaves, Ac; and that they may be more easily handled in slinging 
large blocks of stone, &c. U. S. Govt, expts, 1878, prove that studs weaken the links. 

When so made, their weight per foot run is quite approximately 3X times that of a single bar of the 
round iron of which they are composed. Since each link consists of two thicknesses of bar, it might 
be supposed that a chain would possess about double the strength of a single bar; but the strength of 

the bar becomes reduced about by being formed into links ; so that the chain really has but about 
tV khe strength of two bars. As a thick bar of iron will not sustain as heavy a load in proportion as a 
thinner one, so of course, stout chains are proportionably weaker than slighter ones. In the foregoing 
table. 20 tons per sq inch , is assumed as the average breaking strain of a single straight bar of ordi¬ 
nary rolled iron, 1 inch in diatn; or 1 inch square; 19 tous, from 1 to 2 ins; and 18 tons, from 2 to 3 

ins. Deducting from each of these, we have as the breaking strain of the two bars composing 
each link, as follows : 14 tons per sq inch, up to 1 inch diam; 13.3 tous, from 1 to 2 ins; and 12.6 
tons, from 2 to 3 ins diam : and upon these assumptions the table is based. The wts are approxi¬ 
mate ; depending upon the exactness of diameter of the iron, and shape of link. 


Approximate prices of chains, in cents per pound. Bradlee & Co, 816 
Richmond St, Philadelphia, 1886. 


Diameter of rod from which the links are made; ins 

Ordinary proved or coil chain. 

Crane chain. 

Chain of combined iron and steel... 


% y A i 

6*4 5 34 

144 11 74 


ROLLED LEAD, COPPER, and BRASS: Sheets and Bars. 


Thickness 

or 

Diameter, 
or side, 
in 

Inches. 

LEAD. 

COPPER. 

BRASS 


Thickness 

or 

Diameter, 
or side, 
in 

Inches. 

Sheets, 

per 

Square 

Foot. 

Square 
liars ; 

1 Foot 
long. 

Round 
Bars ; 

1 Foot 
long. 

Sheets, 

per 

Square 

Foot. 

Square 
Bars; 

1 Foot 
long. 

Round 
Bars; 

1 Foot 
long. 

Sheets, 

per 

Square 

Foot. 

Square 
Bars; 

1 Foot 
long. 

Round 
Bars; 

1 Foot. 
long. 


Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

Lbs. 


1-32 

1.86 

.005 

.004 

1.44 

.004 

.003 

1.36 

.004 

.003 

1-32 

1-16 

3.72 

.019 

.015 

2.89 

.015 

.012 

2.71 

.014 

.011 

1-16 

3-32 

5.58 

.044 

.034 

4.33 

.034 

.027 

4.06 

.032 

.025 

3-32 

X 

7.44 

.078 

.061 

5.77 

.060 

.047 

5.42 

.056 

.044 

X 

5 32 

9.30 

.121 

.095 

7.20 

.094 

.074 

6.75 

.088 

.069 

5-32 

3 16 

11.2 

.174 

.137 

8.66 

.135 

.106 

8.13 

.127 

.100 

3-16 

7-32 

13.0 

.237 

.187 

10.1 

.184 

.144 

9.50 

.173 

.136 

7-32 

K 

14.9 

.310 

.244 

11.5 

.240 

.189 

10.8 

.226 

.177 

34 

5 16 

18.6 

.485 

.381 

11.4 

.376 

.295 

13.5 

.353 

.277 

5 16 

% 

22.3 

.698 

.518 

17.3 

.541 

.425 

16.3 

.508 

.399 

% 

7-16 

26.0 

.950 

.746 

20.2 

.736 

.578 

19.0 

.691 

.543 

7-16 

X 

29.8 

1.24 

.974 

23.1 

.962 

.755 

21.7 

.903 

.709 

X 

9 16 

33.5 

1.57 

1.23 

26.0 

1.22 

.955 

24.3 

1.14 

.900 

9-16 

% 

37.2 

1.94 

1.52 

28.9 

1.50 

1.18 

27.1 

1.41 

1.11 

% 

11-16 

40.9 

2.34 

1.84 

31.7 

1.82 

1.43 

29.8 

1.70 

1.34 

11-16 

K 

44.6 

2.79 

2.19 

34.6 

2.16 

1.70 

32.5 

2.03 

1.60 

K 

13-16 

48.3 

3.27 

2.57 

37.5 

2.55 

1.99 

35.2 

2.38 

1.87 

13-16 

% 

52.1 

3.80 

2.98 

40.4 

2.94 

2.31 

37.9 

2.76 

2.17 

Vs 

15-16 

56.0 

4.37 

3.42 

43.3 

3.38 

2.65 

40.6 

3.18 

2.49 

15-16 

1 . 

59.5 

4.96 

3.90 

46.2 

3.85 

3.02 

43.3 

3.61 

2.84 

1 . 

IX 

66.9 

6 27 

4.92 

52.0 

4.87 

3.82 

48.7 

4.57 

3.60 

IK 

1 X 

74.4 

7.75 

6.09 

57.7 

6.01 

4.72 

54.2 

5.64 

4.43 

134 


81.8 

9.37 

7.37 

63.5 

7.28 

5.72 

59.6 

6.82 

5 37 

IK 

IX • 

89.3 

11.2 

8.77 

69.3 

8.65 

6.80 

65.0 

8.12 

6.38 

134 

1 % 

96.7 

13.1 

10.3 

75.1 

10.2 

7.98 

70.4 

9.53 

7.49 

IK 

IK 

lu4. 

15.2 

11.9 

80.8 

11.8 

9.25 

75.9 

11.1 

8.68 

IK 

1 'A 

112. 

17 5 

13.7 

86.6 

13.5 

10.6 

81.3 

12.7 

9.97 

1)6 

2. 

119. 

19.8 

15.6 

92.3 

15.4 

12.1 

86.7 

14.4 

11.3 

2. 


Approximate prices, Phila, 1886. Copper Sheets, Nos. 1 to 25, 16 cts per 
lb. Ingots, 10^ to 12 cts. Brass Sheets, No. 28 and heavier, 16 to 50 cts per lb, ac¬ 
cording to size. 






































416 


WEIGHT OF METALS 


Roof copper is usually in sheets of 2% ft X 5 ft; or 12% 

square feet, weighing 10 to 14 lbs per sheet; and Is laid on boards.* No solder 
is used in the horizontal joints as it is in tin roofs; but both the horizontal and 
the sloping joints are formed by only overlapping and bending the sheets, much 
as shown by the figs on page 418 ; except that the horizontal joiuts are bent 
or locked together, as in this figure; and theu flattened down close. 

Sheet lead. Price, Philadelphia, 1886, about 5%cts per lb. 
Tatham Bros, 226 S Fifth St. List of standard wts in lbs per 
eq ft. Thicknesses in decimals of an inch. 


Wt. 

Th. 

Wt. 

Th. 

Wt. 

Th. 

Wt. 

Th. 

Wt. 

Th. 

Wt. 

Th. 

2.5 

.042 

4 

.068 

6 

.102 

8 

.136 

10 

.170 

14 

.237 

3 

.051 

5 

.085 

7 

.119 

9 

.153 

12 

.203 

16 

.271 


WEIGHT OF BATES. 



Diameter 

in 

Inches. 

Cast 

Lead. 

Cast 

Copper. 

Cast 

Brass. 

Cast 

Ikon. 

Diameter 

in 

Inches. 

Cast 

Lead. 

Cast 

Copper. 

Cast 

Brass. 

Cast 

Iron. 


Lbs. 

Lbs. 

Lbs. 

Lbs. 


Lbs. 

Lbs. 

Lbs. 

Lbs. 

X 

.026 

.021 

.019 

.017 

5X 

30.1 

24.1 

21.5 

19.8 

X 

.088 

.070 

.063 

.058 

X 

34.7 

27.7 

24.7 

22.7 

1 . 

.209 

.167 

.148 

.136 

X 

39.6 

31.7 

28.3 

25.9 

% 

.408 

.325 

.290 

.266 

6. 

45.0 

36.0 

32.0 

29.4 

X 

.705 

.562 

.501 

.460 

X 

57.2 

45.8 

to 8 

37.4 

% 

1.12 

.893 

.795 

.731 

7. 

71.5 

57.2 

50.9 

46.8 

2. 

1.67 

1.33 

1.19 

1.07 

X 

88.0 

70.3 

62.6 

57.5 

X 

2.38 

1.90 

1.69 

1.55 

8. 

106. 

85.3 

76.0 

69.8 

X 

3.25 

2.60 

2.32 

2.13 

X 

127. 

102. 

91.2 

83.7 

X 

4.34 

3.47 

3.09 

2.83 

9. 

151. 

121. 

108. 

99.4 

3. 

5.63 

4.50 

4.01 

3.68 

X 

178. 

143. 

127. 

117. 

X 

7.15 

5.72 

5.10 

4.68 

10. 

208. 

167. 

148. 

136. 

X 

8.94 

7.14 

6.36 

5.85 

X 

241. 

193. 

172. 

158. 

X 

11.0 

8.79 

7.83 

7.19 

li. 

277. 

222. 

198. 

182. 

4. 

13.4 

10.7 

9.50 

8.73 

X 

317. 

253. 

226. 

207. 

X 

16.0 

12.8 

11.4 

10.5 

12. 

360. 

288. 

257. 

236. 

Vi 

18.9 

15.2 

13.5 

12.4 






X 

22.7 

17.9 

15.9 

14.6 

The weight* of balls are as the cubes of their 

5. 

26.0 

20.8 

18.6 

17.0 

diams. 






Eead pipe. Price, Philadelphia, 1886, about 5% cts per lb. Tin-lined, 15 
cts. Tatham Bros, mfrs, 226 S. Fifth St. List of standard sizes. 


Inner 

diam. 

a 

A 

!s« 

H 

WtS per ft (F) 
and per rod (R) 
of 16X ft. 

Inner 

<liam. 

Thick¬ 

ness. 

W IS per ft (F) 
and per rod (R) 
of 16X ft. 

Inner 

diam. 

Thick¬ 

ness. 

CO 

a 

s- 

if a 

Inner 

4li;i in. 

Thick. 

ness. 

W t in lbs 
per ft. 

Ins. 

Ins. 



Ins. 

Ins. 




Ins. 

Ins. 


Ins. 

Ins. 


% 

.06 

7 

lbs R 

X 

.08 

16 

lbs 

R 

IX 

.14 

3.5 

3 

3-16 

9 

4 • 

.08 

10 

oz F 

4 l 

.10 

IX 

tbs 

F 

44 

.17 

4.25 

41 

X 

12 

«« 

.12 

1 

lb F 

4 l 

.12 

1?4 

lbs 

F 

4 4 

.19 

5. 

44 

5-16 

16 

( 1 

.16 

IX 

lbs F 

44 

.16 

2X 

lbs 

F 

14 

.23 

6.5 

14 

% 

20 

<4 

.19 

IX 

lbs F 

44 

.20 

3 

lbs 

F 

44 

.27 

8 

3X 

3-16 

9 5 


.07 

9 

lbs R 

44 

.23 

*x 

lbs 

F 

IX 

.13 

4 


h 

15 

“ 

.09 

X 

1b F 

44 

.30 

4X 

lbs 

F 

.4 

.17 

5 

14 

5-16 

18.5 


.11 

1 

lb F 

1 

.10 

24 % 

lbs 

R 

44 

.21 

6.5 

44 

% 

22 

14 

.13 

ix 

lbs F 

41 

.11 

2 

lbs 

F 


.27 

8 5 

4 

3-16 

12 5 

44 

.16 

IX 

lbs F 

44 

.14 

2X 

lbs 

F 

2 

.15 

4.75 

<4 

X 

16 

44 

.19 

2 

lbs F 

44 

.17 

3X 

lbs 

F 

44 

.18 

6 

44 

5-16 

21 


.25 

3 

lbs F 

44 

.21 

4 

lbs 

F 

44 

.22 

7 

41 

% 

25 

X 

.08 

12 

lbs R 

44 

.24 

*x 

lbs 

F 

44 

.27 

9 

4X 

3-16 

14 


.09 

1 

lb F 

ix 

.10 

2 

lbs 

F 

2X 

3-16 

8 

4 4 

X 

18 

44 

.13 

IX 

lbs F 

44 

.12 

2X 

lbs 

F 

44 

X 

11 

5 

X 

20 


.16 

2 

lbs F 

44 

.14 

3 

lbs 

F 

44 

5-16 

14 

44 

% 

31 

4 4 

.20 

2X 

lbs F 

4 4 

.16 

3X 

lbs 

F 

44 

% 

17 




44 

.22 

nx 

lbs F 

<< 

.19 

4X 

lbs 

F 







44 

.25 

3X 

lbs F 

44 

.25 

6 

lbs 

F 








Eead service pipes for single dwellings in Philadelphia are usually of from 
% inch bore, wt 1 to 2% lbs; to y-g inch bore, wt 1% to 3 lbs per ft run, according 
to head. They rarelj hurst from sudden closing ol stopcocks ; but sometimes 
do so from the freezing of the contained water. 


*To which it is held by copper cleats; as at Fig y, page 418. 
Price ot roofing copper in 1886, about 16 cts ; and copper nails, 30 cts per ft). 




















































































BRASS AND COPPER TUBES, 


417 


Seamless drawn brass and copper tubes, from *4 inch to 8 inches 
outer diameter, and in lengths up to 30 feet, are made by American Tube Works 
Boston Mass. 








V : . 










, 


ti . ' 






(I • 










■ ; 








• ■ 

, 




f> 1 


I )_ ,,U '• • , 






■ 


I 










,1 .1 I . '> 

: r . t 

I I I ■ . . r ' > 


■ 












‘ 






































■I - i< ,i h»o *»-!»;# to 1-01 ‘»*i i 












418 


TIN AND ZINC. 


TIN AND ZINC. 

The pure metal Is called bloeh tin. When perfectly pure, (which it 

rarely Is, being purposely adulterated, frequently to a large proportion, with the 
cheaper metals lead or zinc,) its sp grav is 7.29; and its weight per cub ft is 455 lbs. 
It is sufficiently malleable to be beaten into tin foil, only yjjlyg' an thick. 
Its tensile strength is but about 4600 lbs per sq inch; or about 7000 lbs when made 
into wire. It melts at the moderate temperature of 442° Fah. Pure block tin is 
not used for common building purposes ; but thin plates of sheet iron, covered with 
it on both sides, constitute the tinned plates , or, as they are called, the tin, used for 
covering roofs, rain pipes, and many domestic utensils. For roofs it is laid on boards. 

The sheets 
of tin are uni¬ 
ted as shown in 
this fig. First, sev¬ 
eral sheets are 
joined together in 
the shop, end for 
end, as at 11 ; by 
being first bent 
over, then ham¬ 
mered flat,and then 
__ soldered. These are 
then formed into a 
roll to be carried 
to the roof; a roll 

being long enough to reach from the peak to the eaves Different rolls being spread 
up and down the roof, are then united along their sides by simply being bent as at a 
and s , by a tool for that purpose. The roofers call the bending at s a double groove, 
or double, lock; and the more simple ones at t , a single groove, or lock. 

To hold the tin securely to the sheeting boards, pieces of the tin 3 or 4 ins long, 
by 2 ins wide, called cleats, are nailed to the boards at about every 18 ins along the 
joints of the rolls that are to be united, and are bent over with the double groove s. 
This will be understood from y, where the middle piece is the cleat, before being 
bent over. The nails should be 4-penny slating nails, which have broader heads 
than common ones. As they are not exposed to the weather, they may be of plain iron. 

Much use is made of what is called leadeil till, or ternes, for rooting. It is 
simply sheet-iron coated with lead, instead of the more costly metal tin. It is not 
as durable as the tinned sheets, but is somewhat cheaper. 

The best plates, both for tinning and for ternes, are made of charcoal iron ; which, 
being tough, bears bending better. Coke is used for cheaper plates, but inferior as 
regards bending. In giving orders, it is important to specify whether charcoal 
plates or coke ones are required;* also whether tinned plates, or ternes. 

Tinned and leaded sheets of Bessemer and other cheap steel, are now much used. 
They are sold at about the price of charcoal tin and terne plates. 

There are also in use for roofing, certain compound metals which resist tarnish 
better than either lead, tin, or zinc; but which are so fusible as to be liable to be 
melted by large burning cinders faliiug on the roof from a neighboring conflagration. 

A roof covered with tin or other metal should, if possible, slope not much less than 
five degrees, or about an inch to a foot; and at the eaves there should be a sudden 
fall into the rain-gutter, to prevent rain from backing up so as to overtop the double¬ 
groove joint s, and thus cause leaks. Where coal is used for fuel, tin roofs should 
receive two coats of paint when first put up, and a coat at every 2 or 3 years after. 
Where wood only is used, this is not necessary; and a tin roof, with a good pitch, 
will last 20 or 30 years.f 

Two good workmen can put on, and paint outside, from 250 to 300 sq ft of tin roof, 
per day of 8 hours. 

Tinned iron plates are sold by the box. These boxes, unlike glass, have not equal 
areas ot contents. They may lie designated or ordered either by their names or 
sizes. Many makers, however, have their private brands in addition; .and some of 
these have a much higher reputation than others. See table of sizes, etc., p. 419. 

* Prices, Phila, 1886. Tinned plates, charcoal, IC —10 x 14 and 14 x 20, the standardsizes(see 
table, p 419) $5.50 to $7 per box of 112 lbs, according to grade. Coke, IC—14 x 20, $5 to $5 50 
Hooting ternes, IC — 14 X 20, $o to $7. 1C, 20 X 28, $10 to $15.50 per l.ox of 224 lbs. Wm. F Potts 
Son & Co, 1225 Market St, Phila: Hall & Carpeuter, 709 Market St; Merchant & Co, 525 Arch St ' 

t Tile cost of till-rooting;, so-called, hut actually ternes, in Philadelphia, 1886, is about 7 
or 8 cts per sq ft of roof, including ternes. all labor, and oue coat of paint ou each side. It is often laid on 
old shingle roofs. Gaivauiied iron ruill Water-pipes, 8 ins diaiu, about 20 cts per ft run, 
put up. 














TIN AND ZINC 


419 




Table of Tinned and Terue Plates. 


The marls indicate the thicknesses, approximately as follows: 


Mark. 

Number 
Birmingham 
wire gauge. 

Ins. 

Lbs 

per sq ft. 

Mark. 

Number 
Birmingham 
wire gauge. 

Ins. 

Lbs 

per sq ft. 

IC 

30 

.012 

.48 

DC 

27 

.016 

.64 

IX 

28 

.014 

.56 

DX 

25 

.020 

.80 

] XX 

27 

.016 

.64 

DXX 

23 


1 00 

IXXX 

26 

.018 

.72 

DXXX 

22 

.0*28 

1 13 

IXXXX 

25 

.020 

.80 

DXXXX 

21 

.032 

1/29 


Size, 

inches. 

Mark. 

No. of 
sheets 
iu a box 

1 Weight 

1 per box, 
| pounds. 

Size, 

inches. 

Mark. 

No. of 
sheets 
in a box 

Weight 
per box, 
pounds. 

Size, 

inches. 

Mark. 

No. of | 

sheets 
in a box 

Weight 

per box, 

pounds. 

9 X 18 

IG 

225 

130 

13 X 13 

IXX 

225 

194 

16 X 16 

IC 

225 

205 


IX 


162 

13 X 26 

IC 

112 

135 

<4 

IX 

44 

256 

10 X 10 

IO 

44 

80 

4 » 

IX 

4 4 

169 

44 

IXX 

44 

294 


IX 


100 

4 4 

IXX 

44 

194 

16 X 19 

IX 

112 

152 

10 X 14 

IC 

44 

112 

14 X 14 

IC 

225 

156 

17 X 17 

IX 

4 l 

141 


1X 


HO 


IX 

44 

196 

4 4 

IXX 

4 4 

166 

1 ‘ 

IXX 

44 

161 

44 

IXX 

44 

225 

17 X 25 

DX 

100 

252 


IXXX 

1 4 

182 

4 4 

IXXX 

44 

254 

( < 

DXX 

4 4 

294 


IXXXX 

• 1 

‘203 

14 X 20 

IC 

112 

112 

18 X 18 

IX 

112 

162 

10 X 20 

IC 

4 4 

160 


IX 

44 

140 

4 4 

IXX 

4 i 

180 


IX 

• 4 

200 

tl 

IXX 

44 

161 

44 

IXXXX 

44 

235 

11 X 11 

IC 

4 4 

97 

4 4 

IXXX 

44 

182 

20 X 20 

IX 

I 4 

200 


IX 

14 

121 

4 4 

IXXXX 

41 

203 

<4 

IXX 

44 

230 

11 X 22 

IC 

112 

97 

14 X 22 

IX 

14 

154 

44 

IXXX 

<• 

260 


IX 

4 • 

121 

4 » 

IXX 

14 

177 

4 ( 

IXXXX 


290 


IXX 

44 

139 

14 X 24 

IX 

44 

168 

20 X 28 

IC 

44 

224 

12 X 12 

IC 

225 

11‘2 

“ 

IXX 


193 


IX 

44 

280 


IX 

»< 

no 

14 X 25 

1C 

14 

140 

** 

IXX 

44 

322 

4 4 

IXX 

4 4 

161 

4 4 

IX 

44 

175 


IXXX 

<* 

364 

12 X 24 

IC 

112 

115 

4 4 

IXX 

44 

201 

** 

IXXXX 

44 

406 

>4 

IX 

44 

111 

14 X 26 

IXXX 

44 

237 





44 

IXX 

>4 

166 

14 X 28 

1C 

44 

157 





44 

IXXX 

4 4 

187 

44 

IX 

44 

196 

Teraie I*lates. 

12 % X 17 

DC 

100 

98 

44 

IXX 

(4 

225 





“ 

DX 


120 

14 X 30 

[XX 

44 

241 

10 X 20 

IC 

112 

80 

44 

DXX 

44 

147 

14 X 31 

IX 

44 

217 

4 l 

IX 

44 

100 

44 

DXXX 

ii 

168 

4k 

IXX 

44 

249 

14 X 20 

IC * 

44 

112 

“ 

DXXXX 

44 

189 

15 X 15 

IX 

225 

225 

4 ( 

IX 

44 

140 

13 X 13 

IC 

rib 

135 


IXX 

4< 

259 

20 X 28 

IC 

4 4 

224 


IX 

4 1 

169 


IXXX 

(4 

326 


IX 

44 

280 


Sheets of larger size may he made to special order; those of tinned iron, in England; but leaded 
ternes are made in Phiiada also, and elsewhere. 

A box of 225 sheets of 13% by iO. contains 214.84 sq ft, but, allowing for overlapping, it will cover 
but about 150 sq ft of roof; even without any allowance for the waste which occurs in cutting away 
portions in order to fit at angles, &c. 

To find the area of roof covered by any sheet, first deduct 2 ins from its w'idth, and 1 inch from its 
length. 

Zinc, in sheets, and laid in the same manner as slates, is much used in some 
parts of Europe for roofing. By exposure to the weather, it soon becomes covered by a thin film of 
white oxide, which protects it from further injury, and renders the roof very durable.* Corrugated 
sheet zinc is also used. See Galvanized Sheet Iron, page 403. 

Zinc sheets are usually about 3 ft by 7 or 8 ft. The gauge differs from that of iron ; thus No 13 is 
-032 of an inch thick, or 1.22 lbs per sq ft; No 14 — .035 inch, and 1.35 tbs; No 15= .042 inch, and 
1 49 tbs; No 16 = 049 inch, and 1.62 lbs per sq ft. Any of these numbers may be used on roofs 1 , for 
which purpose it should be very pure. 

Water kept in zinc vessels is said to become injurious to health; and 
recently an outcry has on that account arisen against galvanized-iron service-pipes in dwellings. 
Yet such have been in use for many years in New England, Phiiada, and elsewhere, without as yet 
any deleterious effects. This is possibly owing to the fact that service-pipes being short, the water 
is usually all drawn through them several times a day; and hence does not remain in contact with 
the zinc or lead long enough to acquire a poisonous character. In taking possession of a house in 
which the water has remained stagnant iu the service-pipes for some considerable time, such water 
should all be run to waste; otherwise sickness may eusue from its use. 


*Tlio price of sheet zinc does not ordinarily differ much from that of 

6heet lead, which in Phiiada in 1886 is about 6 to 7 cts per Tb ; or in pigs, from 4 to 6 cts. 

The price of block tin, made into either pipes or sheets, about 35 cts per tb. Messrs Tatbam 
& Bros, 226 South fifth M, make both. In bars, 22 to 25 cts, in 1886. 

28 





































































420 


BOARD MEASURE, 


BOARD MEASURE. 


Remark on following' table. The table extends to 12 ins by 24 ins, but 

it is easy to find for greater sizes; thus, for example, the board measure in a piece of 19 by 22, will 
be twice that of a piece of 19 by 11, or 17.42 X 2 — 34.84 ft board meas ; or that of 19 % by 22, will be 
that of 10)4 by 22 added to that of 9 by 22, or 18.79 16.50 — 35.29. A foot of board meas is equal to 

1 foot square and 1 inch thick, or to 144 cub ins. Hence 1 cub ft = 12 ft board meas. 


13 

ai 

& £ 

Feet of Board Measure contained in one running foot of 
of different dimensions. (Original.) 

1000 ft board measure = 83)4 cub ft. 

Scantlings 











’O o 



THICKNESS IN INCHES. 



£> M 

1 

IX 

IX 

1 X 

2 

2 M 

IX 

iX 

3 


Ft. Bd.M. 

Ft. Bd.M. 

Ft. Bd.M. 

Ft. Bd.M. 

Ft. Bd.M. 

Ft. Bd.M. 

Ft. Bd.M. 

Ft. Bd.M. 

Ft. Bd.M. 

X 

.0208 

.0260 

.0313 

.0365 

.0417 

.0469 

.0521 

.0573 

.0625 

X 

.0417 

.0521 

.0625 

.0729 

.0833 

.0938 

.1042 

.1140 

.1250 

% 

.0625 

.0781 

.0938 

.1094 

.1250 

.1406 

.1563 

.1719 

.1875 

1. 

.0833 

.1042 

.1250 

.1458 

.1667 

.1875 

.2083 

.2292 

.2500 

X 

.1012 

.1302 

.1563 

.1823 

.2083 

.2344 

.2604 

.2865 

.3125 

X 

.1250 

.1563 

.1875 

.2188 

.2500 

.2813 

.3125 

.3438 

.3750 

X 

.1458 

.1823 

.2187 

.2552 

.2917 

.3281 

.3646 

.4010 

.4375 

2. 

.1667 

.2083 

.2500 

.2917 

.3333 

.3750 

.4166 

.4583 

.5000 

X 

.1875 

.2344 

.2813 

.3281 

.3750 

.4219 

.4688 

.5156 

.5625 

X 

.2083 

.2604 

.3125 

.3646 

.4167 

.4688 

.5208 

.5729 

.6250 

% 

.2292 

.2865 

.3438 

.4010 

.4583 

.5156 

.5729 

.6302 

.6875 

8. 

.2500 

.3125 

.3750 

.4375 

.5000 

.5625 

.6250 

.6875 

.7500 

X 

.2708 

.3385 

.4063 

.4739 

.5416 

.6094 

.6771 

.7448 

.8125 

X 

.2917 

.3646 

.4375 

.5104 

.5833 

.6563 

.7292 

.8021 

.8750 

% 

.3125 

.3906 

.4689 

.5469 

.6250 

.7031 

.7813 

.8594 

.9375 

4. 

.3333 

.4167 

.5000 

.5833 

.6667 

.7500 

.8333 

.9167 

1.000 

X 

.3542 

.4427 

.5312 

.6198 

.7083 

.7969 

.8854 

.9740 

1.063 

x 

.3750 

.4688 

.5625 

.6563 

.7500 

.8438 

.9375 

1.031 

1.125 

% 

.3958 

.4948 

.5938 

.6927 

.7917 

.8906 

.9896 

1.086 

1.188 

6. 

.4167 

.5208 

.6250 

.7292 

.8333 

.9375 

1.042 

1.146 

1.250 

X 

.4375 

.5469 

.6563 

.7656 

.8750 

.9844 

1.094 

1.203 

1.313 

x 

.4583 

.5729 

16875 

.8020 

.9167 

1.031 

1.146 

1.260 

1.375 

x 

.4792 

.5990 

.7188 

.8385 

.9583 

1.078 

1.198 

1.318 

1.438 

6. 

.5000 

.6250 

.7500 

.8750 

1.000 

1.125 

1.250 

1.375 

1.500 

X 

.5208 

.6510 

.7813 

.9115 

1.042 

1.172 

1.302 

1.432 

1.563 

x 

.5417 

.6771 

.8125 

.9479 

1.083 

1.219 

1.354 

1.490 

1.625 

% 

.5625 

.7031 

.8438 

.9844 

1.125 

1.266 

1.406 

1.547 

1.688 

7. 

.5833 

.7292 

.8750 

1.021 

1.167 

1.312 

1.458 

1.604 

1.750 

X 

.6042 

■ 7552 

.9063 

1.057 

1.208 

1.359 

1.510 

1.661 

1.813 

X 

.6250 

.7813 

.9375 

1.094 

1.250 

1.406 

1.563 

1.719 

1.875 

X 

.6458 

.8073 

.9688 

1.130 

1.292 

1.453 

1.615 

1.776 

1.938 

8. 

.6667 

.8333 

1.000 

1.167 

1.333 

1.500 

1.667 

1.833 

2.000 

X 

.6875 

.8594 

1.031 

1.203 

1.375 

1.547 

1.719 

1.891 

2.063 

X 

.7083 

.8854 

1.063 

1.210 

1.417 

1.594 

1.771 

1.948 

2.125 

X 

.7292 

.9114 

1.094 

1.276 

1.458 

1.641 

1.823 

2.005 

2.188 

9. 

.7500 

.9375 

1.125 

1.313 

1.500 

1.688 

1.875 

2.062 

2.250 

X 

.7708 

.9635 

1.156 

1.349 

1.542 

1.734 

1.927 

2.120 

2.313 

Yz 

.7917 

.9895 

1.188 

1.385 

1.583 

1.781 

1.979 

2.177 

2.375 

X 

.8125 

1.016 

1.219 

1.422 

1.625 

1.828 

2.031 

2.234 

2.438 

10. 

.8333 

1.042 

1.250 

1.458 

1.667 

1.875 

2.083 

2.292 

2.500 

X 

.8542 

1068 

1.281 

1.495 

1.708 

1.922 

2.135 

2.349 

2.563 

Y 

.8750 

1.094 

1.313 

1.531 

1.750 

1.969 

2.188 

2.406 

2.625 

X 

.8958 

1.120 

1.344 

1.568 

1.792 

2.016 

2.240 

2.463 

2.688 

11. 

.9167 

1.146 

1.375 

1.604 

1.833 

2.063 

2.292 

2.521 

2.750 

X 

.9375 

1.172 

1.406 

1.641 

1.875 

2.109 

2.344 

2.578 

2.813 

Yi 

.9583 

1.198 

1.438 

1.677 

1.917 

2.156 

2.396 

2 635 

2.875 

X 

.9792 

1.224 

1 469 

1.714 

1 958 

2.203 

2.448 

2.693 

2.938 

12. 

1.000 

1.250 

1.500 

1.750 

2.000 

2.250 

2.500 

2.750 

3.000 

Y 

1.042 

1.302 

1.563 

1.823 

2.083 

2.344 

2.604 

2.865 

3.125 

13. 

1.083 

1.354 

1.625 

1.896 

2.167 

2.438 

2.708 

2.979 

3.250 

Y 

1.125 

1.406 

1.688 

1.969 

2.250 

2.531 

2.813 

3.094 

3.375 

14. 

1,167 

1.458 

1.750 

2.042 

2 333 

2.625 

2.917 

3.208 

3.500 

Y 

1.208 

1.510 

1.813 

2.115 

2.417 

2.719 

3.021 

3.322 

3.625 

15. 

1.250 

1.563 

1.875 

2.188 

2.500 

2.813 

3.125 

3.438 

3.750 

X 

1.292 

1.615 

1.938 

2.260 

2.583 

2.906 

3.229 

3.552 

3.875 

16. 

1.333 

1.667 

2.000 

2.333 

2.667 

3.000 

3.333 

3.667 

4.000 

X 

1 ..>73 

1.719 

2.063 

2.406 

2.750 

3.094 

3.438 

3.7s I 

4.125 

17. 

1.41 1 

1.771 

2.125 

2.479 

2.833 

3.188 

3.542 

3.89$ 

4 250 

Y 

1 .458 

1.823 

2.187 

2.552 

2.917 

3.281 

3.646 

4.010 

1 57 5 

18. 

1.500 

1.875 

2.250 

2.625 

3.000 

3.375 

.‘{.750 

4 |2B 

4 500 

19. 

1.583 

1.979 

2.375 

2.771 

3.167 

3.563 

3.958 

4.354 

4 750 

20. 

1.667 

2.083 

2.500 

2.917 

3.333 

3.750 

4.167 

4.583 

5 < w u) 

21. 

1.750 

2.188 

2.625 

3.063 

3.500 

3.938 

4.375 

4 g I 2 

5 250 

22. 

1.833 

2.292 

2.750 

3.208 

3.667 

4.125 

4.583 

5.042 

5 500 

23. 

1.917 

2.396 

2.875 

3.354 

3.833 

4.313 

4.792 


5 750 

24. 

2.000 

2.500 

3.0(H) 

3.500 

4.000 

4.500 

5.000 

5.500 

6.000 

1 


1 . 


2 . 


4. 


6 . 


7. 


8 . 


9. 


10. 


11 . 



































BOARD MEASURE 


421 


Table of Hoard Measure — (Continued.) 


0 . 

•h QQ 

UsS 

Feet of Board Measure contained in one running foot of Scantlings 
of different dimensions. (Original.) 

a- ■ —• 

0 . 

•H QQ 

^ 0 

O 

£ H 

*X 

, 3X 

THICKN 

3X 4 

ESS IIS 

*x 

r INCH 

IX 

tES. 

*X 

5 

5X 

«! 

x 

Ft. Bd. M. 

.0677 

Ft.Bd.M. 

.0729 

Ft.Bd.M. 

.0781 

Ft.Bd.M. 

.0833 

Ft.Bd.M. 

.0885 

Ft.Bd.M. 

.0938 

Ft.Bd.M. 

.0990 

Ft.Bd.M 

.1042 

Ft.Bd.M. 

.1094 

X 

x 

.1354 

.1457 

.1562 

.1667 

.1770 

.1875 

.1979 

.2083 

.2188 

X 

x 

.2031 

.2187 

.2341 

.2500 

.2656 

.2813 

.2969 

.3125 

.3281 

X 

i. 

.2708 

.2917 

.3125 

.3333 

.3542 

.3750 

.3958 

.4167 

.4375 

1. 

x 

.3385 

.3646 

.3906 

.4167 

.4427 

.4688 

.4948 

.5208 

.5469 

X 

X 

.4063 

.4375 

.4688 

.5000 

.5313 

.5625 

.5938 

.6250 

.6563 

X 

X 

.4740 

.5104 

.5469 

.5833 

.6198 

.6563 

.6927 

.7292 

.7656 

X 

2. 

.5417 

.5833 

.6250 

.6667 

.7083 

.7500 

.7917 

•8333 

.8750 

2. 

X 

.6094 

.6563 

.7031 

.7500 

.7969 

.8438 

.8906 

.9375 

.9844 

X 

X 

.6771 

.7292 

.7813 

.8333 

.8854 

.9375 

.9896 

1.042 

1.094 

X 

X 

.7448 

.8021 

.8594 

.9167 

.9740 

1.031 

1.089 

1.146 

1.203 

X 

3. 

.8125 

.8750 

.9375 

1.000 

1.062 

1.125 

1.188 

1.250 

1.313 

3. 

x 

.8802 

.9479 

1.016 

1.083 

1.151 

1.219 

1.286 

1.354 

1.422 

X 

x 

.9479 

1.021 

1.094 

1.167 

1.240 

1.313 

1.385 

1.458 

1.531 

X 

x 

1.016 

1.094 

1.172 

1.250 

1.327 

1.406 

1.484 

1.563 

1.641 

X 

4. 

1 083 

1.167 

1.250 

1.333 

1.416 

1.500 

1.583 

1.667 

1.750 

4. 

x 

1.151 

1240 

1.328 

1.417 

1.504 

1.594 

1.682 

1.771 

1.859 

X 

x 

1.219 

1.313 

1.406 

1.500 

1.593 

1.688 

1.781 

1.875 

1.969 

X 

X 

1.286 

1.384 

1.484 

1.583 

1.681 

1.781 

1.880 

1 979 

2.078 

X 

5. 

1.354 

1.457 

1.566 

1.666 

1.770 

1.875 

1.979 

2.083 

2.188 

5. 

X 

1.422 

1.530 

1.644 

1.750 

1.858 

1.969 

2.078 

2.188 

2 297 

X 

X 

1.490 

1.603 

1.722 

1.833 

1.947 

2 063 

2.177 

2.292 

2.406 

X 

X 

1.557 

1676 

1.800 

1.917 

2.035 

2.156 

2.276 

2.3% 

2.516 

X 

6 . 

1.625 

1.750 

1.875 

2.000 

2.125 

2 250 

2.375 

2.500 

2.625 

6. 

X 

1.693 

1.823 

1.953 

2.083 

2.214 

2.344 

2.474 

2 604 

2.734 

X 

X 

1.760 

1.896 

2.031 

2.167 

2.302 

2.438 

2.573 

2.708 

2.843 

X 

X 

1.828 

1.969 

2.109 

2.250 

2.391 

2.531 

2.672 

2.813 

2 953 

X 

7. 

1.896 

2.042 

2.188 

2.333 

2.479 

2.625 

2.771 

2.917 

3.063 

7. 

X 

1.964 

2.115 

2-266 

2.416 

2.568 

2 719 

2.870 

3 021 

3.172 

X 

X 

2.031 

2 187 

2.344 

2.500 

2 656 

2 813 

2.969 

3.125 

3.281 

X 

X 

2.099 

2 260 

2 422 

2.583 

2.745 

2.906 

3.068 

3.229 

3 .391 

X 

8. 

2.167 

2 333 

2.500 

2.667, 

2.833 

3.000 

3.167 

3.333 

3.500 

8. 

X 

2.234 

2.406 

2.578 

2.750 

2.922 

3.094 

3.266 

3.438 

3.609 

X 

X 

2.302 

2.479 

2 656 

2.833 

3.010 

3.188 

3.365 

3.542 

3.718 

X 

X 

2.370 

2.552 

2 734 

2.916 

3.099 

3.281 

3.464 

3.646 

3.828 

X 

9. 

2.438 

2.625 

2.813 

3.000 

3.187 

3.375 

3.563 

3.750 

3.938 

9. 

X 

2.505 

2.698 

2.891 

3.083 

3.276 

3.469 

3.661 

3.854 

4.047 

X 

X 

2.573 

2.771 

2.969 

3.167 

3.365 

3.563 

3.760 

3.958 

4.156 

X 

X 

2.641 

2.844 

3.047 

3.250 

3.453 

3.656 

3.859 

4.063 

4.266 

X 

10. 

2.708 

2.917 

3.125 

3.333 

3.542 

3.750 

3.958 

4.167 

4.375 

10. 

X 

2.776 

2.990 

3.203 

3.416 

3.630 

3.844 

4.057 

4.271 

4.484 

X 

X 

2.844 

3.063 

3.281 

3.500 

3.719 

3.938 

4.156 

4.375 

4.594 

X 

X 

2.911 

3.135 

3.359 

3.583 

3.807 

4.031 

4.255 

4.479 

4.703 

X 

11. 

2.979 

8.208 

3.438 

3.666 

3.896 

4.125 

4.354 

4.583 

4.813 

11. 

y 4 

3.047 

3.281 

3.516 

3.750 

3.984 

4.219 

4.453 

4.688 

4.922 

X 

X 

3.115 

3.354 

3.594 

3.833 

4.073 

4.313 

4.552 

4.792 

5.031 

X 

X 

3.182 

3.427 

3.672 

3.916 

4.161 

4.406 

4.651 

4.896 

5.141 

X 

12. 

3.250 

3.500 

3.750 

4.000 

4.250 

4.500 

4.750 

5.000 

5.250 

12. 

X 

3.385 

3.646 

3.906 

4.167 

4.427 

4.688 

4.948 

5.208 

5.469 

X 

13. 

3.521 

3.792 

4.063 

4.333 

4.604 

4.875 

5.146 

5.417 

5.688 

13. 

X 

3.656 

3.938 

4.219 

4.500 

4.781 

5.063 

5.344 

5.625 

5.906 

X 

14. 

3.792 

4.083 

4.375 

4.667 

4.958 

5.250 

5.542 

5.833 

6.125 

14. 

X 

3.927 

4.229 

4.531 

4.833 

5.135 

5.438 

5.740 

6.042 

6.344 

X 

15. 

4.063 

4.375 

4.688 

5.000 

5.313 

5.625 

5.938 

6.250 

6.563 

15. 

X 

4.198 

4.521 

4.844 

5.166 

5.490 

5.813 

6.135 

6.458 

6.781 

H 

16. 

4.333 

4.667 

5.000 

5.333 

5.667 

6.000 

6.333 

6.667 

7.000 

16. 

X 

4.469 

4.813 

5.156 

5.500 

5.844 

6.188 

6.531 

6.875 

7.219 

X 

17. 

4.604 

4.958 

5.313 

5.667 

6.021 

6.375 

6.729 

7.083 

7.438 

17. 

X 

4.740 

5.104 

5.469 

5.833 

6.198 

6.563 

6.927 

7.292 

7.656 


18. 

4.875 

5.250 

5.625 

6.000 

6.375 

6.750 

7.125 

7.500 

7.875 

18. 

19. 

5.146 

5.542 

5.938 

6.333 

6.729 

7.125 

7.521 

7.917 

8.313 

19. 

20. 

5.417 

5.833 

6.250 

6.667 

7.083 

7.500 

7.917 

8.333 

8.750 

20. 

21. 

5.688 

6.125 

6.563 

7.000 

7.438 

7.875 

8.313 

8.750 

9.188 

21. 

22. 

5.958 

6.417 

6.875 

7.333 

7.792 

8.250 

8.708 

9.167 

9.625 

22. 

23. 

6.229 

6.708 

7.188 

7.667 

8.145 

8.625 

9.104 

9.583 

10.06 

23. 

24. 

24. 

6.500 

7.000 

7.500 

8.01X1 

8.500 

9.IIU0 

9.5(H) 

10.00 

10.50 























































422 


BOARD MEASURE 


Table of Board Measnre — (Continued.) 


0 

f-t CO 
. ° 
A A 
** o 

2 ci 


34 

34 

% 

1. 

2 . 

54 

34 

34 

3. 
34 
34 
34 

4. 

34 

34 

34 

5. 
34 
34 
34 

6 . 
34 
34 
34 

7. 
34 
34 
34 

8. 

34 

34 

34 

9. 

34 

34 

34 

10 . 

34 

34 

34 

11 . 

34 

34 

34 

12 . 

34 

13. 

34 

14. 

34 

15. 

34 

16. 

34 

17. 

34 

18. 

19. 

20 . 
21 . 
22 . 

23. 

24. 


of different dimensions. (Original.) 


THICKNESS IN INCHES. 


534 

5X 

6 

6M 

6X 

6 X 

7 

Ft.Bd.M. 

Ft.Bd.M. 

Ft.Bd.M. 

Ft. Bd.M. 

Ft. Bd.M. 

Ft. Bd.M. 

Ft. Bd.M. 

.1146 

.1198 

.1250 

.1302 

.1354 

.1406 

.1458 

.2292 

.2396 

.2500 

.2604 

.2708 

.2813 

.2917 

.3438 

.3594 

.3750 

.3906 

.4063 

.4219 

.4375 

.4583 

.4792 

.5000 

.5208 

.5417 

.5C25 

.5833 

.5729 

.5990 

.6250 

.6510 

.6771 

.7031 

.7292 

.6875 

.7188 

.7500 

.7812 

.8125 

.8438 

.8750 

.8021 

.8385 

.8750 

.9115 

.9479 

.9844 

1.020 

.9167 

.9583 

1.000 

1.042 

1.083 

1.125 

1.167 

1.031 

1.078 

1.125 

1.172 

1.219 

1.266 

1.313 

1.146 

1.198 

1.250 

1.302 

1.354 

1.406 

1 458 

1.260 

1.318 

1.375 

1.432 

1.490 

1.547 

1.604 

1.375 

1.438 

1.500 

1.562 

1.625 

1.688 

1.750 

1.490 

1.557 

1.625 

1.693 

1.760 

1.828 

1.896 

1.604 

1.677 

1.750 

1.823 

1.896 

1.969 

2.042 

1.719 

1.797 

1.875 

1.953 

2.031 

2.109 

2.188 

1.833 

1.917 

2.000 

2.083 

2.167 

2.250 

2.333 

1.948 

2.036 

2.125 

2.214 

2.302 

2.391 

2.479 

2.063 

2.156 

2.250 

2.344 

2.438 

2.531 

2 625 

2.177 

2.276 

2.375 

2.474 

2.573 

2.672 

2.771 

2.292 

2.396 

2.500 

2.604 

2.708 

2.813 

2.917 

2.406 

2.516 

2.625 

2.734 

2 844 

2.953 

3.063 

2.521 

2.635 

2.750 

2.865 

2.979 

3.094 

3.208 

2.635 

2.755 

2.875 

2.995 

3.115 

3.234 

3.354 

2.750 

2.875 

3.000 

3.125 

3.250 

3.375 

3.500 

2.865 

2.995 

3.125 

3.255 

3.385 

3.516 

3.646 

2.979 

3.115 

3.250 

3.385 

3.521 

3.656 

3.792 

3.094 

3.234 

3.375 

3.516 

3.656 

3.797 

3.938 

3.208 

3.354 

3.500 

3.646 

3.792 

3.938 

4.083 

3.323 

3.474 

3.625 

3.776 

3.927 

4.078 

4.229 

3.438 

3.594 

3.750 

3.906 

4.063 

4.219 

4.375 

3.552 

3.714 

3.875 

4.036 

4.198 

4.359 

4.521 

3.667 

3.&33 

4.000 

4.167 

4.333 

4.500 

4.667 

3.781 

3.953 

4.125 

4.297 

4.469 

4*641 

4.813 

3.896 

4.073 

4.250 

4.427 

4.604 

4.781 

4.957 

4.010 

4.193 

4 375 

4.557 

4.740 

4.922 

5.103 

4.125 

4.313 

4.500 

4.687 

4.875 

5.063 

5.249 

4.240 

4.432 

4.625 

4.818 

5.010 

5.203 

5.395 

4.354 

4.552 

4.750 

4.948 

5.146 

5.344 

5.541 

4.469 

4.672 

4.875 

5.078 

5.281 

5.484 

5.687 

4.583 

4.792 

5.000 

5.208 

5.417 

5.625 

5.833 

4.698 

4.911 

5.125 

5.339 

5.552 

5.766 

5.979 

4.813 

5.031 

5.250 

5.469 

5.688 

5.906 

6.125 

4.927 

5.151 

5.375 

5.599 

5.823 

6.047 

6.271 

5.012 

5.271 

5 500 

5.729 

5.958 

6.188 

6.417 

5.156 

5.391 

5.625 

5.859 

6.094 

6.328 

6.563 

5.271 

5.510 

5.750 

5 990 

6.229 

6.469 

6.708 

5.385 

5.630 

5.875 

6.120 

6.365 

6.609 

6.854 

5.500 

5.750 

6.000 

6.250 

6.500 

6.750 

7.000 

5.729 

5.990 

6.250 

6.510 

6.771 

7.031 

7.292 

5.958 

6.229 

6.500 

6.771 

7.042 

7.313 

7.583 

6.188 

6.469 

6.750 

7.031 

7.313 

7.594 

7.875 

6.417 

6.708 

7.000 

7.292 

7.583 

7.875 

8.167 

6.646 

6.948 

7.250 

7.552 

7.854 

8.156 

8.458 

6.875 

7.188 

7.500 

7.812 

8.125 

8.438 


7.104 

7.427 

7.750 

8.073 

8.396 

8.719 

9.042 

7.333 

7.667 

8.000 

8.333 

8.667 

9.000 

9.333 

7.563 

7.906 

8.250 

8.594 

8.9.38 

9.281 

9.625 

7.792 

8.146 

8.500 

8.854 

9.208 

9.563 

9.917 

8.021 

8.385 

8.750 

9.115 

9.479 

9.844 

10.21 

8.250 

8.625 

9.000 

9.375 

9.750 

10.13 

10.50 

8.708 

9.104 

9.500 

9.896 

10.29 

10.69 

11.08 

9.167 

9.583 

10.00 

10.42 

10.83 

11.25 

11 67 

9.625 

10.06 

10.50 

10.94 

11.38 

11 81 

12 ‘>5 

10.08 

10.54 

11.00 

11.46 

11.92 

12.38 

12 83 

10.54 

11.02 

11.50 

11.98 

12.46 

12.94 

13 42 

11.00 

11.50 

12.00 

12.50 

13.00 

13.50 

14.00 


Scantlings 

Width in 

Inches. 

?54 

IX 

Ft. Bd.M. 

Ft. Bd.M. 


.1510 

.1563 

X 

.3021 

.3125 

X 

.4531 

.4688 

X 

.6042 

.6250 

i. 

.7552 

.7813 

x 

.9062 

.9375 

X 

1.057 

1.094 

X 

1.208 

1.250 

2. 

1.359 

1.406 

x 

1.510 

1.563 

X 

1.661 

1.719 

x 

1.813 

1.875 

3. 

1.964 

2.031 

X 

2.115 

2.188 

x 

2.266 

2.344 

X 

2.417 

2 500 

4. 

2.568 

2.656 

X 

2.719 

2.813 

X 

2.870 

2.969 

X 

3.021 

3.125 

5. 

3.172 

3.281 

X 

3.323 

3.438 

X 

3.474 

3.594 

X 

3.625 

3.750 

6. 

3.776 

3.906 

X 

3.927 

4.063 

X 

4.078 

4.219 

X 

4.229 

4.375 

7. 

4.380 

4.531 

X 

4.531 

4.688 

X 

4.682 

4.844 

X 

4.833 

5.000 

8. 

4.984 

5.156 

X 

5.135 

5 313 

X 

5.286 

5.469 

X 

5.438 

5.625 

9. 

5.589 

5.781 

X 

5 740 

5.938 

X 

5.891 

6.011 

X 

6.042 

6.250 

10. 

6.193 

6.406 

X 

6.344 

6.563 

X 

6.495 

6.719 

X 

6.646 

6.876 

11. 

6.797 

7.031 

X 

6.948 

7.188 

X 

7.099 

7.344 

X 

7.250 

7.500 

12. 

7.552 

7.813 

X 

7.854 

8.125 

13. 

8.156 

8.438 

X 

8.458 

8.750 

14. 

8.760 

9.063 

X 

9.063 

9.375 

15. 

9.365 

9.688 

X 

9.667 

10.09 

16. 

9.969 

10.31 

X 

10.27 

10.63 

17. 

10.57 

10.94 

X 

10.88 

11.25 

18. 

11.48 

11.88 

19. 

12.08 

12.50 

20. 

12.69 

13.13 

21. 

13.29 

13.75 

22. 

13.90 

14.38 

23. 

14.50 

15.00 

24. 































BOARD MEASURE, 


423 


Table of Board Measure — (Continued.) 


a ~ 

•H CO 
- 


Feet of Board Measure contained in one running foot of Scantlings 
of different dimensions. (Original.) 


•r* 

|>M 



THICKNESS IN INCHES. 



T3 t> 


794 

8 


»X 

s% 

9 

9M 

9X 

9% 



Ft.BtLM. 

Ft.Bd.M. 

Ft.Bd.M. 

Ft.Bd.M 

Ft.Bd.M. 

Ft.Bd.M 

Ft.Bd.M 

Ft.Bd.M. 

Ft.Bd.M. 


X 

.1615 

.1667 

.1719 

.1771 

.1823 

.1875 

.1921 

.1979 

.2031 

X 


.3229 

.3333 

.3438 

.3542 

.3646 

.3750 

.3854 

.3958 

.4063 

x 

% 

.4844 

.5006 

.5156 

.5313 

.5467 

.5625 

.5781 

.5938 

.6094 

X 

1. 

.6458 

.6667 

.6875 

.7083 

.7292 

.7500 

.7708 

.7917 

.8125 

l. 

34 

.8073 

.8333 

.8594 

.8854 

.9115 

.9375 

.9635 

.9896 

1.016 

X 

3^ 

.9688 

1.000 

1.031 

1.063 

1.094 

1.125 

1.156 

1.188 

1.219 

X 


1.130 

1.167 

1.203 

1.240 

1.276 

1.313 

1.349 

1.385 

1.422 

X 

2. 

1.292 

1.333 

1.375 

1.417 

1.458 

1.500 

1.542 

1.583 

1.625 

2. 


1.453 

1.500 

1.547 

1.594 

1.641 

1.688 

1.734 

1.781 

1.828 

A 


1.615 

1.667 

1.719 

1.771 

1.822 

1.875 

1.927 

1.979 

2.031 

X 

x 

1.776 

1.833 

1.891 

1.948 

2.005 

2.063 

2.120 

2.177 

2.234 

X 

3. 

1.938 

2.000 

2.063 

2.125 

2.188 

2.250 

2.313 

2.375 

2.438 

8. 


2.099 

2.167 

2.234 

2.302 

2.370 

2.438 

2.505 

2.573 

2.641 

X 


2.260 

2.333 

2.406 

2.479 

2.552 

2.625 

2.698 

2.771 

2.844 

X 


2.422 

2.500 

2.578 

2.656 

2.734 

2.813 

2.891 

2.969 

3.047 

X 

4. 

2.583 

2.667 

2.750 

2.833 

2.917 

3.000 

3.083 

3.167 

3.250 

4. 


2.745 

2.833 

2.922 

3.010 

3.099 

3.188 

3.276 

3.365 

3.453 

X 

34 

2.906 

3.000 

3.094 

3.188 

3.281 

3.375 

3.469 

3.563 

3.656 

X 

% 

3.068 

3.167 

3.266 

3.365 

3.464 

3.563 

3.661 

3.760 

3.859 

X 

5. 

3.229 

3.333 

3.438 

3.542 

3.646 

3.750 

3.854 

3.958 

4.063 

5. 

34 

3.391 

3.500 

3.609 

3.719 

3.828 

3.938 

4.047 

4.156 

4.266 

X 

34 

3.552 

3.667 

3.781 

3.896 

4.010 

4.125 

4.240 

4.354 

4.469 

X 

% 

3.714 

3.833 

3.953 

4.073 

4.193 

4.313 

4.432 

4.552 

4.672 

X 

6. 

3.875 

4.000 

4.125 

4.250 

4.375 

4.500 

4.625 

4.750 

4.875 

6. 

34 

4.036 

4.167 

4.297 

4.427 

4.557 

4.688 

4.818 

4.948 

5.078 

X 

X 

4.198 

4.333 

4.469 

4.604 

4.740 

4.875 

5.010 

5.146 

5.281 

X 

X 

4.359 

4.500 

4.641 

4.781 

4.922 

5.063 

5.203 

5.344 

5.484 

X 

7. 

4.521 

4.667 

4.813 

4.958 

5.104 

5.250 

5.360 

5.542 

5.688 

7. 

34 

4.682 

4.833 

4.984 

5.135 

5.286 

5.438 

5.590 

5.740 

5.891 

X 

34 

4.844 

5.000 

5.156 

5.313 

5.469 

5.625 

5.782 

5.938 

6.094 

X 

X 

5.005 

5.167 

5.328 

5.490 

5.651 

5.813 

5.975 

6.135 

6.297 

X 

S. 

5.167 

5.333 

5.500 

5.667 

5.833 

6.000 

6.167 

6.333 

6.500 

8. 

34 

5.328 

5.500 

5.672 

5.844 

6.016 

6.188 

6.359 

6.531 

6.703 

X 

34 

5.490 

5.667 

5.844 

6.021 

6.198 

6.375 

6.552 

6.729 

6.906 

X 

94 

5.651 

5.833 

6.016 

6.198 

6.380 

6.563 

6.745 

6.927 

7.109 

X 

9. 

5.813 

6.000 

6.188 

6.375 

6.563 

6.750 

6.938 

7.125 

7.313 

9. 

34 

5.974 

6.167 

6.359 

6.552 

6.745 

6.938 

7.130 

7.323 

7.516 

X 

34 

6.135 

6.333 

6.531 

6.729 

6.927 

7.125 

7.323 

7.521 

7.719 

X 

94 

6.297 

6.500 

6.703 

6.906 

7.109 

7.313 

7.516 

7.719 

7.922 

X 

10. 

6.458 

6.667 

6.875 

7.083 

7.292 

7.500 

7.708 

7.917 

8.125 

10. 

34 

6.620 

6.833 

7.047 

7.260 

7.474 

7.688 

7.901 

8.115 

8.328 

X 

34 

J.78I 

7.000 

7.219 

7.438 

7.656 

7.875 

8.094 

8.313 

8.531 

X 

94 

..943 

7.167 

7.391 

7.615 

7.839 

8.063 

8.286 

8.510 

8.734 

X 

li. 

7.104 

7.333 

7.563 

7.792 

8.021 

8.250 

8.479 

8.708 

8.938 

11. 

34 

7.266 

7.500 

7.735 

7.969 

8.203 

8.438 

8.672 

8.906 

9.141 

X 

34 

7.427 

7.667 

7.906 

8.146 

8.386 

8.625 

8.865 

9.104 

9.344 

X 

94 

7.589 

7.833 

8.078 

8.323 

8.568 

8.813 

9.057 

9.302 

9.547 

X 

12. 

7.750 

8.000 

8.250 

8.500 

8.750 

9.000 

9.250 

9.500 

9.750 

12. 

34 

8.073 

8.333 

8.594 

8.854 

9.115 

9.375 

9.635 

9.896 

10.16 

X 

13. 

8.396 

8.666 

8.938 

9.208 

9.479 

9.750 

10.02 

10.29 

10.56 

13. 

34 

8.719 

9.000 

9.281 

9.563 

9.844 

10.13 

10.41 

10.69 

10.97 

X 

14. 

9.042 

9.333 

9.625 

9.917 

10.21 

10.50 

10.79 

11.08 

11.38 

14. 

34 

9.365 

9.666 

9.969 

10.27 

10.57 

10.88 

11.18 

11.48 

11.78 

X 

15. 

9.688 

10.000 

10.31 

10.63 

10.94 

11.25 

11.56 

11.88 

12.19 

15. 

34 

10.01 

10.33 

10.66 

10.98 

11.30 

11.63 

11.95 

12.27 

12.59 

X 

16. 

10.33 

10.67 

11.00 

11.33 

11.67 

12.00 

12.33 

12.67 

13.00 

16. 

34 

10.66 

11.00 

11.34 

11.69 

12.03 

12.38 

12.72 

13.06 

13.41 

X 

17. 

10.98 

11.33 

11.69 

12.04 

12.40 

12.75 

13.10 

13.46 

13.81 

17. 

34 

11.30 

11.66 

12.03 

12.40 

12.76 

13.13 

13.49 

13.85 

14.22 

X 

* 18. 

11.63 

12.00 

12.38 

12.75 

13.13 

13.50 

13.88 

14.25 

14.63 

18. 

19. 

12.27 

12.67 

13.06 

13.46 

13.85 

14.25 

14.65 

15.04 

15.44 

19. 

20. 

12.92 

13.33 

13.75 

14.17 

14.58 

15.00 

15.42 

15.83 

16.25 

20. 

21. 

13.56 

14.00 

14.44 

14.88 

15.31 

15.75 

16.19 

16.63 

17.06 

21. 

22. 

14.21 

14.66 

15.13 

15.58 

16.04 

16.50 

16.96 

17.42 

17.88 

22. 

23. 

14.85 

15.33 

15.81 

16.29 

16.77 

17.25 

17.73 

18.21 

18.69 

23. 

24. 

15.50 

16.00 

16.50 

17.00 

17.50 

18.00 

18.50 

19.00 

19.50 

24. 






























































424 


BOARD MEASURE 


Table of Board Measure — (Continued.) 


09 

A 2 
a A 
T3 O 

£ fl 


Feet of Board Measure contained in one running foot of Scantlings 
of different dimeusious. (Original.) 



10 

10K 

10A 

10 A 

11 

11A 

UA 

119* 

12 



Ft. Bd.M. 

Ft. Bd.M. 

Ft. Bd.M. 

Ft. Bd.M. 

Ft. Bd.M. 

Ft. Bd.M. 

Ft. Bd.M. 

Ft. Bd.M. 

Ft. Bd.M. 



.2083 

.2135 

.2188 

.2240 

.2292 

.2344 

.2396 

.2448 

.2500 

A 

74 

.4107 

.4271 

.4375 

.4479 

.4583 

.4688 

.4792 

.4896 

.5000 

A 

'i» 

A 

] # 

.6250 

.8333 

.6406 

.8542 

.6563 

.8750 

.6719 

.8958 

.6875 

.9167 

.7031 

.9375 

.7188 

.9583 

.7344 

.9792 

.7500 

1.000 

a 

i. 

1/ 

1.042 

1.068 

1.094 

1.120 

1.146 

1.172 

1.198 

1.231 

1.250 

A 

/A 

1.250 

1.821 

1.31.3 

1.344 

1.375 

1.406 

1.438 

1.469 

1.500 

A 


1.458 

1.495 

1.531 

1.568 

1.604 

1.641 

1.677 

1.714 

1.750 

A 

2. 

1.667 

1.708 

1.750 

1.792 

1.833 

1.875 

1.917 

1.958 

2.000 

2. 

\y. 

1.875 

1.922 

1.969 

2.016 

2.063 

2.109 

2.156 

2.203 

2.250 

A 


2.083 


2.188 

2.240 

2.292 

2.344 

2.396 

2.448 

2.500 

A 


2.292 

2.349 

2.406 

2.464 

2.521 

2.578 

2.635 

2.693 

2.750 

A 

3, 

2.500 

2.563 

2.625 

2.688 

2.750 

2.813 

2.875 

2.938 

3.000 

3. 

\A 

2.798 

2.776 

2.844 

2.911 

2.979 

3.017 

3.115 

3.182 

3.250 

A 


2.917 

2.990 

3.063 

3.135 

3.208 

3.281 

3.354 

3.42 / 

3.500 

A 


3.125 

3.203 

3.281 

3.359 

3.438 

3.516 

3.594 

3.672 

3.750 

A 

4. 

3.333 

3.417 

3.500 

3.583 

3.667 

3.750 

3.833 

3.917 

4.000 

4. 

V 

3.542 

3.630 

3.719 

3.807 

3.896 

3.984 

4.073 

4.161 

4.250 

A 


3.750 

3.844 

3.938 

4.031 

4.125 

4.219 

4.313 

4.406 

4.500 

A 


3.958 

4.057 

4.156 

4.255 

4.354 

4.453 

4.552 

4.651 

4.750 

A 

5. 

4.167 

4.271 

4.375 

4.479 

4.583 

4.688 

4.791 

4.896 

5.000 

5. 

V 

4.375 

4.4S4 

4.594 

4.703 

4.813 

4.922 

5.031 

5.141 

5.250 

A 

A 

4.583 

4.698 

4.813 

4.927 

5.042 

5.156 

5.270 

5.385 

5.500 

A 

H 

4.792 

4.911 

5.031 

5.151 

5.271 

5.391 

5.510 

5.630 

5.750 

A 

6. 

5.000 

5.125 

5.250 

5.375 

5.500 

5.625 

5.750 

5.875 

6.000 

6. 

y< 

5.208 

5.339 

5.469 

5.599 

5.729 

5.859 

5.990 

6.120 

6.250 

A 

A 

5.417 

5.552 

5.688 

5.823 

5.958 

6.094 

6.229 

6.365 

6.500 

A 

H 

5.625 

5.766 

5.906 

6.047 

6.188 

6.328 

6.469 

6.609 

6.750 

A 

7. 

5.833 

5.979 

6.125 

6.271 

6.417 

6.563 

6.708 

6.854 

7.000 

7. 

A 

3.042 

6.193 

6.344 

6.495 

6.646 

6.797 

6.948 

7.099 

7.250 

A 

A 

6.250 

6.406 

6.563 

6.719 

6.875 

7.031 

7.188 

7.344 

7.500 

A 

% 

3.458 

6 620 

6.781 

6.943 

7.104 

7.266 

7.427 

7.589 

7.750 

A 

8. 

6.667 

6.833 

7.000 

7.167 

7.333 

7 500 

7.667 

7.833 

8-000 

8. 

A 

6.875 

f.047 

7.219 

7.391 

7.563 

7.734 

7.906 

8.078 

8.250 

A 

H 

7.083 

7.260 

7.438 

7.615 

7.792 

7.969 

8.116 

8-323 

8.500 

A 

% 

7.292 

7.474 

7.656 

7.839 

8.021 

8.203 

8.385 

8.568 

8.750 

A 

9. 

7.500 

7.S88 

7.875 

8.063 

8.250 

8.438 

8.625 

8.813 

9.000 

9. 

A 

7.708 

7.901 

8.094 

8.286 

8.479 

8.672 

8.865 

9.057 

9.250 

A 

A 

A 

7.917 

8.115 

8.313 

8.510 

8.709 

8.906 

9.104 

9.302 

9.500 

A 

8.125 

8.323 

8.531 

8.734 

8.939 

9.141 

9.344 

9.547 

9.750 

A 

JO. 

8.333 

8.542 

8.750 

8.958 

9.167 

9.375 

9.583 

9.792 

10.00 

10. 

A 

8.542 

8.755 

8.969 

9.182 

9.396 

9.609 

9.823 

10.04 

10.25 

A 

A 

8.750 

8.969 

9.188 

9.406 

9.625 

9.844 

10.96 

10.28 

10.50 

A 

H 

8.958 

9.182 

9.406 

9.630 

9.854 

10.08 

10.30 

10.53 

10.75 

A 

11. 

9.167 

9.396 

9.625 

9.854 

10.08 

10 31 

10.54 

10.77 

11.00 

li. 

A 

9.375 

9.609 

9.844 

10 08 

10.31 

10.55 

10.78 

11.02 

11.25 

A 

A 

9.583 

9.823 

10.06 

10.30 

10.54 

10.78 

11.02 

11.26 

11.50 

A 

A 

9.792 

10.04 

10.28 

10.53 

10.77 

11.02 

11.26 

11.51 

11.75 

A 

12. 

10.00 

10.25 

10.50 

10.75 

11.00 

11.25 

11.50 

11.75 

12.00 

12. 

A 

10.42 

10.68 

10.94 

11.20 

11.46 

11.72 

11.98 

12.24 

12.50 

A 

13. 

10.83 

11.10 

11.38 

11.65 

11.92 

12.19 

12.46 

12.73 

13.00 

13. 

** 

11.25 

11.53 

11.81 

12.09 

12.38 

12.66 

12.94 

13.22 

13.50 

A 

14. 

11.67 

11.96 

12.25 

12.54 

12.83 

13.13 

13.42 

13.71 

14.00 

14. 

A 

12.08 

12.39 

12.69 

12.99 

13.29 

13.59 

13.90 

14.20 

14.50 

A 

15. 

12.50 

12.81 

13.13 

13.44 

13.75 

14.06 

14.38 

14.69 

15.00 

15. 


12.92 

13.24 

13.56 

13.89 

14.21 

14.53 

14.85 

15.18 

15.50 

A 

16 

13.33 

13.67 

14.00 

14.33 

14.67 

15.00 

15.33 

15.67 

16.00 

16. 

A 

13.75 

14.09 

14.44 

14.78 

15.13 

15.47 

15.81 

16.16 

16.50 

A 

17, 

14.17 

14.52 

14.88 

15 23 

15.58 

15.94 

16.29 

16.65 

17.00 

17. 

A 

14.58 

14.95 

15.31 

15.77 

16.04 

16.41 

16.77 

17.14 

17.5C 

A 

IV. 

15.00 

15.38 

15.75 

16.13 

16.50 

16.88 

17.25 

17.63 

18.00 

18. 

19. 

15.83 

16.23 

16.63 

17.02 

17.42 

17.81 

18.21 

18.60 

19 00 

19. 

20. 

16.67 

17.08 

17.50 

17.92 

18.33 

18.75 

19.17 

19.58 

20.00 

20. 

21. 

17.50 

17.94 

18.38 

18.81 

19.25 

19.69 

20.13 

20.56 

21.00 

21. 

22. 

18.33 

18.79 

19.25 

19.71 

20.17 

20.63 

21.08 

21.54 

22.00 

22. 

23. 

19.17 

19.65 

20.13 

20.60 

21.08 

21.56 

22.04 

22.52 

23.00 

23. 

24. 

20.00 

20.50 

21.00 

21.50 

22.00 

22.50 

23.00 

23.50 

24.00 

24. 


d . 
at 

a 2 
A 
'O O 





















































TIMBER. 


425 


PRESERVATION OF TIMBER. 

Art, 1. (a) The decay of timber is caused by the fermentation of its 
sap. If dry air circulates freely about the sides and ends of the sticks, the sap 
evaporates. If air is excluded, as when timber is kept constantly and entirely 
immersed in salt or fresh water, the sap cannot ferment. In either case timber 
may resist decay for centuries. Sap, confined iu timber with air, ferments, pro¬ 
ducing <lry rot; as where beams are enclosed air-tight in brickwork etc; and 
where green timber is painted or varnished, or treated with creosote etc. The 
sap then not only prevents the thorough penetration of the oil etc, but may 
cause the greater part of the wood to rot although its firm outer shell gives it a 
deceptive appearance of strength, (b) Sap should therefore be first removed by 
seasoning ; i e, either by drying the wood in air at natural or higher tem¬ 
peratures, or by first steaming the wood under pres so as to vaporize the sap, 
and then removing the latter by means of a vacuum. Thorough seasoning of 
large timbers in dry air at ordinary temperatures may require years; and too 
rapid kiln-drying cracks and weakens the wood. But it is questionable whether 
steaming and vacuum remove sap as thoroughly as do the slower dry processes, 
(c) Alternate exposure to water and air is very destructive. It causes wet rot. 

Art. 2. Sea-worms. The limnoria terebrans works from near high-water 
mark to a little below the surface of mud bottom ; the teredo navalis within some¬ 
what less limits. The teredo is said to be rendered less active by the presence 
of sewage in water. 

Art. 3. (a) The best timber-preserving processes are practically useless 
unless thoroughly well done. If the gain in durability will not war¬ 
rant the expenditure of time and money reqd for this, it is more economical to 
use the wood in its natural state, (b) The woods best adapted to 
treatment are those of an open or porous texture, as hemlock etc. They ab¬ 
sorb the oil etc better than the denser woods; and their cheapness renders the 
use of the treatment more economical, (c) Most of the processes iu common 
use seem to render wood less combustible. (d) After treatment by any process, 
the wood should be well dried before using. 

Art. 4. (a) Creosote oil, or dead oil, is the best known preservative. 
Against sea-worms it is effective for at least 25 years, and is the only known pro¬ 
tection. (b) As temporary expedients, piles are sometimes covered with sheet 
metal or with broad-headed nails driven close together. These rust or wear 
away in a few years. Oak piles, cut in January, and driven with the bark on, 
have resisted the teredo for 4 or 5 years; and cypress piles, well charred, for 9 
years, (c) For ordinary exposures on land, 8 to 10 lbs of creosote oil, per cub 
ft are reqd = say 670 to 830 lbs per 1000 ft board measure = 30 to 40 lbs per cross 
tie of 4 cub ft. For protection against sea-worms 10 to 12 lbs per cub ft suffice in 
climates like those of Great Britain and the Northern U. S.: but in warmer 
waters where the teredo is very active, from 14 to 20 lbs per cub ft are used. 
Large timbers may not require saturation throughout, and thus may take less 
per cvh ft. But see (i) and end of Art. 1 (a), (d) Creosote oil weighs about 
8.8 lbs per U. S. gallon. Its cost (1886) is about 1 ct per lb: that of the process, 
applied to pine and similar woods, including oil, from 18 cts per cub ft of tim¬ 
ber for 10 lbs of oil per cub ft, to 35 cts per cub ft for 18 lbs per cub ft. Special 
prices for hard wood. See (j). It is cheaper in England, (e) Th.' sticks should 
be reduced to their intended final dimensions and framed (if framing is reqd) 
before treatment: especially if for exposure to teredo, which is sure to attack 
any spots which (as by subsequent cutting) are left unprotected, (f) Creosoted 
ties have remained sound after 22 years’ exposure. The creosote protects the 
spikes from rusting, (g) Spruce, owing to its irregular density, is unsuitable 
for creosoting. (h) Creosote renders wood stiffer and slightly more brittle. In 
hot weather it exudes to some extent and discolors the wood. Its smell excludes 
it from dwellings, (i) It does not wash out from the wood, but often fails to 
penetrate the heart-wood. Then, if any sap remains, decay begins at the cen¬ 
ter. See end of Art 1 (a). Burnettizing the cen of the stick (see Art 7) and us¬ 
ing a coating of creosote outside, has long been suggested as the best possible 
method. This is cheaper than thorough creosoting. For cost, etc, of a “zinc- 
creosote ” process, address J. P. Card, Chicago, Ill. ( j) Freosoting is (lone 
by the Eppinger & Russell Creosoting Works, office 160 Water St, New York. 
Their arrangements are such that the process can, if desired, be confined to a 
portion of the length of the stick. For their prices see (d). The Carolina Oil & 
Creosote Co, Wilmington N. C. treat timber with creosote oil obtained from 
wood. 



425 a 


TIMBER. 


Art. 5. (a) Mineral solutions are inferior to creosote, even on land; 
and useless in running water or against sea-worms; but they approximately 
double the life of inferior timber under ordinary land exposures; and their 
cheapness permits their use where that ot creosote is too expensive. (b) They 
render wood harder; and brittle if the solution is too strong. They are liable 
to be washed out by rain etc. Hence the outer wood decays tirst. See Art 4 (i) 
Art 8 (b) (c) (d). (c) A committee of the American Soc of Civ Engrs,* alter coj- 
lating a large number of experiments, recommended Burnettizing; (Art 7) 
for damp exposure, as that of cross ties, damp floors etc ; and liyaniz- 
ing (Art b) for comparatively dry situations with exposure to air 
and sun-light, as in bridge timbers, for which it is better suited than Burnettiz- 
ing because it seems to weaken wood less, in such exposures it preserves wood 
for 20 to 30 years. 

Art. 6. (a) Kyanizing consists in steeping the wood in a warm solu¬ 
tion of 1 lb of bi-chloride of mercury (corrosive sublimate) in 100 

lbs of water, (b) It is usual to allow the wood to soak a day for each inch of the 
thickness, or least dimension, of the piece, and one day in addition, whatever 
the size, (c) With the sublimate at 55 ets per ft) (1886) and 4 to 5 lbs per 1000 ft 
bm, it costs about $7 per 1000 ft bin, or 8£ cts per cub ft, or 34 ets per tie of 4 
cub ft. (d) Gen’l Cram found the process very unhealthy, “ salivating all 
the men”; but Mr. J. B. Francis, at Lowell, and Mr. H. Bissell of the Eastern 
R. R. of Mass, had little or no trouble in this respect. The sublimate, however, 
which is very poisonous, is apt to effloresce, and the use of the timber is thus 
rendered dangerous, (e) The wood decays sooner under than above ground; 
but spruce ties, kyanized in 1840, were perfectly sound in 1855. The sublimate 
readily washes out, and the process is therefore unsuitable for damp situations. 
It is carried on by the Proprietors of the Locks and Canals on Merrimack 
River, at Lowell, Mass. 

Art. 7. (a) Burnettizing consists in immersing the wood for several 
hours in a solution of 2 lbs chloride of zinc in 100 lbs of water, under a 
pres of from 100 to 300 lbs per sq inch, (b) It. seems to render wood more brit¬ 
tle than kyanizing, and is therefore less adapted for timbers bearing tensile or 
transverse strain. Spikes rust away rapidly in Burnettized ties, (c) It costs 
(1886) about $5 per 1000 ft bm = 6 cts per cub ft = 24 cts per tie of 4 cub ft. 

Art. 8. Other preventives, (a) Steeping in a solution of sulphate 
of copper (bine vitriol) has been extensively used, but does not seem to 
have been permanently successful. The blue vitriol washes out readily, (b) 
In the Tliilinany process, as practised by the Wisconsin Wood Preserv¬ 
ing Co., Milwaukee, Wis.,tlie timber is first steamed. The steam and air are 
then exhausted by an air-pump, alter which are injected first a solution of sul¬ 
phate of copper (blue vitriol) or of sulphate of zinc (white vitriol) and then one 
of chloride ot barium, both under pressure. It is claimed that this fills the pores 
with insoluble sulphate of baryta. Railroad ties require about 12 hours. Cost, 
about 20 cents per tie, or from to $5 per 1000 feet, board measure, (e) The 
Wcllliouse process, as employed by the Chicago Tie Preserving Co., 
injects first a solution of chloride of zinc with glue, and then one of tannin 
(both under pres), in order to diminish the subsequent washing out of the 
chloride. The process costs (1886) about $8 per 1000 ft bm = say 10 cts per cub 
ft. It is not recommended for sub-aqueous use. (<1) The “Gypsum pro¬ 
cess” of the American Wood Preserving Co. of St Louis Mo uses gypsum and 
chloride of zinc in order to retain the latter more perfectly. The wood is then 
kiln-dried, (e) Fence-posts etc seem to be preserved to some extent by having 
only their Imver ends dipped in tar well boiled to remove the ammonia, which 
last is destructive to wood. The upper end must be left uutarred to let the sap 
evaporate, (f) Attempts at wood preservation by means of vapor of creo¬ 
sote etc have proved failures, (g) While wood' is thoroughly saturated with 
petroleum it does not decay. But unless the supply is kept up the oil 
evaporates and leaves the wood unprotected. (1») Cottonwood ties laid upon a 
soil containing about 2 per cent carbonate of lime, 1 per cent salt and £ per 
cent each of potash and oxide of iron, on the Union Pacific R. R. in 1868, were 
found in 1882 “as sound and a good deal harder than when first laid,” although 
such ties in other soils lasted hut from 2 to 5 years, (i) The use of solutions of 
limeaud of salt; and charring the surface; are sometimes found useful 
in damp situat ions. 


* See Transactions, July, Aug and Sept 1885, to which we are indebted for many of the above sug- 
gestions. 






TIMBER. 


42 ob 


Price of 1 u m bor . Phila, 1886; Spruce joists, $20 to $24 per 1000 ft board meas. 
Hemlock joists, $18 to $16. Yellow pine floor boards, $20 to $35. White pine boards, 
$18 to $50, according to quality, degree of seasoning, &c. Sawed W pine timbers, $28 
to $35. Heart Y pine, $20 to $35. Hemlock, $16 to $20. Gillingham, Garrison & 
Co., 943 Richmond St. 

Boards of oak or pine, nailed together by from 4 to 16 tenpenny 

common cut nails, and then pulled apart in a direction lengthwise of the boards, and across the nails, 
tending to break the latter in two by a shearing action, averaged about 300 to 400 lbs per nail to sepa¬ 
rate them ; as the result of many trials. 


Table of Cut Ufails, ordinary pattern. 
Base price, Phila, 1886, about $2.50 per keg containing 100 lbs.* 


Name. 

Length. 

Inches. 

Number 
of Nails 
per fi>. 

Extra over 
base price, 
cts per keg. 


Name. 

Length. 

Inches. 

Number 
of Nails 
per lb. 

Extra over 
base price, 
cts per keg. 

3 penny 


557 

150 


10 penny 

3 

60 

0 

4 “ 

ill 

336 

75 


12 “ 

z'A 

50 

0 

5 “ 

m 

210 

75 


20 “ 

4 

32 

0 

6 “ 

2 

163 

50 


30 “ 

4^ 

19 

0 

7 “ 

2 A 

123 

50 


40 “ 

5 

16 

0 

8 “ 


93 

25 


50 “ 


13 

0 


The sizes and weights vary considerably with different makers. Ours are averages. The 
above are machine made, or cut nails, cut by machinery from plates of rolled iron or steel. 
Wrought nails are forged by a blacksmith, or'by machinery, from rolled iron or steel rods. 

Nails cut from Bessemer and similar steels cost about 10 cts per keg more than 
the above. 


# Morris, Wheeler & Co (Pottstown Iron Co), 18th and Market Sts, Phila, manufacturers. 

























426 


PLASTERING 


PLASTERING. 


The plastering of the inside walls of buildings, whether done on laths, bricks, or 
stone, generally cousists of three separate coats ol mortar. The first of these is called 
by workmen the rough or scratch coat; and consists ol about 1 measure ol quicklime, 
to 4 of sand ; (which latter need not be of the purest kind;) and ]/ A measure of bul¬ 
lock or horse hair; the last of which is for making the mortar more cohesive, and 
less liable to split oft in spots. This coat is about % to ^ inch thick; is piit on 
roughly; and should be pressed by the trowel with sufficient force to enter perfectly 
between and behind the laths; which for facilitating this should not be nailed 
nearer together than an inch. In rude buildings, or in cellars, <tc, this is often 
the only coat used. When this first coat has been left for one or more days, accord¬ 
ing to the dryness of the air, to dry slightly, it is roughly scored , or scratched , (hence 
its name,) with a pointed stick, or a lath, nearly through its thickness, by lines run¬ 
ning diagonally across each other, and about 2 to 4 ins apart. This gives a better 
hold to the second coat, which might otherwise peel oft’. If the first coat has be¬ 
come too dry, it is well also to dampen it slightly as the second one is put on. 

The second coat is put on about % to % inch thick, of the same hair mortar, or 
coarse stuff. Before it becomes hard, it is roughed over by a hickory broom, or 
some substitute, to make the third coat adhere to it better. 

The third coat, about % inch thick, contains no hair; and forgiving it a still 
whiter and neater appearance, more lime is used, say 1 of lime, to 2 of sand: and 
the purest sand is used. This mortar is by plasterers called stucco; a name 
also applied to mortar when used for plastering the outsides of buildings. Or in¬ 
stead of stucco, the third coat may be, and usually is, of hard finish, or gauge stuff; 
which consists of 1 measure of ground plaster of Paris, to about 2 of quicklime, 
without sand. Hard finish works easier; but is not as good as stucco, for walls in¬ 
tended to be painted in oil. The plaster of Paris is for hastening the hardening. 

Either of these third coats is smoothed or polished to a greater or less extent, according to whether 
it is to show, or to be papered, painted, &c. The polishing tools are merely, the trowel; the band- 
tioat, (a kind of wooden trowel;) and the water brush, (a short-handled brush for wetting the surface 
part at a time with water, iu order to polish more freely.) For finer polishing, a float made of cork 
is used. The smooth piece of board about 10 to 12 ins square, with a handle beneath, on which the 
plasterer holds his mortar until he puts it on to the wall with his trowel, is called a hawk. 

The more thoroughly each coat is gone over with the water-brush and trowel, (which process is 
called hand floating ,) the firmer and stronger will it be. Frequently only two coats of plastering are 
put on in inferior rooms ; or where great neatness of appearance is not needed. The first is of hair 
mortar, or coarse stuff; this is scratched with the broom, and then covered by the finisbiug coat of 
finer mortar, (stucco.) If this last is nearly all lime, or with but very little sand, to make it work 
easier, it is called a slipped coat. Without any sand it is called fine stuff. Neither is as good as 
stucco, if the wall is to be papered. When this is the case, the third coat also may have a little hair, 
to give it more strength ; but this is not absolutely necessary. 

A very good effect may be produced in station-houses, churches, &c, by only two coats of plaster in 
which fine clean screened gravel is used instead of sand. When lined into regular courses, it resem¬ 
bles a buff-colored sandstone, very agreeable to the eye. 

In purchasing plastering hair, care must be taken that it has not been taken from salted hides; 
inasmuch as the salt will make the walls damp. For the same cause sea-shore sand should not be 
used. It is almost impossible to wash it entirely free from salt. 

In brick walls intended to be plastered, the mortar joints should be left very rough, to let the plas¬ 
ter adhere. If it is put on smooth walls, without first raking out the mortar to the depth of nearly 
an inch, it is very apt to fall off; especially from outside walls; as can be seen daily in any of our 
cities. As this raking out of brick joints is tedious and expensive, it would generally be better to 
use paint rather than plaster. The walls should also be washed clean from all dust; aud should be 
slightly dampened as the plaster is put on. 

To imitate granite on outer walls: after the second or smooth coat of plaster is dry, it receives a 
coat of lime wash, slightly tinted by a little umber, or ochre, &c. After this is dry, in case it appears 
too dark, or too light, another may be applied with more or less of the coloring matter in it. Finally, 
a wash of lime and mineral-black is sprinkled on from a Hat brush, to imitate the black specks of 
granite. By this simple means, a skilful workman can produce excellent imitations. The horizontal 
and vertical joints of the imitation masonry, may be ruled in by a small brush, usiug the same black 
wash, and a long straight-edge. 

The rough surfaces of all walls are more or less warped, or out of line ; and it is not possible for 
the plasterer to rectify this perfectly by eye, as may be seen in almost every house. Even in what 
are called first-class ones, a quick eye can generally detect unsightly undulations of the plastered 
surfaces. 

To prevent this, the process of screening* is resorted to. Screeds are a kind of 

gauge or guide, formed by applying to the first rough coat, when partly dried, horizontal strips of the 
plastering mortar, about 8 ins wide, and from 2 to 4 ft apart all arouud the room- These are made to 
project from the first coat, out to the intended face of the second one; and while soft arc carefully 
made perfectly straight, and out of wind with each other, by means of the plumb-line, straight-edge. 
&c. When they become dry, the second coat is put on. filling up the broad horizontal spaces between 
them; and is readily brought to a perfectly flat surface, corresponding with that of the screeds, by 
means of long straight-edges extending over two or more of the latter. 

A <lay’s work at plastering*. 

A plasterer, aided by one or two laborers to mix his mortar, and to keep his hawk supplied, can 
average from 100 to 200 square yards a day, of first coat; about % as much of secoud; aud half as 



SLATING. 


427 


much of third, which requires more care. The amount will depend upon the number of angles, size 
of rooms, whether on ceilings or on walls, &c, Ac. 


Geu Gillmore's estimate of cost of plastering:* 100 square yarde 
with 2 or with 3 coats. Common labor $1 per day. 


Materials. 

Three Coats. 

Hard finished work. 

Two Coats. 
Slipped coat finish. 

Quicklime. 

4 casks. 

$4.00 

3K casks. 

$3.33 

“ for fine stuff. 

% “ 

.85 



Plaster of Paris. 

K “ 

.70 



Laths. 

2000 

4.00 

2000. 

4.00 

Hair... 

4 bushels. 

.80 

3 bushels. 

.00 

Common Sand. 

7 loads.t 

2.00 

6 loads. 

1.80 

White Sand. 

2K bushels. 

.25 



Nails. 

13 lbs. 

.90 

13 lbs. 

.90 

Mason’s labor. 

4 days. 

7.00 

3K days. 

6.12 

Laborer. 

3 days. 

3.00 

2 days. 

2.00 

Cartage. 


2.00 


1.20 

Cost of 100 square vards. 


$25.50 


$19.95 






This amounts to 25K cts per sq yd for 3 coats; and say 20 cts for 2 coats. See Art 6, P 674. 


Plastering: laths are usually of split white or yellow pine, in lengths of 
about 3 to 4 feet; and hence called 3 or 4 ft laths. They are about IK ins wide, by K inch thick. 
They are nailed up horizontally, about K inch apart. The upright studs of partitions are spaced at 
such distances apart, (generally about 15 ins from center to center.) that the ends of the laths may 
be uailed to them. Laths are sold by the bundle of 1000 each. A square foot of surface requires IK 
four feet laths ; or 1000 such laths will cover 666 sq ft. Sawed laths may be had to order, of any re¬ 
quired length. A carpenter can nail up the laths for from 40 to 60 sq yds of plastering in a day of 
10 hours; depending on the number of angles in the rooms, Ac. 



SLATING. For prices, see note* p 429. 


Roofing slates are usually from % to ^ inch thick; about y 3 q being a common 
average. They may be nailed either to a sheeting of rough boards (c, g , in the fig) 
from % to 1 % inch thick, (which should be, but rarely are, tongued and grooved,) 



^Average prices of plastering: in Philada, 1886, in cts per sq yard. 

Three coins, iuciudiSg laths, scaffold 1 Ac. 50 to 55 cts- Two coats, 35 to 40.. Three coats on brick or 
stone (no laths reqd). 30 to 40. Outside plastering, 60; or if to imitate marble, 75. Simple plaster 
. i i opts ner inch of frirth, per ft run. Plaster center flowers for parlors, $o to $.15 each, 
put up ’ The plastering of a 20-ft front, 3-story dwelling, with large 3-storv back buildings, $o00 to 
$700. ‘Stipulate expressly to pay only for surfaces actually plastered; and thus avoid extras, even 

if vou have to nav a few cts more per yard. __ ,, _„ . „ ,. 

t A IosmI (one-horse), both in the U. S. and in England, usually means a cub 

but many dealers adopt 20 struck bushels = 25 cub ft = fully a ton. 




























































428 


SLATING 


laid horizontally from rafter to rafter; or sloping, from purlin to purlin, as the cas 
may be; or to stout laths ttt about 2 to 3 ins wide, and from 1 to 1% thick, uaile 
to the rafters at distances apart to suit the gauge of the slates. Two nails are used t 
each slate; one near each upper corner. They may be either of copper, (which is th 
most durable, hut most expensive,) of zinc, or of either galvanized or tinned iror 
The last two are generally used; or in inferior work, merely plain iron ones, pre i 
viously boiled in linseed oil, as a partial preservative from rust. Rust, howevei 
sometimes weakens them so much that they break ; and the slates are blown off ii 
high winds, to the danger of passers by. Since good slate endures for a long seric 
of years, it is true economy to use nails that are equally durable. In iron roots, th 
slates, instead of being nailed to boards, are sometimes tied directly to the iroi 
purlins, by wire. A square of slating, shingling, &c, is 100 sq ft. 

In laboratories, chemical factories, &c, subject to acid fumes, it is difficult t< 

provide a metal fastening that will not be eaten away. In such cases it is best to depend chiefly upo; 
a layer of tnortar between the slates. This will harden before the metal fastenings give way ; am 
will hold the slates in place, while new fastenings are being inserted. 

The least pitch considered advisable for a roof, to prevent rain or snow from being drivei 
through the interstices between the slates, is about 26)4°; or 1 vert to 2 bor; which corresponds to : 
rise of hi the span in a common double pitched roof. But even at steeper pitches, rain, and mor 
particularly snow, will be forced through the roof by violent winds; especially if laths alone be used 
or even boarding alone. To avoid this, a layer of mortar about hi inch thick, may be spread ove 
the touching surfaces of the slates if on laths. If on boards, the same process may be adopted ; o 
the more common one of first covering the boards with a layer of what is called slating felt; bu 
which in reality is merely thick brown paper, soaked in tar. This is sold in long continuous rolls 
28 ins wide, and weighing from 40 to 50 lbs. A 50 R> roll will cover about 300 sq ft of roof. Wit) 
proper precautions against the admission of rain and snow, a pitch as flat as 1 in 2)4. or even 1 it 
3, may be adopted. 

The thickness of slate on a roof is double; except at the laps is, is. kc, where it is triple. Thi 
lap is measured from the nail hole (under i) of the lower slate, to the lower edge or tail, s, of th. 
upper one ; and is usually about 3 ins. In order that the showing lower edges of the slates shall 
when laid, form regular straight lines along the roof, the nail holes are made at equal distances fron 
said lower edges ; so that any irregularity of length is concealed from view at the hidden heads ol 
the slates. The slater estimates the length of his slate from the nail hole to the tail: discarding thi 
narrow strip between the nail hole and the head. If from this reduced length the lap be deducted 
then one-half of the remainder will be the gauge, weathering, or margin, of the slating ; or. in othei 
words, the showing or exposed width of the courses of slates. The gauge in ins multiplied by th( 
width of a slate in ins, gives the area in sq ins of finished roof covered by a single slate ; and if 14‘ 
(the sq ins in a sq foot) be divided by this area, the quotient will be the number of slates required pei 
sq ft of roof. The upper side of a slate is called its hack; the lower one, its led. 

Slating, like shingling, must evidently be commenced at the eaves, and extended upward. Sinct 
the beds of the slates are not exactly parallel to the boarding, and consequently do not rest flat upor 
it, those at the lower edge w would easily be broken. To prevent this, a tilting strip (a stout wid< 
lath, with its upper side planed a little bevelling, to suit the slope of the slates) is first nailed arounc 
near the eaves, for the tails of the lowest course of slates to rest on. This is shown ou a larger scalt 
atT. 

Slate of the best quality has a glistening semi-metallic appearance, somewhat like that of a surfact 
of paper rubbed with black lead pencil. That of a dull earthy aspect, is softer, more absorbent, anc 
consequently more liable to yield to atmospheric influences, fain, frost, &c. Iron pyrites frequently 
occurs in slate; and since it always decomposes and leaves holes, should never be admitted on a roof. 
Of two qualities of slate, that which absorbs the least weight of water, when pieces of equal size art 
soaked for an hour or two, is generally the best; being^least liable to split by frost, and become 
weather-worn. This test is easily applied. 

In England the different sizes are distinguished by absurd names of no meaning. In the 
United States they are called 6 by 12’s; 16 bv 24’s, &c, according to their measures in inches. They 
may be cut to order, of almost any prescribed dimensions, or shape. Those in common use vary from 
about 7 by 14, to 12 by 18. The first forms about 5 to 6 inch courses; and the last about 7 to 8* inch ; 
depending upon how far from the head the nail holes are pierced. The farther this is, the firmer 
will the slating be. 

Slate roofs, like iron ones, heat the rooms immediatelv below them very much. This is somewhat 
diminished when the slates are on hoards, instead of laths; and still more bv a coat of plaster be¬ 
neath. They are also liable to break when walked on ; less so when bedded in mortar. 

Weight of slate roofs. Slate weighs about 175 lbs per cub foot; therefore 
a sq ft, hi inch thick, weighs about 1.8 lbs; y 3 ^, 2.7 lbs; and hi thick, 3.0 lbs. But owing to the 
overlapping, a sqnare foot of roof requires about 2>4 sq ft of slate of ordinary sizes; and if the slate 
is laid on hoards an inch thick, the weight per sq ft of roof will be increased about 2)4 lbs • or wi'h 
Ihi inch boards, 2.8 fts. Laths will weigh about hi lb per sq ft of roof. 

Hence, 


Slate hi inch thick on laths. 

" “ on 1 inch boards. 

“ “ on 1)4 “ “ . 

“ 3-16 “ ou laths. 

“ “ “ on 1 inch boards. 

“ “ on \y A “ “ . 

“ hi “ on laths. 

“ “ “ on 1 inch boards. 

“ “ “ on 1)4 “ “ . 

2f slating felt is used, add hi ^ i or if the slates are bedded in 


Approx Weight 
of one sq ft of 
Slating, in lbs. 
.... 4.75 


6.75 

7.30 

7.00 

9.00 

9.55 

9.25 

11.25 

11.80 


hi inch of mortar, add 3 lbs. 











SHINGLES. 


429 


For the total weight borne by the roof trusses, that of the purlins also must be added. This will 
not, vary much from the limits of to 3 lbs per sq ft iu roofs of moderate span. Add for wind and 
snow, say ~0 lbs per sq ft; * and finally add the weight of the truss itself. 


For Stopping? tlie joints between slates (or shingles, &c) and chimneys, 

dormer w tudows, Ate, a mixtux-e of stirt white-lead paint, as sold by the keg, with sand enough to pre¬ 
vent it from running, is very good; especially if protected by a covering of strips of lead, or copper, 
tin, <fec, nailed to the mortar-joints of the chimneys, after being bent so as to enter said joints ; which 
should he scraped out for an inch in depth, and afterward refilled. Mortar protected in the sain? 
way. or even unprotected, is often used for the purpose; but is not equal to the paint and sand. Mor* 
tar a Tew days^ old, (to allow refractory particles of lime to slack,) mixed with blacksmith’s cinders* 
and molasses, is much used for this purpose , and becomes very hard, and effective 

For prices of slating, see foot note* below. 


- m > m m - 

SHINGLES, 


White cedar shingles are the best in use; and when of good quality will last 40 or 
, 50 years in our Northern States. They are usually 27 ins long; by from 6 to 7 ins 
wide; about % inch thick at upper end; and about % at lower end or butt; and are 
' laid in courses about 8^ ins wide; so that not quite % of a sliingle is exposed to the 
' weather. 

They are usually laid in three thicknesses; except for an inch or two at the upper ends, where there 
, are four. They are nailed to sawed shingling-laths of oak or yellow pine; about 1G ft long; 1% ins 

i wide, and 1 inch thick; placed in horizontal rows about 8% ins apart. These are nailed to the raft¬ 

ers. or purlins ; which, for laths of the foregoing size, should not be more than 2 ft apart from center 
i to center. Two nails are used to each shingle, near its upper end. They should not be of less size 

t than 400 to a lb. Wrought nails being the strongest, are the best; cut ones are apt to break 

by the warping of the shingles. Two pounds of such nails will suffice for 100 sq ft of roof, including 
i waste. An average shingle 7% ins wide, in 8% inch courses, exposes 63% sq ins; making 2% shingles 
( to a sq ft of roof; but to allow for waste, and narrow shingles, it is better in practice to allow about 3 
e shingles to a sq ft. 

Shingling, like slating, must plainly be begun at the eaves; and extended upward. For closing the 
r joints between the shingles, and chimneys, dormer windows, Ac, see at end of Slating, 
e Cypress and white pine are also much used for shingles, being much cheaper, hut scarcely half as 
t| durable.t All shingles wear quite thin in time by rain and exposure. In warm damp climates they 
• all decay within 6 to 12 years. 


i PAINTING. 

t 

The principal material used in house-painting, is either white lead, or oxide of 
j zinc, ground in raw (unboiled) linseed oil, by a mill, to the consistency of a thick 
! paste. In this condition, it is sold by the manufacturers in kegs of 25, 50, and 100 
lbs. To prepare it for actual nse, merely requires the addition of more linseed oil, 
say 3 or 4 pints to 10 lbs of the keg paint, for thinning it sufficiently to flow readily 
: under the brush. 

i j Good painting requires 4 or5 coats; but usually only 4 are used in principal rooms; and 3 in inferior 
i| ones. Each coat must be allowed to dry perfectly before the next one is put on. One fl> of the keg 
i paint will, after being thinned, cover about 2 sq yds of first coat; 3 yds of second; and 4 yds of each 
, subsequent coat; or 1 sq yd of 3 coats will require in all, 1.08 lbs ; of 4 coats, 1% lbs; of 5 coats, 1.58 
! lbs. The reason why the first coats require so much more than the subsequent ones, is that the bare 
surface of the wood absorbs it more. 

; When, as is usual, raw or unboiled oil is used for thinning, dryers must be added to it; otherwise 
the paint might require several weeks to harden; whereas, with dryers, from 1 to 3 days, according 
to the weather, suffice for each coat to become hard enough to receive the next one. The dryers most 
commonly used, are powdered litharge, in the proportion of one heaped teaspoonful; or Japau var¬ 
nish, 1 table-spoonful, to 10 lbs of the keg paiut. Either sugar of lead, or sulphate of zinc, may also 
be used instead of litharge; and in the same proportion. Although both litharge and Japan varnish 
are dark-colored, yet the quantity is so small as not to appreciably affect the whiteness of the paint. 
If the varnish is used in excess, as is often done in the hurry to have work finished, it produces 
cracks all over the surface. No dryer is necessary if painters' boiled oil be used for thinning. Mere 
boiling will not cause oil to harden more rapidly ; hut that intended for painters, has litharge added 
to it previously to boiling, in the proportion of 1% lbs to each 10 gallons of raw oil. In some works 
written for the use of house painters, it is asserted that boiling renders the oil too thick for any but 
coarse outdoor work. But this is entirely a mistake ; for if the boiling be properly done, the oil 
will be quite thin enough for the best inside work ; and will moreover be clearer than while raw ; aud 


* Price of slate, felt, and slating in Philada, in 1886, is from 8 to 9 cts per sq ft, 
according to quality of slate, kiud of nails, Ac ; but exclusive of boarding. With copper nails add l 
ct per sq ft. The slate from Peach Bottom, York Co, Penna, is the best in the State. It commands 
1 or 2 cts per sq ft more than the others. A roof of leaded tin will cost about the same as one of slate; 
aud not much more than half as much as good cedar shingles, (in Philada.) Felt about 2 cts per lb. 

t Price of shingles per 1000, in Philada, in 1886 : Cedar, 6 ins X 30 ins, $25; 6 ins X 24 ins, $20. 
Cypress. $18. Shingling laths, $5. Cedar shingles, laths, nails, and shingling complete, 20 cts per 
sq ft of roof; or about twice as much as slate or leaded tin roofing, exclusive of boards. 















430 


PAINTING 


Will impart to the painted surface a more shining appearance. The heat should be barely sufficient 
to produce boiling; or about 800° Fah. The boiling should continue about 1H hours; the oil being 
thoroughly stirred at short intervals, to preveut the litharge from settling at the bottom. The (ire 
may then be allowed to subside; when the operation will be completed. A sediment will then form 
at the bottom; which must be left behind when the oil is poured off. Although no dryer is necessary 
with this oil, still a little litharge may be added when great expedition demands it. Painters rarely 
use this oil, on account of its trifling increase of cost. 

Another substance much used with the thinning oil, (except for the first coat,) is spirits of turpen¬ 
tine ; called “ turp ” by the workmen. The quantity of oil may be diminished, to the exteut ot the 
added turp. This being more fluid than oil, causes the paint to work more pleasantly under the brush. 
It moreover diminishes the tendency of the paint to become yellow ; especially in rooms kept closed 
for some time. It is also much cheaper than oil. It should not be used, or but sparingly, for exposed 
outdoor work ; inasmuch as its tendency is to impair the firmness of the paint; and although its 
effects are scarcely appreciable indoors, they are quite apparent when the work has to resist the 
weather. As the fashions change in house-paiutiug, the surface is at times required to present a 
shining or glossy finish ; at other times a dead one is in vogue. The glossy one is that which the 
paint will naturally have, provided that no more turp thau oil be used in the thinning. The dead 
finish is obtaiued'by using no oil, but turp alone, for the last coat; which in that case is called a 
flatting coat. Although turp is not properly a dryer, still, as it evaporates quickly, it facilitates ihe 
hardening of the paint. 

In outdoor work it is usually advisable to use more dryer than inside, so that the paint may sooner 
become hard enough not to be injured by dust or rain. Otherwise less would be better. 

When, instead of a white finish, one of some other color is required, the coloring ingredient is 
mixed with the white paint to be used in the last coat only ; although two coloring coats are some¬ 
times fouud to be necessary before a satisfactory effect is produced. The coloring ingredients may be 
indigo, lampblack, terra sienna, umber, ochre, chrome yellow, Venetian red, red lead, Ac, &c; which 
are ground in oil, ready for sale, by the manufacturers of the white-lead and zinc paints. They are 
simply well stirred iuto the white paint. 

All surfaces to be paiuted, should first be thoroughly dry, and free from dust. If on wood, all 
plane-marks, and other slight irregularities, should first be smoothed off by sand-paper, when the 
neatest finish is required. Also, all heads of nails must be puuched to about % inch below the sur¬ 
face. To prevent knots from showing through the finished work, (as those in white or yellow pine 
would do, on account of the contained turpentine,) they must first be killed, as it is termed. A usual 
and effective way of doing this, is by covering them with two oats of shellac varnish ; which, when 
dry, should be smoothed by sand-paper. Another mode, not quite so certain, is by one or two coats 
of white lead mixed with thin glue-water, or size, as it is called. 

After these preparations, the first, or priming coat, is put on ; in which there should be no turp; 
because it would sink at once into the bare wood, leaving the white lead behind it, in a nearly dry 
friable condition. After this the nail holes, cracks, Ac, must be filled with common glaziers' putty, 
made of whiting (fine clean washed chalk) and raw linseed oil; boiled oil will not answer; the putty 
would be friable. The putty would be apt to fall out, if put in before priming; because the wood 
would absorb the oil, and the putty would then shrink. After the first coat is perfectly dry, the 
second one is put on ; and for it about 1 measure of turp may be mixed with 3 measures of the thin¬ 
ning oil. In the third, and any subsequent coats, equal measures of turp and oil, may be used for 
thinning, if the work is required to dry with a gloss; but if it is to finish dead, the last coat must 
be a flatting one; or one in which the thinning oil is entirely omitted, and turp alone substituted 
for it.* 

Painters generally clean their brushes by merely pressing out most of the paint with a knife; and 
then keep them in water until further use. If to be put away for some time, they may be thoroughly 
cleaned by turp; or by soap and water. To prevent a hard skin from forming on the top of their 
paint when not used for some days, they pour on a little oil. 

The best paints for preserving - iron exposed to the weather, 

appear to be pulverized oxides of iron, such as yellow and red iron ochres; or brown hematite iron 
ores finely ground ; and simply mixed with linseed oil, and a dryer. White lead applied directly to 
the iron, requires incessant renewal; and indeed probably exerts a corrosive effect. It may, how¬ 
ever, be applied over the more durable colors, when appearance requires it. Red lead is said to be 
very durable, when pure. An instance is recorded of pump-rods, in a well 200 ft deep, near London, 
which, having first been thus painted, were in use for 45 years; and at the expiration of that time, 
their weight was found to be precisely the same as when new; thus showiug that rust had not 
affected them. See p 403. 

When the size of the exposed iron admits of it, its freedom from rust may be very much promoted 
by first heating it thoroughly; and then dipping it iuto. or washing it well with, hot linseed oil; 
which will then penetrate into the interior of the iron. For tinned iron exposed to the weather, on 
roofs, rain pipes, Ac, Spanish brown is a very durable color. The tin is frequently found perfectly 
bright ami protected, when this color has been used, after an exposure of 40 or 50 years. White 
paint washes off in a few years by rain. 

Plastered walls should if possible be allowed to dry for at least a year, before being painted in oil; 
otherwise the paint will be liable to blister. They may, if preferred, be frescoed (water-colors, 
mixed with size) to the desired tint during the interval. 

The painting of unseasoned wood hastens its decay. If the surface to be painted is greasy, the 
grease must first be removed by water in which is dissolved some lime. 

Washes lor outside work. Downing, in his work on country houses, 

recommends the following: For wood-work; in a tight bushel, slack half a bushel of fresh lime, by 
pouring over it boiling water sufficient to cover it 4 or 5 ins deep; stirring it until slacked. Add 2 
lbs of sulphate of zinc (white vitriol) dissolved in water. Add water enough to bring all to the con¬ 
sistence or thick whitewash. Apply with a whitewash brush. This wash is white; but it may b« 
colored by adding powdered ochre, Indian rod, umber, &c. If lampblack is added to water-colors, it 


♦Average cost of Painting in Philada, 1886, including scaffold, &c, per 
square yard. Four coats in plain colors, 30 cts; 3 coats, 25. Graining in imitation of oak, walnut, 
Ac, 50. White lead ground in oil, in kegs, 8 cts per lb. The cost of painting and glazing a 20-U front 
8-story dwelling, with large 3-story back buildiugs, $300 to $400. A church of 00 by 80 rt., with base¬ 
ment story, and galleries, $900 to $1000. Avoid extras ; or stipulate for them in advance. 






GLASS, AND GLAZING 


431 


should first be thoroughly dissolved in alcohol. The sulphate of zinc causes the wash to become hard 
in a few weeks. 

For brick, masonry, or rough-cast. Slack % a bushel of lime as 

before; then fill the barrel % full of water, and add a bushel of hydraulic cement. Add 3 lbs of sul¬ 
phate of zinc, previously dissolved iu water. The whole should be of the thickness of paint; and 
may be put ou with a whitewash brush. The wash is improved by stirring iu a peck of white sand, 
just before using it. It may be colored, if desired, like the preceding. 

He also gives the following cheap oil-paint for outside work on wood, brick, stone, &c : and says it 
becomes far harder and more durable than common paint: One measure of ground fresh quicklime; 
add the same quautity of fine white sand, or fine coal ashes; and twice as much fresh wood ashes; 
all the foregoing to be passed through a fine sieve. Mix well together dry. Mix with as much raw 
linseed oil as will make the mixture as thin as paint. Apply with a painter's brush. It may be col¬ 
ored like the foregoing, taking care to mix the colors well with oil before adding them. It is best to 
put on two coats ; the first thin, and the second thick. 

Also, another, said to stand 15 to '20 years : 50 lbs best white lead ; 10 quarts raw linseed oil; y. lb 
dryer; 50 lbs finely sifted sharp clean sand; 2 lbs raw umber. Add very little, say y pint of tur¬ 
pentine. Apply with a large brush. 

Cement for stopping joints, such as around chimneys, &c, &c. White 

lead ground in oil, as sold by the keg; mixed with enough pure sand to make a stiff paste that will 
not run. It grows hard by exposure, and resists heat, cold, and water. Pieces of stone may ba 
strongly cemented together by it, allowing a few months for proper hardening. 

Whitewash for inside work, according to Mr. Downing, “is made more 

fixed and permanent, by adding 2 quarts of thin size to a pailful of the wash, just before using. 
The best size for this purpose is made of shreds of glove leather; but any clean size of good quality 
will answer," as thin glue-water. We will add, that the common practice of mixing salt with white¬ 
wash, should not be permitted. Paper pasted on a wall which has previously been covered with salt 
whitewash, is very’ apt to become wet, and loose, and to fall off during damp weather. The white¬ 
wash should be scraped off, and the wall or partition covered with a coat or two of thin size, to pro- 
i tect the paper from the effect of the salt that may still adhere to the plaster. 


GLASS, AND GLAZING. 

Window glass is sold by the box. Whatever may be the size of the panes, a box 
contains as nearly 50 sq ft of glass as the dimensions ot the panes will admit of. 

Panes of any size may be made to order by the manufacturers. The sizes given in the following 
1 table, as well as many others, are generally to be had ready made. Ordinary window glass of all the 
sizes in the table, is about one-sixteenth of an inch thick ; and this is the thickness supposed to be 
intended when a greater one is not specified. Double-thick glass is nearly % inch; and its price is 
50 per ct more than the single thick. It is of course much stronger than the single. 

The panes are confined to the sash by glaziers’ putty, made of whiting (powdered chalk) and raw 
linseed oil ; and by small triangular pieces of thin tin, about % * nc h on a side, which uphold the 
| glass while the putty is being put on; and are allowed to remain afterward, as a protection while the 
putty continues soft. 

TABLE OF NUMBERS OF PANES IN A liOX. 


Size in 

Panes 

Size in 

Panes 

Size in 

Panes 

Size in 

Panes 

to 

Size in 

Panes 

to 

ins. 

a box. 

ins. 

a box. 

ins. 

a box. 

ins. 

a box. 

ins. 

a box. 

6X8 

150 

12 X 36 

17 

16 X 42 

11 

24 X 24 

12 

30 X 66 

4 

7X9 

115 

13 X 14 

40 

48 

9 

26 

12 

70 

3 

8 X 10 

90 

16 

35 

54 

8 

30 

10 

32 X 34 

7 

12 

75 

18 

31 

60 

8 

36 

9 

36 

6 

9 X 12 

67 

20 

28 

18 X 20 

20 

42 

7 

42 

6 

14 

57 

24 

23 

22 

18 

48 

6 

48 

5 

16 

50 

32 

17 

24 

17 

54 

6 

60 

4 

18 

45 

14 X 16 

32 

30 

14 

60 

5 

66 

3 

10X12 

60 

18 

29 

36 

11 

66 

5 

34 X 36 

6 

14 

52 

20 

26 

42 

10 

26 X 28 

10 

44 

5 

16 

45 

24 

22 

50 

8 

32 

9 

48 

5 

18 

40 

30 

17 

60 

7 

36 

8 

54 

4 

20 

36 

36 

14 

20 X 22 

17 

42 

7 

60 

4 

24 

30 

42 

12 

24 

15 

48 

6 

66 

3 

30 

24 

46 

11 

30 

12 

64 

5 

36 X 40 

5 

11 X 12 

55 

15 X 16 

30 

38 

10 

60 

5 

44 

5 

14 

47 

18 

27 

42 

9 

28 X 30 

9 

48 

4 

16 

41 

20 

24 

48 

8 

36 

7 

54 

4 

18 

37 

33 

24 

20 

54 

7 

42 

6 

60 

3 

2D 

30 

16 

64 

6 

56 

5 

70 

3 

24 

27 

36 

13 

22 X 24 

14 

66 

4 

38 X 44 

4 

12 X 14 
16 

43 

38 

40 

16 X 18 

12 

25 

30 

36 

11 

9 

30 X 34 
36 

7 

7 

52 

40 X 46 

4 

4 

18 

34 

20 

23 

42 

8 

42 

6 

54 


20 

24 

30 

25 

24 

30 

19 

15 

48 

56 

7 

6 

48 

54 

5 

4 

72 

44 X 50 

3 

28 

22 

36 

13 

60 

5 

60 

4 

56 

3 

30 

20 













































432 


GLASS. 


The best qualities of American glass made in the vicinity of Philadelphia, 
Boston, Pittsbur g, Ac, are for most purely useful purposes, as good as those from 
foreign countries; but when the highest degree of beauty is required, as in the 
lower front windows of tirst-class dwellings, fancy stores, Ac. polished plate- 
glass of England, France, or Germany, must be used: although the price for 
moderate sized panes is from 5 to 8 times as great as that of the best quality 
single-thick American, as given in the following table.* Its perfectly smooth 
surface, free from distorted reflections, also makes it the best for covering pic¬ 
tures; still, if carefully selected American panes be used for this purpose, few 
except critics in glass will detect the difference. 

A thick glass is made expressly for flooring-, up to 1 inch thick, 

and up to 50 inches by 9 feet dimensions. Also, for skylights, from l /i to % inch 
thick. This can be furnished to order of any size up to 40 inches by 8 or 10 feet. 
The smaller sizes can also be had ground. Grinding prevents the entrance of 
the full glare of the sun; and, moreover, diffuses the light over a much greater 
width of space below. 

Strengtli of glass. Tensile 2500 to 9000 lbs per square inch. Boston rods 
by author, 3500 to 5200. Crushing strength, 6000 to 10000 lbs per square inch. 
Transversely, (by the writer’s trials,) flooring glass, 1 inch square, and 1 foot 
between the end supports, breaks under a center load of about 170 lbs; con- I 
sequently, it is considerably stronger than granite, except as regards crushing ; 
in which the two are about equal. 

Remark. Window and other glass which contains an excess of potash or of 
soda is very liable to become dull in time, owing to the decomposition of those 
ingredients by atmospheric influences. 


* Price list of American single-thick glass, adopted by Ameri¬ 
can Window Glass Manufacturers’ National Association, July 22, 1886. Double¬ 
thick about 50 per cent. more. For ground glass add about $2.50 per box. 8>is- 
connt, 1886, about 75 per cent. Benj. II. Shoemaker, dealer in French and 
American window glass, 205 N. 4th St, Philadelphia; also Malaga Glass and 
Manufacturing Co., office 400 Chestnut St, Philadelphia. 



Size in 

Inches. 


1st Quality. 

2d Quality. 

3d Quality. 

4th Quality. 

From 


to 


Per box. 

Per box. 

Per box. 

Per box. 

6 

X 

8 

10 

X 

15 

$8.75 

$8.00 

$7.50 

$7.00 

11 

X 

14 

16 

X 

24 

10.00 

9.25 

8.75 

8.00 

18 

X 

22 

20 

X 

30 

12.50 

11.50 

10:25 

9.00 

15 

X 

36 

24 

X 

30 

13.25 

12.00 

10.75 

9.50 

26 

X 

28 

24 

X 

36 

14.50 

13.00 

11.50 

10.25 

26 

X 

36 

26 

X 

44 

15.00 

13.50 

12.25 

11.00 

26 

X 

46 

30 

X 

50 

16.75 

15.25 

13.75 

11.75 

30 

X 

52 

30 

X 

54 

17.50 

16.00 

14.25 


30 

X 

56 

34 

X 

56 

19.25 

17.59 

15.75 


34 

X 

58 

34 

X 

60 

20.75 

18.75 

16.75 


36 

X 

60 

40 

X 

60 

22.25 

20.75 

18.50 



In small quantities, the following are also approximate prices for American 
glass: Large plates of X A inch thick, rough, 40 cents per square foot. One inch 
thick, 75 cents to $1; it either is ground, 10 to 15 cents additional per square 
foot. Ribbed glass, % inch thick, 20 cents; % inch, 25 cents per square foot. 
Stained glass, siugle thickness, (^ 5 inch,) or figured white enameled glass, (single 
thickness,) 20 to 2.5 cents per square foot. Superior thicker strong figured glass, 
first ground, and the transparent figures then formed by polishing away por¬ 
tions of the ground surface, $1.00 per square toot. 

Mufted glass is an inferior article of fanciful colored patterns, attached by 
some imperfect process which allows them to peel oil' after a year or two of ex¬ 
posure to the weather. 

The charge by glaziers for putting the glass into new windows, including 
putty, tins, and two coats ol paint to the sash, (one of which is a priming coat) 
is (1886) equal to the cost of the glass at the above prices. 

For reglazing old sash aud removing the broken panes, the charge is about 
twice as great. 























PAPER. 


433 


PAPEK.* 

24 sheets 1 quire. 20 quires 1 ream. 

Sizes of drawing- papers. 


Ins. Ins. 


Antiquarian. 31 X 52 

Double Elephant. 20 X 40 

Atlas. 26 X 34 

Imperial. 21 X 30 


Super Royal. 

Royal. 

Medium. 

Demy. 

Cap. 


Ins Ins 

10 X 27 
19 X 24 
17 X 22 
15 X 20 
13 X 17 


The English drawing-papers are stronger and superior to the American. Those 
by Whatman have a high reputation; they are, however, of different qualities. When 
paper is pasted on muslin, the difference in quality is not so important. Of paper 
in rolls, the German makes are the best. There is but little of other makes imported. 

Rotli white and tinted papers, for the use of engineers, are made in 
continuous roll6, without seams. Widths 36, 42, 54, 58, and 62 ins; usual lengths 40 
yds: but can be had to order to 400 yds or more. These may also be purchased ready 
pasted on muslin, in rolls 10 to 40 yds long. This last, on account of its strength, 
should be used for all drawings which undergo frequent rolling and unrolling; or 
other hard usage; particularly working-drawings. For the last purpose, strong 
cartridge or pattern paper answers very well. It is for sale in long rolls, of same 
lengths as white paper, mounted or not; widths up to 54 ins. Color, a light buff. 

Tracing paper. Most of that sold, whether domestic or foreign, tears so 
readily as to be of comparatively little service, except for tracings to be enclosed in 
letters for mailing. Some of what is called French vegetable tracing-paper , is, how¬ 
ever, quite stout and strong,and good for line drawings; but it shrivels' badly under 
broad washes of color, even when stretched, forming little puddles, which make it 
difficult to produce a uniform tint. Sizes 19 X 25, 21 X 26, 28 X 40 ins; also in 
rolls of 11 and 22 yds. Parchment paper, 37 and 38 ins wide, rolls of 20 aud 33 yds, 
is better, but does not take ink perfectly. 

Tracing cloth, usually called tracing muslin , and sometimes vellum cloth , is 
altogether preferable to tracing paper, on account of its great strength. Widths 18, 
30, 36, and 42 ins; lengths to 24 yds. 

Common inks dry pale on either tracing muslin or tracing paper; therefore use 
India ink. Neither the muslin nor the paper takes colors as kindly as drawing 
paper. 

l*rofile paper is made in widths of 9 ins and 20 ins, and in single sheets or 
in long, continuous rolls. 

Ruled squares, or cross-section paper. Paper carefully ruled in 
small squares, so that the divisions answer for a scale for the drawing, is exceedingly 
useful for sketching out plans, &c. It is sometimes ruled on both sides of the sheet. 

Colors. A good draughtsman needs but few colors; say India ink, Prussian 
blue, lake, or carmine, light red, burnt umber, burnt sienna, raw sienna, gamboge, 
Roman ochre, sap green. Winsor & Newton’s colors are among the best in use. 
Purchase none but the very best India ink. Cakes of colors should always be wiped 
dry on paper,after being rubbed in water; and but littie water should be used while 
rubbing; more being added afterward. 

JLcad pencils. Genuine A. W. Faber’s Nos. 2, 3, and 4, are very good. The 
hardness increases with the number. Nos. 3 and 4 are good for field-book use : which 
to prefer, will depend on the character of the paper; No. 3 for smooth, and No. 4 for 
the coarser or more granular papers. Ilis lettered pencils are of a higher grade and 
better suited for draughting. “ II ” stands for “ hard,” “ B ” for “soft.” The degree 
of hardness or of softness is indicated by the number of H’s or of B’s. One H cor¬ 
responds with No. 3. Dixon’s American pencils are good, the office draughtsman 
should have a fiat file, or a piece of fine emery paper glued to a strip of wood, upon 
which to rub bis lead to a fine point readily, after using the knife. 


* James W. Queen & Co, No 924 Chestnut St, Philadelphia. 

29 























434 


STRENGTH OF MATERIALS, 


STRENGTH OF MATERIALS. 


The mo<lill ns of Elasticity, and its nse. within the limit or elasticity, a 

uniform rod of given material lengthens or shortens equally under equal additions of load. If this 
were also the case beyond said limit, it is plain that there would be some load which would stretch a 
uniform bar to twice its original length, or shorten it to zero or 0. And this load in lbs, for a bar of 
one inch square cross section, is the mod of elas for the given material; or is the E of authors on 
Strength of Materials. For example, a one-inch square bar of wrought iron will, within the limit 
of elas, stretch on an average about 1 part in 12000 of its length under each additional load of 2240 lbs. 
Consequently, if the same rate of stretching continued beyond the limit of elas, it is evident that 
12000X 2240, or 26880000 lbs, would stretch the bar to twice its original length. Hence these 26880000 
lbs are the mod of elas for average bar iron. And so with any other material. Hence the mod of 
elas is a load which bears the same proportion to the origiual length of a uniform bar, as the load 
which will produce any given amount of stretch, is to the length of said stretch. This fact facilitates 
certain calculations; thus, 

Load in lbs reqd 3 Req<i stretch 

to produce a given I _ >n 1DClie8 mod cross sec 

stretch within j orig length * elas. * in sq in. 
elas limit, in ins, J in inches 

Load in lbs X orig length in ins 


mod of elas X cross sec in sq ins See Table, below. 

E may also be found thus. At the center of a rectang beam supported hor at both ends, ap¬ 
ply any load within its elas limit, and measure its deflection. Then as expressed by writers E is = 
W13-j- 4Abd3. Which means 

Coef const, op Mod - (^oad * n ® IS "H -625 wt of clear span of beam ) X cube of span ins, _ 

E in tbs per sq iucn 1 X Def, ins X breadth, ins X cube of depth ins. 

Moduli of elasticity for different materials. Authors differ 

considerably in their data on this subject. Where it was possible to adopt au average without error 
of practical importance, we have done so; but where the differences were too great for this, we have 
omitted the material entirely. Fortunately it is a matter of but little importance to the engineer: for, 
with the exception of iron and steel, he rarely need care to know the exact degree to which bis mate¬ 
rials are lengthened or shortened by their loads. 

It must be carefully remembered that in practice the principle of mod of elas applies only within 
the elas limit of materials. This limit may ordinarily be safely taken at about one-third of the 
breaking tensile strengths given on pages 463 to 466. This is done in our 5th column. It is, how¬ 
ever, but a tolerable approximation. There is reason to doubt the accuracy of mauy of the items in 
this table. 


Stretch in ins 7 
produced by any 
load within elas , 
limit, J 


MATERIAL. 

Modulus 
or Coeflf 
of 

Elasticity. 

Stretch or C 
in a lengt 
UDder a 
1000 lbs per 
sq iu. 

compression 

3 of 10 ft, 
load of 

1 ton per 
sq in. 

Approx elas 
limit. 


Ibs per sq in. 

Ins. 

Ins. 

Bis per sq in. 

Ash. 

1 600 000 

.075 

.168 

4500 

Beech. 

1 300 000 

.092 

.207 

4000 

Birch. 

1 400 000 

.086 

.192 

5000 

Brass, cast. 

9 200 000 

.013 

.029 

6000 

“ wire. 

14 200 000 

.009 

.019 

16000 

Chestnut. 

1 000 000 

.120 

.269 

4500 

Copper, cast. 

18 004) 000 

.007 

.015 

6300 

“ wire. 

18 000 000 

.007 

.015 

10000 

Elm. 

1 000 000 

.120 

.269 

2000 

Glass. 

8 000 000 

.015 

.034 

3200 

Iron, cast .. 

12 000 000 

.010 

.022 

4500 


to 

to 

to 

to 


23 000 000 

.005 

.012 

8000 

“ “ average.. 

17 590 000 

.007 

.015 

6250 

“ wrought, in either « 

18 000 000 

.006 

.015 

20000 

bars, sheets or plates. 1 

to 

to 

to 

to 

/ 

40 000 000 

.003 

.007 

40000 

“ average. 

29 000 000 

.004 

.009 

300(0 

“ wire, hard. 

26 000 000 

.005 

.010 

27000 

“ wire ropes. 

15 000 000 

.008 

.018 

13000 

Larch. 

1 100 000 

.109 

.244 

2300 

Lead, sheet. 

720 000 

.167 


1100 

** wire. 

1 000 000 

1*20 



Mahogany. 

l 400 000 

.086 

.192 

2700 


1 000 000 

120 




to 

to 

to 



2 000 000 

.060 

.134 


“ average. 

1 500 000 

.080 

.179 

3300 

Pine, white or yellow. 

1 600 000 

.075 

.168 

3300 

Slate. 

14 500 000 

.008 

.018 

3700 

Spruce. 

1 600 000 

.075 

.168 

3300 

Steel bars. 

29 000 000 

.004 

.009 

34000 


to 

to 

to 

to 


42 000 000 

.003 

.006 

44000 

11 " average.... 

35 500 000 

.003 

.007 

39000 

Sycamore. 

1 000 000 

.120 

.269 

4000 

Teak. 

2 000 000 

.060 

.134 

5000 

Tin.cast. 

4 600 000 

.026 


1500 


























































STRENGTH OF MATERIALS. 


435 


Fatigue of Materials. In the following articles on Strength of Mate¬ 
rials, the ultimate or breaking load is that which will,during its first application 
rupture the given piece within a short time. But Wohler’s and Spaugenberg’s 
experiments show that a piece may be ruptured by repeated applica¬ 
tions of a load much less than this; and that theoftener the load is applied the 
less it needs to be in order to produce rupture. Thus, wrought iron which re¬ 
quired a tension of 53U00 lbs per sq inch to break it in 800 applications, broke 
with 35000 lbs per sq inch applied about 10 million times ; the stress, after each 
application, returning to zero in both cases. 

The diff between the maximum and minimum tension in a piece subjected to 
tension only, or between the max and min compression in a piece subjected to 
vomp only; or the sum of the max tension and max comp in a piece subjected 
alternately to tension and comp; is called the range of stress in the piece. 
When this is less than the elastic limit, the application may be repeated an 
“enormous” number (say about 40 million) of times without rupture.* 

For a given number of applications, the load required for rupture is least when 
the range of stress is greatesr. If the stress is alternately comp and tension, 
rupture takes place more readily than if it is always comp or always tension. 
That is, it takes place with a less range of stress applied a given number of 
times, or with a less number of applications of a given range of stress. For a 
given range of stress and given number of applications, the most unfavorable 
condition is where the tension and comp are equal. 

The above facts are now generally taken into consideration in designing 
members of important structures subject to moving loads. For instance, Mr. 
Jos. M. Wilson, C. E., Mem. Inst. C. E. (London Eng.), Mem. Am. Soc. C. E., uses 
the following formulae for determining the “ permissible stress” in iron bridges, 
in lbs per sq inch; in order to provide the proper area of cross section for eacli 
member. 

For pieces subject to one kind of stress only (all comp or all tension) 


, /„ min stress in the piece \ 

a = U t ( 1-1-:-;- - -- I 

\ max stress in the piece/ 

For a piece subject alternately to comp and tension , find the max comp and the 
max tension in the piece. Call the lesser of these two maxima “max lesser”, 
and the other or greater one, “ max greater ”. Then 

. /„ max lesser \ 

a = w f ( 1 — --) 

\ 2 max greater/ 

For a piece whose max comp and max tension are equal , this becomes 

« = 4 ) 

The above a is the permissible tensile stress in lbs per sq inch on any mem¬ 
ber; but the permissible compressive stress is found by “Gordon’s formula” for 
pillars, p 439, using a (found as above) as the numerator, instead of/. Font in 
the divisor or denominator of Gordon’s formula (which must not be confounded 
with the a of the foregoing formulae) Mr. Wilson uses for wrought iron : 

when both ends are fixed. 36000 

when one end is fixed and one hinged. 24000 

when both ends are hinged. 18000 

Experiments show that materials may fail under a long continued 
stress of much less intensity than that produced by the ult or bkg load. 


* This does not always hold in cases where the elastic limit has been artificially raised 
by process of manufacture, etc. Oft-repeated alternations between tension and compres¬ 
sion below such a limit reduce it to the natural one. A slight flaw may cause rupture 
under comparatively few applications of a range of stress but little greater, or even less, 
than the elastic limit. Rest between stresses increases the resisting power of a piece. 
In many cases, stresses a 1 ittie beyond the elastic limit, even if oft-repeated, raise that 
limit and the strength, but render the piece brittle and thus more liable to rupture from 
shocks; and a little further increase of stress rapidly lessens, or may entirely destroy, 
the elasticity. A tensile stress above the elastic limit greatly lowers, or may even destroy, 
the compressive elasticity, and vice versa. If a tensile stress, by stretching a piece, reduces 
its resisting area, it may thus reduce its total strength, even though the strength per 
iq in has increased. Mr. B. Baker finds that hard steel fatigues much faster under re¬ 
peated loads than soft steel or iron. 

t u - 6500 lbs per sq inch for rolled iron in compression 

= 7000 tbs “ “ “ tension (plates or shapes). 

= 7500 His “ for double rolled iron in tension (links or rods). 















436 


STRENGTH OF MATERIALS. 


Art. 1. Compressive sirens'*Its of American woods, u>her 
slowly and carefully seasoned. Approximate averages deduced from many experi¬ 
ments made with the U S Govt testing machine at Watertown, Mass, by Mr. S. P. 
Sharpies, for the census of 1880. Seasoned woods resist crushing much better 
than green ones; in many cases, twice as well. This must be allowed for when 
building bridges, &c, of timber recently cut. Different specimens of the same wood 
vary greatly; frequently as 5 to 8, 9, or more. See Rem and foot-note p 611. 


The strengths in all these 
tables may readily vary as 
much as one-third part more 
or less than our average. 

End¬ 

wise.* 

lbs per 
sq in. 

Side- 

wise. 11 

lbs per 
sq in. 

The strengths in all these 
tables may readily vary as 
much us one-third part more 
or less than our average. 

End¬ 

wise.* 

lbs per 
sq in. 

Side- 

wise.11 

lbs per 
sq in. 


.01 

.1 


.01 

.1 

Ash, red and white.. 

6800 

1300 

3000 

Maple, broad - leafed, 




Aspen . 

4400 

800 

1400 

Oregon. 

5300 

1400 

2600 

Beech . 

7000 

1100 

1900 

“ sugar and black.. 

8000 

1900 

4300 

Birch . 

8000 

1300 

2600 

“ white and red. 

6800 

1300 

2900 

Buckeye. . 

4400 

600 

1400 

Oak, white, post (or iron) 




Butternut . 

5400 

700 

1600 

swamp white, red 




Buttonwood (sycamore).. 

6000 

1300 

2600 

and black. 

7000 

1600 

4000 

Cedar , red. 

6000 

700 

1000 

“ scrub and basket... 

6000 

1700 

4200 

“ white (arbor vitae) 

4400 

500 

900 

“ chestnut and live... 

7500 

1600 

4500 

Catalpa (Indian bean)... 

5000 

700 

1300 

“ pin. 

6500 

1300 

3000 

Cherry , wild. 

8000 

1700 

2600 

Pine , white . 

5400 

600 

1200 

Chestn ut . 

5300 

900 

1600 


6300 

600 

1400 

Coffee, tree , Kentucky. 

5200 

1300 

2600 

“ pitch and Jersey 



Cypress, bald. 

6000 

500 

1200 

scrub.. 

5000 

1000 

2000 

Elm, Am’n or white. 

6800 

1300 

2600 

“ Georgia . 

8500 

1300 

2600 

“ red. 

7700 

1300 

2600 

Poplar . 

5000 

600 

1100 

Jlemloclc . 

5300 

600 

1100 

Sassafras . 

5000 

1300 

2100 

Hickory .. 

8000 

2000 

4000 

Spruce, black. 

5700 

700 

1300 

Lignum vitie . 

10000 

1600 

13000 


4500 

600 

D' 

Linden, American . 

5000 

500 

900 

Sycamore (buttonwood).. 

6000 

1300 

2600 

Locust , black and yellow 

9800 

1900 

4400 

Walnut, black. 

8000 

1300 

2600 

“ honey. 

7000 

1600 

2600 

“ white (butter- 




Mahogany . 

9000 

1700 

5300 


5400 

700 

1600 




| 

Willow . 

4400 

700 

1400 


Hence it appears that seasoned white and yellow pines, spruce, and ordinary oaks, 
which are the woods most employed in the United States for bridges, roofs, etc., crush 
endwise with from 5000 to7000ftsper sq inch, in short blocks; average, 6000. 

But it is well to bear in mind that in practice perfectly equable pressure is rarely 
secured. In a few trials on sidewise compression, with fairly seasoned white pine 
blocks, 6 ins high, 5 ins long, and 2 ins wide, we found that under an equally dis¬ 
tributed pressure of 5000 lbs total or 500 lbs per sq inch, they compressed about fron. 
Yu to % inch; which is equal to from % to inch per foot of height; or from ^ 
to ^ of the height; the mean being about % inch to a foot, or X of the height. 
Under 10000 lbs total, or 1000 lbs per sq inch, they split badly; and in some cases 
large pieces flew off. Nee Rem, p 500. 

The tensile or cohesive strengths of pine and oak average about 10000 lbs per sq 
inch, or % as much as average cast-iron, or nearly double their resistance to crush¬ 
ing. The tensile strength does not change with the length of the piece; so that in 
practice we may take its safe strain at from 1000 to 2000 lb* per sq inch, depending 
upon the character of the structure, Ac., without regard to the length, except when 
this is so great that two or more pieces have to be spliced together to.make it; thus 
weakening the piece very much. 

* Specimens 4 centimetres (1.57 inch) square, 32 centimetres (12.6 ins) long 
When the length exceeds 10 times the least side, see Wooden Pillars, p 458. 

U Specimens 4 centimetres (1.57 inch) square, 16 centimetres (6.3 ins) long; lai<, 
upon platform of testing machine. Pressure applied at their mid-length, by means; 
of an iron punch 4 centimetres square, or just covering the entire width of < * 
specimen, and one-fourth of its length. The first column (headed “.01”) gives 
loads producing an indentation of .01 inch. The second column (headed “.1”) git * 
those producing an indentation of .1 inch. 



































































STRENGTH OF MATERIALS. 


437 


Art. 2. 
foot, for 

of all to be 
split under 
brickwork 
cording to 
stones lose 


cr « sh '»ST loads in tons, per square 
stones, dec. 1 he stones are supposed to be on bed, and the heights 
lrom 1.5 to 2 times the least side. Stones generally begin to crack or 
about one-half ot their crushing loads. In practice, neither stone nor 
should be trusted with more than % to jjyth of the crushing load, ac- 
circumstances. When thoroughly wet some absorbent sand- 
tully half their strength. See head of next page. 


Granites and Syenites. 

Basalt. 

Limestones and Mar¬ 
bles *. 

Oolites, good. 

Sandstonesfitfor build¬ 
ing*. 

Sandstone, red, of Con¬ 
necticut and N. Jer¬ 
sey, to crack. 

Brick*. 

Brickwork, ordinary, 

cracks with*. 

Brickwork, good, in ce¬ 
ment* . 

Brickwork, first-rate, 

in cement... 

Slate . 

Caen Stone. 

“ “ to crack. 

Chalk, hard,. 

Plaster of Paris, 1 day 
5 old. . 


Tons per 
sq. ft. 


300 to 1200 


250 to 1000 
100 to 250 

150 to 550 


40 to 300 

20 to 30 

30 to 40 

50 to 70 
400 to 800 
70 to 200 


20 to 30 


Mean. 

Tons. 


750 

700 

625 

175 

350 


200 

170 

25 

35 

60 

600 

135 

70 

25 

40 


Cement, Portland, 
neat,U. S. or foreign, 

7 days in water. 

Common U.S.cements, 
neat, 7 days in water 
Concrcteof Port, 
cement, sand, and 
gravel or brok stone 
in theproper propor¬ 
tions, rammed 1 mold 

6 months old. 

12 months old. 

With good common 
hyd cements, 
abt .2 to .25 as much 
Coignet beton, 3 

months old. 

Rubble masonry, 

mortar, rough. 

Glass,green,crown and 
flint. 


Tons per 
sq. ft. 


75 to 150 
15 to 30 


12 to 18 
48 to 72 
74 to 120 


100 to 150 
15 to 35 
1300to2300 


Mean. 

Tons. 


112.5 

22.5 


15 

60 

97 


125 

25 

1800 


or 3 times that of granite. 
Ice, firmf. | 12 to 18 | 15 


Crushing' height of Brick and Stone. 


If we assume the wt of ordinary brickwork at 112 lbs per cub ft, and that it would 
crush under 30 tons per sq ft, then a vert uniform column of it 600 ft high, would 
crush at its base, under its own wt. Caen stone, weighing 130 lbs per cub ft, would 
require a column 1376 ft high to crush it. Average sandstones at 145 lbs per cub ft, 
would require one 4158 ft high ; and average granites, at 165 lbs per cub ft, one 
of 8145 feet. But stones begin to crack and splinter at about half their ultimate 
crushing load; and in practice it is not considered expedient to trust them with more 
than %th to j^th part of it. especially in important works; inasmuch as settlements, 
and imperfect workmanship, often cause undue strains to be thrown on certain 
parts. 

n The Merchants’ shot-tower at Baltimore is 246 ft high ; and its base sustains 6^ 
tons per sq ft. The base of the granite pier of Saltash bridge, (by Brunei,) of solid 
masonry to the height of 96 ft, and supporting the ends of two iron spans of 455 ft 
each, sustains 9 }/ 2 tons per sq ft. The base of a brick chimney at Glasgow, Scotland, 
468 ft high, bears 9 tons per sq ft; and Professor Rankine considers that in a high 
gale of wind, its leeward side may have to bear 15 tons. The highest pior of Rocque- 
favour stone aqueduct, Marseilles, is 305 ft, and sustains a pressure at base of 13}^ 
tons per sq ft. For greater pressures on arch-stones, see p 694. 


* Trials at St. Louis bridge, by order of Capt James B. Eads, C. E., 
showed that some magnesian limestone did not yield under less than 1100 tons per sq ft. A column 
8 ins high and 2 ins diam shortened % inch under pressure; and recovered when relieved. 

Experiments made with the Govt testing machine at Water- 
town, Mass, 1882-3, gave 1400 tons per sq ft ultimate crushg load for white 
and blue marble from Lee, Mass, 700 for blue marble from Montgomery Co, Pa, 960 for limestone from 
. Conshohocken, Pa, 500 for limestone from Indiana, S40 for red sandstone from Hunimelstowu, Pa, 
260 to 1000 for yellow Ohio sandstone ; Phila bricks, flatwise; hard, machine-made, 350 to 700 tons ; 
-'‘hand-made, 700 to 1300; pressed, machine-made, 450 to 580; Brickwork columns, 13 ins sq and 13 ins 
ivigh; in lime, 100 tons ; in cement, 150. 

1 1 Experiments by Col. Wm. Ludlow, C. S. A., with Govt testing machines, in 1881, gave from 21 
64 tons per sq ft for pure, hard ice; and 16 to 59 tons for inferior grades. The specimens (6 and 
1 J-inch cubes) compressed % to 1 inch before crushing. 











































438 


STRENGTH OF MATERIALS, 


Sheet lead is sometimes placed at the joints of stone col¬ 
umns, with a view to equalize the pressure, and thus increase the strength of the column. But 
experiments have proved that the effect is directly the reverse, and that the column is materially 
weakened thereby. Does this singular fact apply to cast iron and other materials ? 

Art. 3. Average crushing load Tor Metals. 

It must be remembered that these are the loads for pieces but two or three times their least side in 
height. As the height increases, the crushing load diminishes. See “ Strength of Pillars,” p 439. 


The crushing load per sq inch, of any material, is frequently 
called its constant, coefficient, or modulus, of crushing or of com¬ 
pression. 


Pounds per Tons per 

sq. inch. sq. inch. 


Cast Iron, usually. 

It is usually assumed at 100000 lbs, or say 45 tous per sq inch. Its 
crushing strength is usually from 6 to 7 times as great as its tensile. 
Within its average elastic limit of about 15 tons per sq inch, avetage 
cast iron shortens about 1 part in 5555; or % inch in 58 ft under each 
ton per sq inch of load; or about twice as much as average wrought 
iron. Hence at 15 tons per sq inch it will shorten about 1 part in 370; 
or full % inch in 4 feet. Different cast irons may however vary 10 to 
15 per ct either way from this. 

U. S. Ordnance, or gun metal ; Some. 

Wrought iron, within elastic limit. 

Its elastic limit under pressure averages about 13 tons per sq inch. 
It begins to shorten perceptibly under 8 to 10 tons, but recovers when 
the load is removed. With from 18 to 20 tons, itshortens permanently, 


85000 to 125000 


.175000. 

22400 to 35840 

. 29120. 


about g^-th part of its length; and with from 27 to 30 tons, about yLth 
part, as averages. The crushing weights therefore in the table are 
not those which absolutely mash wrought iron entirely out of shape, 
but merely those at which it yields too much for most practical build¬ 
ing purposes. About 4 tous per sq inch is considered its average safe 
load, in pieces not more than 10 diams long; and will shorten it ^ inch 
in 30 ft. average. 


Brass, reduced y^th part in length, by 51000; and by 

Copper, (cast,) crumbles. 

(wrought) reduced )^th part in length, by. 

Till, (cast,) reduced yLth in length, by 8800; and X / A by 
Lead, (cast,) reduced ]/, of its length, by 7000 to 7700.... 

“ By writer. A piece finch sq, 2 ins high, at 1200tbs the com¬ 
pression was 1-200 of the ht; at 2000, 1-29; at 3000, 1-8; at 
5000, 1-3; at 7000, 1-2 of the ht. 


165000. 

117000. 

103000. 

..15500. 

....7350. 


Spelter or Zinc, (cast.) By writer. A piece 1 inch 

square, 4 ins high, at 2000 lbs was compressed 1-400 of its ht; at 4000, 
1-200; at 6000, 1-100; at 10000, 1-38; at 20000, 1-15; at 40000 yielded 
rapidly, and broke into pieces. 


Steel, 224000 lbs or 100 tons shorten it from .2 to .4 part. 
“ American. Black Diamond steel-works, Pittsburg, Penn, 
experiments by Lieut W. H. Shock, U. S. N., on pieces in 
square; and ins, or 7 sides long. 

“ Untempered. 100100 to 104000. 

“ Heated to light cherrv red, then plunged into oil of 82° Fah, 

173200 to 199200. 

“ Heated to light cherry red, then plunged into water of 79° 
Fah ; then tempered on a heated plate, 325400 to 340800.... 
“ Heated to light oherry red, then plunged into water of 79° 
Fah. 275600 to 400000. 

“ Elastic limit, 15 to 27 tons.. 

“ Compression, within elas limit averages abt 

1 part in 13300, or .1 of an inch in 111 ft per ton per sq inch ; 
or .1 of an inch in 5.3 ft under 21 tons per sq inch. 

Best steel knife ed$?es, of large R R weigh scales 

are considered safe with 7000 lbs pres per lineal iuch of edge; and 

solid cylindrical steel rollers under bridges, and 

rolliiifi on steel, safe with I diam in ins X 3 100 000, in lbs per lineal 
inch of roller parallel to axis. And per the same, for 


102050. 

186200. 

333100. 

337800. 

.47040, 


38 to 56 


78.1 

10 to 16 

13 


73.6 

52.2 

46.0 

6.92 

3.28 


45.5 

83.1 

148.7 

150.8 

21 


Solid cast Iron wheels rolling on wrought iron, |/Dium ins X 352 000. 

" “ “ “ “cast iron, )/Diam ins X 222 222. 

Solid Steel “ “ “ steel, ) DiaminsX 1300000. 

“ “ “ wrought iron, y/Diam ins X 1024 000. 

“ “ " “ cast iron, |/ Diam ins X 850 000. 

From “ Specifications for Iron Drawbridge at Milwaukee,” by Don J. Whittemore, C. B. 









































STRENGTH OF PILLARS. 


439 


STRENGTH OF PIEEARS. 

The foregoing remarks on crushing or compressive strength refer to that of 
pieces so short as to be incapable of yielding except by crushing proper. Pieces 
longer in proportion to their diameter of cross section are liable to yield by 
bending sideways. They are called pillars or columns. 

The law governing the strength of pillars is but imperfectly understood; and 
the best formulae are rendered only approximate by slight unavoidable and un¬ 
suspected defects in the material, straightness and setting of the column. A 
very slight obliquity between the axis of a pillar and the line of pressure may 
reduce the strength as much as 50 per cent; and differences of 10 per cent or 
more in the bkg load may occur between two pillars which to all appearances 
are precisely similar and tested under the same conditions. Hence a liberal 
factor of safety should be employed in using any formulae or tables for pillars. 
See “ factor of safety ” p 442, and at foot of p 446. 

Iu our following remarks on this subject, the pillars are supposed to sustain 
a constant load ; and the ultimate or breaking load referred to is that one which 
would, during its first application, cripple or rupture the pillar in a short time. 
But struts iu bridges etc often have to endure stresses which vary greatly in 
amount from time to time. Their ultimate load is then less. For such cases 
see p 435. 

Long pillars with rounded ends, as in Fig 1, have less strength than 
those with lint ends, whether free or firmly fixed. See table p 442, which 
also shows that iu short columns the difference in this respect is 
but slight. 

In iron bridges and roofs, the ends of the pillars and of oblique 
struts are frequently sustained by means of pins or bolts passing 
through (across) them, at either one or both ends, as at p. Fig 1. 

See also p 612. These we will call hinged ends. Our table 
Fig. 1. p 442, shows that pillars so fixed are about intermediate in 
strength between those with flat and those with round ends. 
There is much uncertainty about this and all such matters. The 
strength of a given hinged-end pillar is increased to an important 
P extent by increasing t he diameter of the pin. 

The formula in most general use for the strength of pillars, 
is that attributed to Prof. Lewis Gordon of Glasgow, and 
called by his name. With the use of the proper coefficients for the given case, 
it gives results agreeing approximately with averages obtained in practice with 
pillars of such lengt hs (say from 10 to40diams) as are commonly used. 

It is as follows 

/ 


Breaking load in lbs per sq inch __ 
of area of cross sect ion of pillar 


1 + 


l* + r* 
a 


in which 

f is a coefficient depending upon the nature of the material and (to 
some extent) upon the shape of cross section of the pillar. It is olten taken, 
approxiinatelv enough, as being the ult crushing strength of short blocksof the 
given material. For good American wrought iron, such as is used for pillars 
40000 is generally used ; for cast iron 80000. Mr Cleeman* found form.hi steel 
( 15 per cent carbon) 52000; and for hard steel (.36 per cent carbon) 83000 lbs. 
Mr. C. Shaler Smith gives 5000 for Pine. See p 458. 

a, for wrought iron, is usually taken as follows: a = 

when both ends of the pillar are flat or fixed. 36000 to 40000 

when both ends of the pillar are hinged........... 18000 to 20000 

when one end is flat or fixed, and the other hinged... 24000 to 30000 

For cast iron about one eighth of these figures is generally used ; and for pine 
about one twelfth. 

I is the length of the pillar. If the pillar has, between its ends, supports 
which prevent it from yielding side-ways, the length is to be measured 
between such supports. See lines 11 to 18, p 457. 
r is the least radius of gvrationf of the cross section of the pillar. I and r 
must be in the same unit; as both in feet, or both in inches._ 


* Proceedings Engineers' Club of Phila, Nov 1884. 
t See p 440. 















440 


STRENGTH OF PILLARS. 


Radius of gyration. Suppose a body free to revolve around an axis which 
passes through it in any direction ; or to oscillate like a pendulum hung from a point 
of suspension. Then suppose in either case, a certain given amount of force to he 
applied to the body, at a certain given dist from the axis, or from the point of sus¬ 
pension, so as to impart to the body an angular vel; or in other words, to cause it 
to describe a number of degrees per sec. Now, there will be a certain point in the 
body, such that if the entire wt of the body were there concentrated, then the same 
force as before, applied at the same dist from the axis, or from the point of suspen¬ 
sion as before, would impart to the body the same angular motion as before. This 
point is the center of gyration ; and its dist from the axis, or from the point of sus¬ 
pension, is the Radius of gyration, of the body. In the case of areas, as of cross- 
sections of pillars or beams, the surface is supixised to revolve about an imaginary 
axis; and, unless otherwise stated, this axis is the neutral axis of the area, which 
passes through its center of gravity. Then 

Radius of gyration = l/Moment of inertia -i- Area 

Square of radius of gyration = Moment of inertia Area 

For moment of inertia, see p 486. 

In a circle, the radius of gyration remains the same, no matter in what direc¬ 
tion the neutral axis may be drawn. In other figures its length is different for 
the different neutral axes about which the figure may be supposed to be capable 
of revolving. Thus, in the I beam Fig 18, p 521, the radiusof gyration about the 
neutral axis X Y is much greater than that about the longer neutral axis WZ. 
In rules for pillars the least radius of gyration must be used. 

The following formulse enable us to find the least radius of gyration, and the 
square of the least radius of gyration, for such shapes as are commonly used for 
pillars. 


Shape of cross section 
of pillar 



Solid square 


Least radius 
of gyration 

side 

V'12 :fr 


Square of 
least radius 
of gyration 

side 2 

12 ~ 


—d—»; 




lU. 

1 

I 


Hollow square of uniform 
thickness 


/l>2 + 


D 2 4- d? 



Solid reetangle 


l east sid e least side 2 

l/~w* 



* Vvi —- about 3.4011. 

































STRENGTH OF PILLARS. 


441 


The following are only approximate: 


Shape of cross section Least radius , s< I uar o of 
ol pillar of g'vration loast ratlins 

of gyration 



Phoenix column. (See p 449) D X .3636 


D 2 X .1322 






► -F-4 


trr77Z%\ 



I beam. (See p 521) 


F 

4.58 


Channel. (See p 521) 


F 

3.54 


Deck beam. (See p 521) 


F 

6 


F2 

21 


F2 

12.5 


F2 

36^5 


y? Angle, with equal legs F F 2 

(See p 525) 5 "25 

4 V 


F 2 / 2 

13 (FS + / 2 ) 



Angle, with unequal legs 
(See p 525) 


F / 


2.6 (F + /) 


v>fc 



T, with F =/ 

(See p 525) 




JF 

4.74 


Cross, with F — / 


F 2 

22.5 



























442 


STRENGTH OF IRON PILLARS. 


Table of approximate average ultimate loads in lbs per 
square inch, as found by experiment with carefully prepared specimens. In 
practice, allowance must be made for the rougher character of actual work, for 
jarrings etc etc. 


Length least radius of 
gyration. 

Pencoyd Angles, Tees, 

I beams 

and Channels.* 
See pp 521 to 527. 

Phoenix 
columns.! 
See p 449. 

Length least radius of 

gyration. 

Steel. 

Iron. 

Iron. 

Hard; .36 
per cent 
carbon 

Mild; .12 
per cent 
carbon 

Flat 

ends 

Flat 

ends 

Fixed 

ends 

Flat 

ends 

Hinged 

ends 

Round 

ends 

Flat 

ends 

3 







57200 

3 

17 







50400 

17 

20 

100000 

70000 

46000 

46000 

46000 

44000 

4SOOO 

20 

30 

74000 

51000 

43000 

43000 

43000 

40250 

40000 

30 

40 

62000 

46000 

40000 

40000 

40000 

36500 

37000 

40 

50 

60000 

44000 

38000 

38000 

38000 

33500 

37000 

50 

60 

58000 

42000 

36000 

36000 

36000 

30500 

37000 

60 

70 

55500 

40000 

34000 

34000 

33750 

27750 

37000 

70 

80 

53000 

38000 

32000 

32000 

31500 

25000 

36000 

80 

90 

49700 

36000 

31000 

30900 

29750 

22750 

35000 

90 

100 

46500 

34000 

30000 

29800 

28000 

20500 

35000 

100 

120 

40000 

30000 

28000 

26300 

24300 

16500 

34500 

120 

140 

33500 

26000 

25500 

23500 

21000 

12800 


140 

160 

28000 

22000 

23000 

20000 

16500 

9500 


160 

200 

19000 

14800 

17500 

14500 

10 S00 

6000 


200 

300 

8500 

7200 

9000 

7200 

5000 

2800 


300 


The following simple formula, by Mr. D. J. Whittemore, was found to 
agree very closely with the results of the experiments on Phoenix columns :f 

Breaking load in lbs wnnn 

per sq inch of area = [(1200 —II) X 30] + 
of cross section of pillar H 2 


where H = 
See also p 443. 


length of pillar 
diam D, fig p 449 


both in the same unit. 

< 


Mr. Christie* adopts the following formulae for obtaining the proper factor 
of safety for pillars of wrought iron or steel: 


For flat and fixed ends, 


For hinged and round ends, 


Factor of safety = 3 -f 


Factor of safety — 3 -f 


(.01 

(.015 


length 

least rad of gyr 


) 


length 
least rad of 



It will be noticed that the factor of safetv. as found by these formulae, in¬ 
creases with the ratio of the length of the pillar to the least radius of gyration 
of its cross section: and is greater for round and hinged ends than for flat and 
fixed ends. See foot of p 446. 


* See “ Wrought Iron nnd Steel in Construction", bv Pencoyd Iron Works; published bv John 
Wiley & Sons, New York, 1S84. ' 

t See Transactions, American Society of Civil Engineers; Jan, Feb and March 1882. 






























































STRENGTH OF IRON PILLARS 


443 


r intimate crippling- strengths in lbs per s<j inch of metal 
n section of the four wrought iron pillars below. These formulas 
rjare deduced by Chs. Shaler Smith, from many tests by G. Bouscaren, C. E., of 
large pillars of good American iron. The lower Table is an abridgment of the 
full ones by C. L. Gates, C. E., in the Trans. Am. Soc. C. E., Oct., 1880. 


H = 


length between end bearing s 
least diameter d 


both in the same measure; and is to be squared. 


For safety take from ^ to %, according to circumstances. 


Flat ends 


One pin end.. 


Two pin ends. 





, H 2 

1 H- 

^ 2700 

36500 


H 2 
+ 1500 
36500 
H 2 

+ 1200 


Ultimate ami safe loads in lbs per sq inch, of the above four pillars, with 
flat ends, and equally loaded. Coef of Safety — 4 + .05 H. By C. L. Gates, C. E. 


II. 

A. Square Col. 

B. Phoenix Col. 

C. American Col. 

D. Common Col. 


Ult. 

Safe. 

Ult. 

Safe. 

Ult. 

Safe. 

Ult. 

Safe. 

15 

37067 

7822 

40476 

8521 

34434 

7249 

33693 

7093 

16 

36876 

7683 

40212 

8377 

34167 

7118 

33339 

6946 

18 

36470 

7443 

39645 

8091 

33597 

6856 

32589 

6651 

20 

36024 

7205 

39030 

7806 

32982 

6596 

31790 

6358 

22 

35544 

6970 

38373 

7524 

32327 

6338 

30952 

6069 

25 

34767 

6622 

37317 

7110 

31285 

5959 

29639 

5646 

30 

33344 

6063 

35424 

6440 

29435 

5352 

27375 

4977 

35 

31806 

5531 

33406 

5810 

27512 

4789 

25108 

4367 

40 

30198 

5033 

31352 

5226 

25584 

4264 

22919 

3820 

45 

28562 

4570 

29310 

4690 

23701 

3792 

20857 

3337 

50 

26932 

4143 

27321 

4203 

21900 

3369 

18952 

2916 

55 

25333 

3728 

25415 

3765 

20203 

3004 

17214 

2550 

60 

23787 

3398 

23611 

3373 

18621 

2660 

15643 

2235 






































































444 


STRENGTH OF IRON PILLARS 


TABLE OF BREAKING LOADS OF IRON PILLARS, 

in tons per square inch of metal area. Deduced from Gordon. The ends are ! 
supposed to be planed to form perfectly true bearings; and all parts to be equallyt 
pressed. The last is rarely the case in practice. li'tlie pillar is rectailgu- j 
lar instead of square, use the least side for a measure of length. (Original.)^ 


Hollow 

Round. 

Leugth 
measd 
in sides 
or 

diams. 

Hollow 

Square. 

Solid 

Round. 

Length 
measd 
in sides 
or 

diams. 

Solid 

Square. 

Breakg loads per 
sq inch of metal 
area of 

transverse section. 

Breakg loads per 
sq inch of metal 
area of 

transverse section. 

Breakg loads per 
sq inch of metal 
area of 

transverse section. 

Breakg loads per 
sq inch of metal 
area of 

transverse section. 

Cast. 

Wrt. 

Cast. 

Wrt. 

Cast. 

Wrt. 

Cast. 

Wrt. 

Tons. 

Tons. 


Tons. 

Tons. 

Tons. 

Tons. 


Tons. 

Tons. 

35.7 

16.1 

1 

35.7 

16.0 

35.6 

16.1 

1 

35.6 

16.1 

35.5 

16.1 

2 

35.5 

16.0 

35.2 

16.1 

2 

35.4 

16.0 

35.3 

16.1 

3 

35.3 

16.0 

34.6 

16.0 

3 

34.9 

16.0 

34.8 

16.0 

4 

35.0 

16.0 

33.9 

15.9 

4 

34.3 

16.0 

34.3 

16.0 

5 

34.6 

16.0 

33.0 

15.9 

5 

33.6 

16.0 

33.7 

16.0 

6 

34.2 

16.0 

31.9 

15.8 

6 

32.8 

15.9 

33.0 

15.9 

7 

33.7 

16.0 

30.6 

15.7 

7 

31.8 

15.3 

32.3 

15.8 

8 

33.1 

15.9 

29.3 

15.6 

8 

30.8 

15.7 

31.5 

15.8 

9 

32.4 

15.9 

28.0 

15.5 

9 

29.7 

15.7 

30.6 

15.7 

10 

31.7 

15.8 

26.7 

15.4 

10 

28.6 

15.6 

29.7 

15.6 

11 

31.0 

15.8 

25.4 

15.2 

11 

27.4 

15.5 

28.8 

15.5 

12 

30.3 

15.7 

24.1 

15.1 

12 

26.3 

15.3 

27.9 

15.5 

13 

29.5 

15.7 

22.8 

14.9 

13 

25.1 

15.2 

27.0 

15.4 

14 

28.7 

15.6 

21.6 

14.8 

14 

24.0 

15.1 

26.0 

15.3 

15 

27.9 

15.5 

20.5 

14.6 

15 

22.9 

15.0 

25.1 

15.2 

16 

27.1 

15.5 

19.4 

14.4 

16 

21.8 

14.8 

24.2 

15.1 

17 

26.3 

15.4 

18.3 

14.2 

17 

20.7 

14.7 

23.3 

15.0 

18 

25.4 

15.3 

17.2 

14.0 

18 

19.7 

14.5 

22.3 

14.9 

19 

24.6 

15.2 

16.2 

13.8 

19 

18.8 

14.3 

21.4 

14.8 

20 

23.8 

15.1 

15.3 

13.6 

20 

17.9 

14.2 

20.6 

14.7 

21 

23.0 

15.0 

14.5 

13.4 

21 

17.0 

14.0 

19.8 

14.6 

22 

22.2 

14.9 

13.7 

13.2 

22 

16.2 

13.8 

19.0 

14.4 

23 

21.5 

14.8 

12.9 

13.0 

23 

15.4 

13.6 

18.3 

14.3 

24 

20.8 

14.7 

12.2 

12.8 

24 

14.6 

13.5 

17.5 

14.2 

25 

20.1 

14.6 

11.6 

12.6 

25 

13.9 

13.3 

16.8 

14.0 

26 

19.4 

14.5 

11.0 

12.4 

26 

13.3 

13.1 

16.2 

13 9 

27 

18.7 

14.4 

10.4 

12.1 

27 

12.7 

12.9 

15.5 

13.7 

28 

18.0 

14.3 

9.90 

11.9 

28 

12.1 

12.7 

14.9 

13.5 

29 

17.4 

14.2 

9.41 

11.7 

29 

11.5 

12.6 

14.3 

13 4 

30 

16.8 

14.0 

8.93 

11.5 

30 

11.0 

12.4 

13.8 

13.2 

31 

16.3 

13.9 

8.50 

11.3 

31 

10.5 

12.2 

13.2 

13.1 

32 

15.7 

13.8 

8.10 

11.1 

32 

10.0 

12.0 

12.7 

12.9 

33 

15.2 

13.6 

7.70 

10.8 

33 

9.59 

11.8 

12 3 

12.8 

34 

14.6 

13.5 

7.36 

10.6 

34 

9.18 

11.6 

11.7 

12.6 

35 

14.1 

13.3 

7.04 

10.4 

35 

8.79 

11.4 

11.3 

12.5 

36 

13.6 

13.2 

6.71 

10.2 

36 

8.42 

11.2 

10.9 

12.3 

37 

13.2 

13.1 

6.41 

10.0 

37 

8.07 

11.0 

10.5 

12.2 

38 

12.7 

13.0 

6.11 

9.79 

38 

7.75 

10.8 

10.1 

12.0 

39 

12.3 

12.9 

5.85 

9.59 

39 

7.44 

10.7 

9.75 

11.9 

40 

11.9 

12.7 

5.62 

9.39 

40 

7.14 

10.5 

9.40 

11.7 

41 

11.5 

12.6 

5.40 

9.20 

41 

6.86 

10.3 

9.07 

11.6 

42 

11.1 

12.4 

5.19 

9.00 

42 

6.60 

10.1 

8.76 

11.4 

43 

10.8 

12.3 

4.99 

8.81 

43 

6.35 

9.95 

8.46 

11.3 

44 

10.4 

12.2 

4.79 

8.64 

44 

6.11 

9.77 

8.16 

11.1 

45 

10.1 

12.0 

4.59 

8.46 

45 

5.89 

9.59 

7.88 

10.9 

46 

9.80 

11.9 

4.42 

8.27 

46 

5.68 

9.42 

7.61 

10.8 

47 

9.50 

11.7 

4.26 

8.10 

47 

5.48 

9.25 

7.36 

10.6 

48 

9.20 

11.6 

4.11 

7.94 

48 

5.28 

9.09 

7.13 

10.4 

49 

8.94 

11.4 

3.97 

7.76 

49 

5.10 

8.92 

6.91 

10.3 

50 

8.66 

11.3 

3.84 

7.60 

50 

4.92 

8.77 

5.90 

9.61 

55 

7.47 

10.7 

3.22 

6.86 

55 

4.16 

8.00 

5.10 

8.92 

60 

6.49 

10.0 

2.75 

6.18 

60 

3.57 

7.30 

4.44 

8.29 

65 

5.68 

9.43 

2.37 

5.59 ‘ 

65 

3.09 

6.67 

3.90 

7.69 

70 

5.01 

8.84 

2.06 

5.06 

70 

2.70 

610 

3.44 

7.14 

75 

4.44 

8.30 

1.80 

4.59 

75 

2.37 

559 

3.08 

6.64 

80 

3.97 

7.77 

1.59 

4.18 

80 

2.10 

5.13 

2.46 

5.73 

90 

3.21 

6.88 

1.27 

3.49 

90 

1.68 

4.34 

2.02 

4.99 

100 

2.65 

6.02 

1.04 

2.95 

100 

1.37 

3.71 














































STRENGTH OF IRON PILLARS, 


445 


Table 1. HOLLOW CTLIND CAST IRON PILLARS. 

Breaking- loads, flat ends, perfectly true,and firmly fixed; 
and the loads pressing equally on every part of the top. 

By Gordon’s formula. 

For diams or lengths intermediate of those in the table, the 

loads may be found near enough by simple proportion. 

For thicknesses less than those in the table, the breaking loads 

may safely be assumed to diminish in the same proportion as the thickness, while the outer diam 
remains the same. But for greater thicknesses than those in the table, the loads do not increase 
as rapidly as the new thickness. Still, in practice, they may be assumed to do so approximately, 

if the new thickness does not exceed about % part of the 
.outer diam. 


ength in 
feet. 


CAST IRON. 

THICKNESS % INCH. (Original.) 


a 

Outer Diameter in inches. 


2 

2* 

2K 

2 H 

3 

3X 

4 

*X 

6 

$X 

6 



Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 


1 

45 2 

52.5 

59.8 

66.8 

74.1 

88.5 

103.0 

117.1 

131.3 

145.5 

159.6 

1 

2 

36.2 

43.4 

51.0 

58.6 

66.5 

81.5 

96.7 

111.3 

125.9 

140.5 

155.2 

2 

3 

27.2 

34.1 

41.4 

48.8 

56.7 

71.9 

87.6 

102.7 

117.8 

132.9 

148.1 

3 

4 

20.2 

26.3 

32.9 

39.7 

47.0 

61.9 

77.5 

92.9 

108.3 

123.7 

139.1 

4 

5 

15.1 

20.2 

25.9 

32.0 

38.6 

52.6 

67.4 

82.6 

97.9 

113.4 

129.1 

5 

6 

11.6 

15.8 

20.7 

25.9 

31.6 

44.3 

58.2 

72.7 

87.7 

103.0 

118.7 

6 

7 

9.1 

12.5 

16.6 

21.0 

26.1 

37.4 

50.1 

63.7 

78.1 

92.9 

108.3 

7 

8 

7.3 

10.1 

13.5 

17.3 

21.7 

31.6 

43.2 

55.8 

69.3 

83.4 

98.4 

8 

9 

5.9 

8.2 

11.1 

14.3 

18.2 

26.9 

37.3 

48.8 

61.5 

74.9 

89.2 

9 

10 

4.9 

6.9 

9.3 

12.0 

15.4 

23.1 

32.4 

42.9 

54.6 

67.1 

80.7 

10 

11 

4.1 

5.8 

7.9 

10.3 

13.2 

20.0 

28.3 

37.8 

48.6 

60.3 

73.0 

11 

12 

3.5 

5.0 

6.8 

8.9 

11.4 

17.4 

24.9 

33.5 

43.3 

54.2 

66.1 

12 

13 

3.0 

4.2 

5.8 

7.6 

9.9 

15 2 

21.9 

29.7 

38.8 

48.8 

60.0 

13 

14 

2.6 

3.6 

5.1 

6.7 

8.7 

13.5 

19.5 

26.6 

34.9 

44.1 

54.5 

14 

15 

2.3 

3.2 

4.5 

6.0 

7.7 

11.9 

17.4 

23.9 

31.4 

39.9 

49.7 

15 

16 

2.0 

2.8 

4.0 

5.3 

6.9 

10 7 

15.6 

21.5 

28.4 

36.4 

45.6 

16 

18 

1.6 

2.3 

3.2 

4.2 

5.5 

8.6 

12.7 

17.5 

23.4 

30.2 

38.0 

18 

20 

1.3 

1.8 

2.6 

3.4 

4.5 

7.1 

10.5 

14.7 

19.7 

25.6 

32.3 

20 

25 

.... 

....... 

1.7 

2.3 

3.0 

4.7 

7.0 

9.8 

13 3 

11.6 

22.5 

25 

30 


....... 

12 

1.6 

2.1 

3.2 

5-0 

7.0 

9.4 

12.6 

16.1 

30 

35 





1.5 

2.4 

3.7 

5.2 

7.1 

9.4 

12.2 

35 

40 





1.2 

1.9 

2.8 

4.0 

5.4 

7.2 

9.5 

40 




Weight of 1 foot of length of pillar in pounds. 

4.31 | 4.91 | 5.53 | 6.13 | 6.75 | 7.97 | 9.22 | 10.4 | 11.7 | 12.9 | 14.1 


1.38 I 


1.57 


Area of ring of solid metal in square inches. 

| 1.77 | 1.96 | 2.16 1 2.55 1 2.95 | 3.34 | 3.73 | 4. 


12 | 4.52 


CAST IKON. THICKNESS X INCH. (Original.) 



2 

4 

6 

8 

10 
12 
14 
16 
18 
4 20 

22 
25 
30 
35 


40 

45 

50 

60 

70 

80 


Outer Diameter in inches. 


5 

oX 

6 

6X 

7 

IX 

8 

8X 

9 

10 

11 

12 

Tons. 

Tons. 

Tons. 

Ton8. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

239 

268 

297 

325 

354 

383 

412 

440 

469 

526 

583 

640 

205 

235 

266 

296 

326 

356 

387 

416 

445 

504 

563 

622 

166 

196 

227 

257 

288 

319 

350 

380 

410 

471 

532 

593 

131 

159 

188 

218 

248 

279 

310 

340 

371 

432 

494 

557 

103 

127 

154 

182 

210 

240 

270 

300 

330 

391 

454 

517 

82 

103 

126 

151 

177 

205 

233 

262 

292 

351 

413 

475 

66 

84 

104 

126 

149 

174 

200 

227 

255 

313 

372 

434 

54 

69 

87 

106 

126 

149 

173 

198 

224 

277 

334 

394 

45 

58 

73 

90 

108 

128 

140 

172 

196 

246 

300 

357 

37 

48 

62 

77 

93 

111 

130 

151 

173 

219 

270 

323 

32 

41 

53 

66 

80 

96 

113 

132 

151 

194 

241 

292 

25 

34 

43 

54 

65 

79 

93 

109 

126 

164 

206 

252 

18 

24 

31 

39 

48 

59 

70 

83 

96 

126 

160 

199 

13 

18 

24 

30 

36 

44 

53 

63 

73 

98 

126 

459 

10 

14 

18 

23 

28 

35 

42 

50 

59 

78 

102 

129 

8 

11 

15 

19 

23 

28 

34 

41 

48 

64 

84 

107 

6 

9 

12 

15 

19 

23 

28 

33 

39 

53 

70 

89 

4 

6 

9 

11 

14 

17 

20 

24 

28 

38 

50 

65 

3 

4 

6 

8 

10 

12 

15 

18 

21 

29 

38 

50 

3 

4 

5 

6 

8 

9 

11 

13 

16 

22 

30 

38 


22.1 | 


Weight of 1 foot of length of pillar, in pounds. 

24.5 \ 27.“ I 29 4 | 31.9 | 34.4 | 36.9 | 39.4 | 41.9 | 46.6 | 51.6 | 


56.6 


7.07| 7.85 


Area of rin 

8.64 | 9.43 


f of solid metal, in square inches. 

10.2 | 11.0 | 11.8 1 12.6 | 13.4 | 14.9 


I 16.5 | 18.1 


a 



cu 


►J 


2 

4 

6 

8 

10 

12 

14 

16 

18 

20 

22 

25 

30 

35 

40 

45 

50 

60 

70 

80 






















































































446 STRENGTH OF IRON PILLARS. 


Table 1. HOLLOW €YLIN5> CAST IRON PILLARS. 

BREAKING LOADS.— (Continued.) By Gordon’s Rule. 


Length in 
feet. 

CAST IRON. THICKNESS 1 INCH. (Original.) 

a 1 

J3 ^ <S 

n 

Outer Diameter in Inches. 

12 

13 

14 

15 

16 

18 

20 

22 

24 

27 

30 

86 


Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 


4 

1188 

1301 

1415 

1530 

1645 

1874 

2103 

2330 

2557 

2896 

3236 

3915 

4 

6 

1138 

1253 

1368 

1484 

1601 

1833 

2066 

2295 

2525 

2866 

3208 

3890 

6 

8 

1065 

1184 

1303 

1423 

1543 

1779 

2015 

2247 

2479 

2824 

3170 

3860 

8 

10 

989 

1110 

1231 

1355 

1475 

1716 

1957 

2193 

2430 

2780 

3128 

3823 

10 

12 

909 

1030 

1152 

1275 

1399 

1644 

1889 

2129 

2369 

2723 

3076 

3778 

12 

14 

829 

949 

1071 

1195 

1320 

1566 

1813 

2056 

2300 

2659 

3016 

3726 

14 

16 

756 

873 

992 

1114 

1237 

1484 

1733 

1979 

2226 

2589 

2951 

3668 

16 ; 

18 

683 

796 

913 

1034 

1155 

1401 

1651 

1899 

2147 

2515 

2879 

3604 

18 

20 

618 

727 

840 

958 

1077 

1320 

1568 

1817 

2066 

2437 

2805 

3536 

20 

22 

559 

663 

772 

887 

1002 

1241 

1486 

1734 

1982 

2356 

2726 

3464 

22 

24 

508 

606 

709 

818 

929 

1163 

1404 

1650 

1899 

2272 

2644 

3387 

24 

26 

459 

553 

651 

756 

863 

1090 

1326 

1570 

1816 

2188 

2560 

3308 

26 

28 

418 

506 

598 

697 

800 

1020 

1250 

1489 

1733 

2103 

2475 

3226 

28 

30 

380 

462 

519 

644 

743 

954 

1178 

1412 

1653 

2020 

2392 

3143 

30 

32 

347 

424 

506 

595 

689 

893 

1110 

1338 

1575 

1938 

2308 

3059 

32 

34 

318 

390 

467 

552 

641 

836 

1046 

1268 

1500 

1857 

2225 

2974 

34 

36 

292 

359 

432 

511 

596 

783 

984 

1199 

1427 

1779 

2143 

2889 

36 

38 

268 

331 

400 

475 

556 

734 

928 

1136 

1361 

1704 

2063 

2804 

38 

40 

247 

305 

370 

441 

518 

687 

874 

1076 

1296 

1630 

1984 

2720 

40 

42 

229 

283 

344 

411 

484 

645 

825 

1019 

1232 

1560 

1909 

2636 

42 

41 

212 

263 

321 

383 

452 

605 

776 

963 

1169 

1491 

1834 

2552 

44 

46 

197 

246 

299 

358 

423 

569 

734 

915 

1114 

1428 

1765 

2474 

46 

48 

183 

229 

280 

335 

397 

536 

694 

869 

1060 

1367 

1696 

2396 

48 

50 

170 

213 

261 

314 

373 

505 

656 

824 

1008 

1306 

1627 

2319 

50 

55 

144 

181 

222 

269 

320 

438 

573 

725 

893 

1172 

1474 

2135 

55 

60 

124 

157 

192 

233 

278 

383 

503 

641 

795 

1054 

1333 

1964 

60 

65 

105 

134 

165 

202 

242 

335 

443 

568 

709 

951 

1211 

1808 

65 

70 

93 

118 

146 

178 

213 

297 

394 

507 

636 

853 

1099 

1664 

70 

80 

73 

92 

113 

139 

168 

235 

315 

408 

516 

705 

914 

1414 

80 

90 

58 

73 

91 

112 

136 

191 

257 

335 

426 

586 

768 

1209 

90 

100 

48 

60 

75 

92 

112 

157 

213 

279 

356 

491 

651 

1040 

100 


Weight of one foot of length of pillar, in pounds. 

108 | 118 | 128 | 138 | 147 | 167 | 187 | 206 | 226 | 255 | 285 | 344 


Area of ring of solid metal, in square inches. 

34.6 | 37.7 | 40.8 | 44.0 | 47.1 | 53,4 | 59.7 [ 66.0 | 72,2 | 81.7 [ 91.1 | 110.0 


CAST IRON. THICKNESS 3 INCHES. (Original.) 


ngth 

feet 

Outer Diameter in Feet. 

ngth 

feet. 


3 

3 K 

4 


5 


6 

7 

8 

10 

12 

0> 


Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 


10 

10806 

12878 

14908 

16967 

18986 

21038 

23052 

27143 

31196 

39254 

47375 

10 

15 

10453 

12564 

14628 

16712 

18754 

20825 

22855 

26972 

31045 

39133 

47273 

15 

20 

9996 

12150 

14250 

16369 

18440 

20533 

22585 

26737 

30837 

38962 ' 

47131 

20 

30 

8884 

11102 

13275 

15459 

17593 

19743 

21847 

26084 

30257 

38486 

46729 

30 

40 

76,88 

9907 

12113 

14344 

16531 

18735 

20891 

25224 

29476 

37838 

46175 

40 

50 

6554 

8702 

10888 

13126 

15341 

17580 

19780 

24107 

28532 

37037 

45484 

50 

60 

5553 

7575 

9690 

11892 

14100 

163+9 

18570 

23049 

27460 

36103 

44666 

60 

70 

4704 

6570 

8576 

10702 

12870 

15097 

17319 

21824 

26287 

35058 

43737 

70 

80 

3998 

5699 

7570 

9595 

11692 

13873 

16070 

20566 

25048 

33924 

42712 

80 

90 

3417 

4953 

6683 

8588 

10594 

12706 

14856 

19:304 

23791 

32725 

41670 

90 

100 

2940 

4321 

5909 

7665 

9588 

11614 

13700 

18065 

22521 

31481 

40438 

100 

110 

2547 

3788 

5238 

6888 

8677 

10606 

12613 

16869 

21270 

30213 

39220 

no 

125 

1950 

2954 

4400 

5864 

7483 

9257 

11132 

14650 

19448 

27664 

36692 

125 

150 

1532 

2350 

3353 

4547 

5900 

7417 

9058 

12701 

16668 

25182 

34127 

150 


Weight of one foot of length of pillar, in pounds. 


972 | 1150 | 1325 | 1503 | 1678 | 1856 | 2031 | 2388 | 27 40 | 3444 | 4153 

Coefficients of safety for hollow east iron pillars. 


Mr. James B. Francis, of Lowell, Mass., a high authority, in his “Tables of Cast Iron Pillars ’ 
recommends that in order to allow for unequal loading, imperfect casting, bad end bearings sidt 
blows. &c., we should not take the safe load at more than one-iifteenth of Hodgkinson's breakini 
load, if the pillars are roughly cast, and the ends not perfectly planed aud adjusted; and one-flftli 
when they are so, and the loads about equally distributed. 

It will be seen by the last table ou p 462, how our Gordon's loads differ from Hodgkinson's - bui 
we think that the same proportions of ours may be taken as safe; depending on the above con 
iitions. 
























































































STRENGTH OF IRON PILLARS, 


447 


HOLLOW CYLINDRICAL WROUGHT IRON PILLARS. 

Table 4, of breaking loads in tons of hollow cylindrical 
wrought iron pillars, with flat ends, perfectly true, and 
firmly fixed, and the loads pressing equally on every part 

of the top. Calculated by Gordon’s formula. No pains have been taken to have 
the last figure of the loads perfectly correct in every case. 

(Original.) 


a 

WROUGHT IRON. THICKNESS % INCH. 

.2 

a . 




Outer diameter in inches. 




•O 














X 

1 1 

IX 

IX 


2 

2« / VA 

m 

3 






BREAKING 

LOAD. 






Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 


1 

3.64 

5.27 

6.88 

8.50 

10.1 

11.7 

13.2 

14.8 

16.4 

18.0 

1 

2 

2.94 

4.64 

6.32 

8.00 

9.6 

11.2 

12.8 

14.5 

16.1 

17.8 

2 

3 

2.30 

3.86 

5.57 

7.28 

8.9 

10.6 

12.2 

13.9 

15.6 

17.3 

3 

4 

1.77 

3.13 

4.74 

6.36 

8.1 

9.9 

11.6 

13.3 

15.0 

16.7 

4 

5 

1.36 

2.51 

4.07 

5.66 

7.3 

9.1 

10.8 

12.5 

14.2 

16.0 

5 

6 

1 04 

2.03 

3.46 

4.91 

6.6 

8.3 

9.9 

11.6 

13.4 

15.2 

6 

7 

.81 

1.65 

2.91 

4.24 

5.7 

7.4 

9.1 

10.8 

12.6 

14.4 

7 

8 

.61 

1.36 

2.46 

3.67 

5.1 

6.7 

8.3 

9.9 

11.7 

13.5 

8 

9 

.50 

1.05 

2.03 

3.18 

4.5 

6.0 

7.5 

9.1 

10.8 

12.6 

9 

10 

.41 

.95 

1.75 

2.77 

4.0 

5.4 

69 

8.4 

10.1 

11.8 

10 

11 

.34 

.81 

1.52 

2.41 

3.6 

4.8 

6.2 

7.7 

9.3 

11.0 

11 

12 

.29 

.70 

1.34 

2.14 

3.2 

4.3 

5.6 

7.0 

8.6 

10.2 

12 

13 

.24 

.60 

1.16 

1.88 

2.8 

3.9 

5.2 

6.5 

8.0 

9.5 

13 

14 

.21 

.53 

1.03 

1.69 

2.5 

3.5 

4.7 

6.0 

7.4 

8.9 

14 

15 

.19 

.47 

.91 

1.50 

2.3 

3.2 

4.3 

5.5 

6.9 

8.3 

15 

16 

.18 

.42 

.84 

1.38 

2.1 

2.9 

4.0 

5.1 

6.4 

7.7 

16 

18 

.14 

.33 

.67 

1.11 

1.7 

2.4 

3.4 

4.4 

5.6 

6.8 

18 

20 


.27 

.55 

.91 

1.4 

2.0 

2.8 

3.7 

4.7 

5.8 

20 

25 





.9 

1.4 

2.0 

2.6 

3.4 

4.2 

25 


Weight of one foot of length of pillar, in pounds. 

.820 | 1.15 | 1.47 | 1.80 | 2.13 | 2.45 | 2.78 | 3.11 | 3.43 | 3.77 


Area of ring of solid metal, in square inches. 

.246 | .344 | .442 | .540 | .638 | .736 | .835 | .933 | 1.03 | 1.13 


C v- 
V 


1 

2 

3 

4 

5 

6 

7 

8 
9 

10 

11 

12 

13 

14 

15 

16 
18 
20 
25 
30 
35 
40 
45 
50 


WROUGHT IRON. THICKNESS % INCH. 


□ 


Outer diameter in inches. 

ngt 

feel 












0) 

2 

2 X 

234 

2% 

3 

3>4 

4 

434 

5 

5J4 

1 6 

►J 




BREAKING LOAD. 






Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 


21.9 

25.4 

28.3 

31.4 

34.5 

40 

47 

53 

60 

66 

72 

1 

21.1 

24.3 

27.6 

30.7 

33.9 

•40 

47 

53 

60 

66 

72 

2 

19.9 

23.1 

26.4 

29.7 

33.0 

39 

46 

52 

59 

65 

71 

3 

18.6 

21.8 

25.3 

28.5 

31.9 

38 

45 

51 

58 

64 

71 

4 

17.0 

20.4 

23.5 

27.3 

30.7 

37 

44 

50 

57 

63 

70 

5 

15.4 

18.8 

22.1 

25.7 

29.2 

36 

43 

49 

56 

62 

69 

6 

13.9 

17.3 

20.5 

23.8 

27.8 

34 

41 

47 

54 

61 

68 

7 

12.5 

15.6 

19.1 

22.3 

25.9 

32 

40 

46 

53 

60 

67 

8 

11.2 

14.2 

17.5 

20.6 

24.3 

30 

38 

44 

51 

58 

65 

9 

10.0 

13.0 

16.1 

19.1 

22.7 

29 

37 

43 

50 

57 

64 

10 

9.0 

10.7 

15.7 

17.6 

21.1 

27 

85 

41 

48 

55 

62 

11 

8.1 

10.6 

13.5 

16.4 

19.6 

26 

33 

40 

46 

54 

61 

12 

7.3 

9.6 

12.4 

15.1 

18.2 

24 

31 

38 

44 

52 

59 

13 

6.6 

8.8 

11.3 

14.0 

17.0 

23 

30 

36 

43 

51 

57 

14 

6.0 

8.0 

10.4 

12.9 

15.8 

21 

28 

34 

41 

49 

55 

15 

5.5 

7.3 

9.5 

12.0 

14.6 

20 

27 

33 

40 

47 

54 

16 

4.5 

6.0 

8.0 

10.3 

12.7 

18 

24 

30 

37 

43 

50 

18 

3.8 

5.1 

6.8 

8.7 

11.0 

16 

21 

27 

34 

40 

47 

20 





7.9 

12 

16 

21 

27 

33 

39 

25 






13 

17 

22 

27 

32 

30 







10 

14 

18 

22 

27 

35 









14 

18 

23 

40 









11 

15 

19 

45 









8 

12 

16 

50 


Weight of one foot of length of pillar, in pounds. 

4.60 | 5.23 | 5.90 | 6.53 | 7.20 | 8.50 | 9.83 | 11.1 | 12.4 | 13,7 | 15.0 

Area of ring of solid metal, in square inches. 

1.38 I 1.57 i 1.77 | 1.96 | 2.16 | 2.55 | 2.95 | 3.34 j 3.73 I 4.12 I 4.51 





























































448 STRENGTH OF IRON PILLARS. 

HOLLOW CYLINDRICAL WROUGHT IRON PILLARS 


Table 4, (Continued.) (Original.) 


WROUGHT IRON. THICKNESS H INCH. 


.2 


ra . 

Outer diameter in inches. 

ngth 

feet. 

G 

'll 

1-3 

5 

5^ 

6 

1 634 

1 7 

1 7J4 

1 8 

8 M 1 

9 1 

10 

11 

12 

0 > 






BREAKING LOAD. 







Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 


2 

112 

125 

139 

152 

166 

177 

189 

201 

214 

238 

263 

290 

2 

4 

110 

123 

136 

149 

163 

174 

186 

199 

212 

237 

262 

289 

4 

6 

106 

119 

132 

145 

158 

171 

184 

197 

210 

235 

261 

288 

6 

8 

101 

114 

127 

140 

154 

167 

181 

194 

207 

232 

258 

284 

8 

10 

95 

108 

123 

136 

149 

162 

176 

189 

203 

228 

254 

280 

10 

12 

89 

102 

116 

129 

143 

157 

171 

185 

199 

224 

250 

276 

12 

14 

82 

95 

108 

122 

137 

151 

165 

179 

194 

219 

245 

272 

14 

16 

76 

89 

103 

117 

131 

145 

160 

173 

187 

213 

240 

268 

16 

18 

70 

83 

97 

110 

124 

138 

153 

166 

180 

207 

235 

263 

18 

2C 

64 

77 

91 

104 

117 

131 

145 

159 

173 

201 

227 

257 

20 

22 

58 

70 

83 

96 

109 

123 

138 

151 

165 

192 

220 

250 

22 

25 

52 

64 

76 

89 

102 

115 

129 

143 

157 

183 

212 

241 

25 

30 

42 

52 

63 

74 

87 

100 

113 

127 

141 

167 

195 

224 

30 

35 

34 

43 

53 

64 

75 

87 

99 

112 

125 

151 

178 

207 

35 

40 

27 

35 

44 

53 

64 

75 

86 

98 

110 

135 

163 

190 

40 

45 

23 

30 

38 

46 

55 

65 

76 

87 

98 

123 

148 

174 

45 

50 

19 

24 

32 

38 

47 

56 

66 

76 

67 

109 

133 

158 

50 

60 

15 

19 

24 

29 

36 

43 

51 

60 

69 

88 

109 

132 

60 

70 

11 

14 

18 

23 

28 

34 

40 

48 

56 

73 

91 

111 

70 

80 

9 

11 

14 

18 

22 

27 

32 

37 

44 

57 

74 

93 

80 

90 

7 

1 9 

11 

14 

18 

22 

26 

31 

36 

49 

63 

78 

90 

100 

6 

1 1 

9 

12 

15 

18 

22 

26 

30 

41 

53 

66 

100 


( 

1 I 


I 

I 




Weight of one foot of length of pillar, in pounds. 

23.6 | 26,2 | 28.8 | 31.4 | 34.0 | 36.6 | 39.3 | 42.0 | 44.7 | 49.7 | 55.0 | 60.3 

Area of ring of solid metal, in square inches. 

7.07 | 7.85 | 8.61 | 9.43 | 10.2 | 11.0 | 11.8 | 12.6 | 13.4 | 14.9 16.5 | 18.1 


Table 4, (Continued.) (Original.) 


a 

WROUGHT IRON. THICKNESS 1 INCH. 

a 

5 ** 

Outer diameter in inches. 

-G 

Is 
















13 

14 

15 

16 

17 | 

18 | 

20 | 

22 | 

24 | 

26 | 

28 | 

30 







BREAKING LOAD. 







Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 


i 

603 

653 

704 

753 

805 

854 

955 

1056 

1157 

1257 

1357 

1458 

1 

10 

588 

638 

691 

742 

795 

846 

949 

1049 

1149 

1248 

1354 

1457 

10 

20 

543 

595 

651 

702 

759 

810 

913 

1016 

1120 

1223 

1327 

1430 

20 

30 

479 

538 

594 

645 

699 

758 

866 

973 

1077 

1186 

1289 

1394 

30 

40 

415 

470 

528 

584 

636 

691 

806 

912 

1027 

1130 

1237 

1348 

40 

50 

355 

405 

462 

516 

570 

627 

740 

848 

961 

1067 

1179 

1294 

50 

60 

300 

348 

400 

452 

505 

559 

669 

781 

891 

1005 

1115 

1228 

60 

70 

256 

300 

348 

398 

448 

499 

606 

715 

824 

936 

1046 

1160 

70 

80 

215 

255 

298 

344 

392 

440 

543 

649 

757 

868 

978 

1092 

80 

90 

185 

222 

261 

303 

347 

892 

489 

590 

694 

800 

910 

1023 

90 

100 

157 

190 

225 

262 

303 

345 

436 

532 

631 

735 

843 

955 

100 

110 

134 

162 

193 

227 

264 

302 

386 

474 

568 

666 

770 

877 

110 

125 

111 

135 

162 

192 

225 

259 

336 

416 

505 

598 

697 

799 

125 

150 

82 

101 

122 

145 

171 

198 

262 

328 

405 

485 

574 

666 

150 

175 

62 

78 

95 

112 

133 

155 

208 

266 

331 

400 

478 

560 

175 

200 

49 

60 

74 

89 

106 

124 

168 

216 

269 

328 

395 

467 

200 


126 


Weight of one foot of length of pillar, in pounds. 

136 | 147 1 157 | 168 | 178 | 199 | 220 | 241 | 262 | 283 | 304 


37.7 


Area of ring of solid metal, in square inches. 

40.8 | 44.0 | 47.1 | 50.3 | 53.4 | 59.7 | 66.0 | 72.3 | 78.5 | 84.8 | 91.1 


The breaking 1 loads for less thicknesses may safely be assumed to 

diminish at the same rate as the thickness. 



































































STRENGTH OF IRON PILLARS, 


449 


Table of rolled-iron segment-columns of tbe Phoenix 

Iron Co, 410 Walnut St, Philada. For their 
strengths, seepp 442, 443, or formula, p439, with the 
least radius of gyration, as given below, or as obtained 
by multiplying D by .3636. The dimensions given 
are subject to slight variations which are unavoidable 
in rolling iron shapes. The weights of columns given 
are those of the 4, 6 , or 8 segments, of which they are 
composed. The shanks of the rivets used in joining them 
together, of course, merely make up the quantity of metal 
punched or drilled out, in making the holes; but the rivet- 
heads add fro'm 2 to 5 per cent to the weights given. The 
rivets are spaced 3, 4, or 6 ins apart from cen to 
cen. Prices of the finished columns (1886), from 3 to 
41*2 cts per pound, at the works, according to specifica¬ 
tions. and varying with the quantity ordered, the length, 
the thickness, and the amount of extra work required. 

Any desired thickness between tbe min and 
max for any given size, can be furnished. We give the 
dimensions, wts, Ac, corresponding to the principal thick¬ 
nesses. G columns have 8 segs. E, 6 segs. All others, 4 segs. 






! il 



fL\ 

‘ * \ 

l 

1 1 

1 | 



1 1 

r i ! 

i 

! i 



1 

! i 


w 

1 | 

I | 

i < 

*-■— d—^ 

! ! 
i i 

i i 

! l -- 

- --D,-- 

i i 

i 






• 

X 

U 

ce 

S 

* 

S 

X « 

V = 
"" 

m 

Diameters, ins. 

One column. 

Size of 
Rivets. 

d 

D 

D/ 

Area 

of cross 
sec, 
sq ins. 

W t per 
ft run, 
lbs. 

Eeast 
rad of 

gyr, 

ins. 

A 

* 

3% 

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226.6 

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21 

92. 

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X 

44 


30 




































Table 3, of breaking loads, in tons, of SOLID CYLINDRICAL CAST IRON pillars, wlt!> Rat ends, firmly 
fixed, and equally loaded. By Gordon’s rule. 

No pains have been taken to have the last figure of the loads perfectly correct in every case. 

(Original.) 


450 


STRENGTH OF IRON PILLARS 


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Table 5, of breaking loads, in tons, of SOLID CYLINDRICAL WROUGHT IRON pillars, with flat ends, 
firmly fixed, and equally loaded. By Gordon’s rule. 

No pains have been taken to have the last figure of the loads perfectly correct in every case. 

, (Original.) 


STRENGTH OF IRON PILLARS 


451 


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Weight of one foot of length of pillar, in pounds. 

.655 4 1.48 | 2.62 [ 4.09 | 5.90 | 8.02 | 10.5 | 13.3 | 16.4 | 19,8 | 23 6 | 27.7 | 32.1 | 36.9 | 41.9 | 47.3 | 53.1 | 59.0 | 65.5 | 72.2 [ 79.2 | 86,6 | 94.3 

Area of solid metal, in square inches. 

.196 | .442 | .785 | 1.23 | 1,77 [ 2,41 | 3,14 | 3.98 | 4,91 | 5.94 | 7.07 [ 8,30 | 9.62 | 11.0 | 12.6 | 14,2 | 15.9 | 17.7 | 19.6 1 21.6 | 23.8 | 26.0 | 28.3 












































































Table 3, of breaking load, in tons, of SOLID SQUARE CAST IRON pillars, with fiat ends, firmly fixed 
and equally loaded. By Gordon’s rule. 

No pains have been taken to have the last figure of the loads perfectly correct in every case. 

(Original.) 


452 


STRENGTH OF IRON PILLARS. 


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.781 | 1.76 1 3.13 | 4,88 | 7.03 j 9.57 | 12.5 | 15,8 | 19.5 | 23,6 | 28.1 | 33.0 | 38.3 | 44.0 | 50.0 [ 56.5 [ 63.3 | 70.5 [ 78.1 [ 86.1 | 94.5 [ 103 | 1 13 

Area of solid metal, in square inches. 

.250 | .563 1 1 | 1,56 | 2.25 | 3.06 | 4 | 5.06 | 6.25 | 7.56 | 9 | 10.6 | 12,3 | 14.1 | 16 | 18.1 | 20.3 | 22.6 | 25 | 27.6 | 30.3 | 33.1 | 36 

































































of SOLID SQUARE WROUGHT IRON PILLARS, with flat ends, 




STRENGTH OF IRON PILLARS 


453 


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Weight of one foot of length of pillar, in pounds. 

.83 | 1.88 [ 3.33 | 5.21 | 7.51 | 10.2 | 13.4 | 16,0 | 20.9 [ 25.2 | 30.0 | 35.3 | 40.9 | 46.9 | 53.4 | 60.2 | 67.5 | 75.3 | 83.4 | 91.9 [ 101 | 110 1 120 

Area of solid metal, in square inches. 

.250 [ .563 | 1.00 | 1.56 | 2,25 | 3.06 | 4.00 | 5.06 ) 6.25 | 7.56 | 9.00 | 10.6 | 12.3 | 14.1 | 16,0 | 18,1 [ 20.3 | 22,6 | 25.0 | 27.6 | 30.3 | 33.1 | 36.0 



























































































454 


ROLLED I BEAMS AS PILLARS. 


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ROLLED I BEAMS AS PILLARS, 


455 


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tons. 

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Light.. 
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456 


ROLLED CHANNEL BAR PILLARS 


Tableof quiescent breaking 1 loads in short, tons (2000 lbs) 
of rolled channel iron LJ . as pillars or struts. From Carnegie 

Bros & Cos. (Union Iron Mills, Pittsburgh. Penn) “ Tables and Information ou Wrought Iron." 

For manner of use see the paragraph above the preceding table. Hgd, fxd, mean hinged, lixed. 

The headings 12 Hy, 12 Mm, &c, mean 12 inch heavy, and 12 inch medium 
channels. Sideway "means across the web; Edge way means along the 
web. 


Length of Strut. 

12 Hy. 

12 Mm. 

10 Hy. 

10 Mm. 

Both 


Both 

Side 

Edge 

Side 

Edge 

Side 

Edge 

Side 

Edge 

ends 

1 fxd. 

ends 

Way. 

Way. 

Way. 

Way. 

Way. 

Way. 

Way. 

Way. 

flxd. 

1 hgd. 

hgd. 

Sht 

Sht 

Sht 

Sht 

Sht 

Sht 

Sht 

Sht 

Ft. 

Ft. 

Ft. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

1 



268 

270 

160 

160 

187 

189 

121 

124 

2 

1.5 

1 

260 

268 

154 

160 

181 

189 

118 

123 

4 

3. 

2 

235 

266 

135 

160 

J61 

188 

104 

123 

6 

4.5 

3 

200 

264 

112 

160 

135 

187 

87 

122 

8 

6. 

4 

168 

262 

90 

159 

111 

186 

71 

122 

10 

7.5 

5 

138 

260 

72 

159 

90 

185 

57 

121 

12 

9. 

6 

114 

258 

60 

159 

75 

183 

46 

121 

14 

105 

7 

95 

256 

47 

158 

60 

181 

38 

120 

16 

12. 

8 

79 

254 

38 

156 

50 

178 

31 

118 

18 

13.5 

9 

66 

253 

32 

155 

41 

176 

26 

116 

20 

15. 

10 

56 

252 

27 

153 

35 

174 

22 

115 

22 

16.5 

11 

48 

249 

23 

151 

31 

171 

19 

113 

24 

18. 

12 

41 

246 

20 

149 

27 

167 

16 

111 

26 

19.5 

13 

36 

243 

17 

147 

24 

163 

14 

109 

28 

21. 

14 

32 

239 

15 

145 

20 

159 

12 

107 

30 

22.5 

15 

28 

235 

13 

143 

17 

156 

11 

105 

32 

24. 

16 

25 

230 

11 

140 

15 

153 

10 

103 

34 

25.5 

17 

23 

226 

10 

138 

14 

149 

9 

100 

36 

27. 

18 

21 

222 

9 

136 

13 

146 

8 

98 

38 

28.5 

19 

19 

218 

8 

134 

11 

143 

7 

96 

40 

30. 

20 

17 

213 

7 

132 

10 

140 

6 

94 




9 Hy. 

9 Mill. 

8 Hy. 

8 Mm. 

1 

1.5 


160 

162 

96 

97 

134 

135 

85 

86 

2 

1 

154 

161 

93 

97 

130 

134 

83 

85 

4 

3. 

2 

134 

160 

80 

96 

112 

133 

70 

85 

6 

4.5 

3 

110 

159 

65 

96 

91 

132 

57 

84 

8 

6. 

4 

90 

158 

53 

95 

72 

131 

45 

84 

10 

7.5 

5 

72 

157 

42 

95 

58 

130 

36 

83 

12 

9. 

6 

58 

154 

33 

94 

46 

127 

28 

83 

14 

10.5 

7 

47 

150 

27 

93 

30 

125 

23 

82 

16 

12. 

8 

38 

147 

22 

92 

25 

123 

19 

81 

18 

13.5 

9 

32 

143 

18 

91 

21 

120 

15 

80 

20 

15. 

10 

27 

140 

15 

90 

19 

118 

13 

78 

22 

16.5 

11 

23 

138 

13 

88 

17 

115 

11 

76 

24 

18. 

12 

20 

136 

11 

86 

15 

112 

10 

74 

26 

19.5 

13 

17 

134 

9 

84 

13 

109 

8 

72 

28 

21. 

14 

15 

131 

8 

82 

12 

105 

7 

70 

30 

22.5 

15 

13 

129 

7.3 

80 

10 

102 

6 

68 

32 

24. 

16 

12 

126 

6.6 

78 

9 

99 

5.5 

66 

34 

25.5 

17 

11 

122 

5.9 

76 

8 

96 

5 

64 

36 

27. 

18 

10 . 

118 

5.2 

74 

7.4 

93 

4.5 

62 

38 

28.5 

19 

9 

115 

4.5 

72 

6.7 

89 

4 

60 

40 

30. 

20 

8 

111 

3.8 

70 

6.0 

86 

3.5 

58 

i 



7 Hy. 

7 Min. 

« Hy. 

5 Hy. 

1.5 

1 

105 

106 

75 

76 

54 

55 

53 

54 

2 

103 

106 

72 

75 

51 

54 

50 

54 

4 

3. 

2 

87 

105 

61 

75 

43 

53 

41 

53 

6 

4.5 

3 

70 

104 

49 

74 

; 33 

52 

31 

52 

8 

6. 

4 

55 

104 

39 

74 

25 

51 

24 

51 

10 

7.5 

5 

43 

103 

30 

73 

20 

50 

18 

50 

12 

9. 

6 

34 

100 

24 

72 

16 

49 

14 

49 

14 

10.5 

7 ■ 

27 

98 

19 

71 

13 

48 • 

11 

47 

16 

12. 

8 

23 

96 

16 

70 

10 

47 

8.8 


18 

13.5 

9 

18.2 

94 

13 

68 

8 

46 

7.2 

43 

20 

15. 

10 

15.5 

92 

11 

66 

7 

45 

6 

41 

22 

16.5 

11 

13.5 

89 

9 

64 

6 

44 

5 

40 

24 

18. 

12 

12 

86 

8 

62 

5 

42 

4.3 

38 

26 

19.5 

13 

10.5 

83 

7 

60 

4.5 

40 

3.7 

36 

28 

21. 

14 

9 

80 

6 

58 

4. 

38 

3.2 

34 

30 

22.5 

15 

7.5 

77 

i 5 

56 

3.6 

37 

2.8 

32 

32 

24. 

16 

6.9 

75 

4.6 

54 

3. 

36 

2.5 

31 

84 

25.5 

17 

6.2 

72 

4.2 

52 

2.6 

35 

'2.2 

30 

36 

27. 

18 

5.5 

69 

3.7 

50 

2.3 

34 

2.0 

28 

38 

28.5 

19 

5.0 

66 

3.3 

48 

2.1 

33 

1 8 

20 

40 

30. 

20 

4.5 

63 

3.1 

46 

1.9 

30 

1.6 

24 






































STRENGTH OF IRON PILLARS. 


457 


In arches of cast iron for bridges, &c, it is usual among'English 
engineers not to allow more than 2^ tons (5600 tt>s) of compression, or thrust, per sq 
inch. Brunei never subjected cast-iron pillars to more than 1% tons (3360 lbs) per 
sq inch. C. Slialer Smith employs as maximum working strains, 1. of the calcu¬ 
lated breaking strain for such hollow chords and posts of bridges as are 1 inch or 
more in thickness, and not more than 15 diams long. For posts, only j ; when not 
less than % inch thick, nor more than 25 diams long; or from -jL to when % 
thick or less, and more than 25 diams long. 

The young' engineer must bear in mind that the breakg and the 
safe loads per sq inch, of pillars of any given material, are not constant quantities ; 
but diminish as the piece becomes longer in proportion to its diam. If a very long 
piece or pillar be so braced at intervals as to prevent its bending at those points, 
then its length becomes virtually diminished, and its strength increased. Thus, if a 
pillar 100 ft long be sufficiently braced at intervals of 20 ft, then the load sustained 
may be that due to a pillar only 20 ft long. Therefore, very long pillars used for 
bridge piers, &c, are thus braced; as are also long horizontal or inclined pieces, 
exposed to compression in the form of upper chords of bridges; or as struts of any 
kind in bridges, roofs, or other structures. 

Mistakes are sometimes made by assuming, say 5 or 6 tons per sq inch, as the safe 
compressing load for cast iron; 4 tons for wrought; 1000 pounds for timber; without 
any regard to the length of the piece. 

But although the final crushing loads, as given in tables of strengths of materials, 
are usually those for pieces not more than about 2 diams high, they will not be much 
less for pieces not exceeding 4 or 5 diams. 

Cautions. Remember a heavily loaded cast-iron pillar is easily broken by a 
side blow. Cast-iron ones are subject to hidden voids. All are subject to jars and 
vibrations from moving loads. It very rarely happens that the pres is equally dis¬ 
tributed over the whole area of the pillar; or that the top and bottom ends have per¬ 
fect bearing at every part, as they had in the experimental pillars f Cast pillars are 
seldom perfectly straight, and hence are weakened. 

Hollow pillars intended to bear heavy loads should not be cast 
with such mouldings as a a; or with very 
projecting bases or caps such as g, Fig 19. 

( It is plain that these are weak, and would 
break off under a much less load than 
w r ould injure the shaft of the pillar. When 
such projecting ornaments are required, 

I they should be cast separately, and be at¬ 
tached to a prolongation of the shaft, as 
cd, by iron pins or rivets. 

Ordinarily, it is better to adopt a more 
simple style of base and cap, which, as at 
5, can be cast in one piece with the pillar, 
without weakening it. 

Hodginson states that while the quantity of material is the same 
in both pillars , no sti'ength is gained in hollow ones by making 
the diams greater at the middle than at the 
| ends ; but that in solid ones, with rounded ends, there is a gain 
of about i.th part; and in those with fiat ends , of about i.th or 
Ath part, by making the diam at the middle about 1% or 2 times 
that at the ends. Also that a uniform round pillar has the same 
strength as a moderately tapering one whose diam at half-way up 
is equal to the uniform diam of the cylindrical one. 

Also, that when a flat-ended pillar, Fig. 2, is so irregularly 
fixed, that the pressure upon it passes along its diagonal a a, it 
loses two-thirds of its strength. Hence the necessity for equalizing, as far as possi¬ 
ble, the pressure over every part of the top and bottom of the pillar; a point very 
difficult to secure in practice. 


f In important cases, both ends should be planed perfectly true; 

as is done in iron bridges, &c. 























458 


STRENGTH OF WOODEN PILLARS. 


Steel pillars. Mr. Kirkaldy experimented with a small steel pillar or tube 
of Shortridge, Howell & Co’s homogeneous metal. Its length was 4 it, or 25.6 diams; 
outer diam 1% inch: inner diam thickness % inch. Area of cross-section 
so ins. Flat ends. Under 67300 lbs, or 30 tons total pressure, or 17.14 tons per sq 
inch of solid metal section, it bent very slightly. On increasing the pressure con¬ 
siderably, the pillar sprang out from under the load. Our table, page 444, gives 24)4 
tons total, or 13 tons per sq inch, as the ultimate load for a wrouglit-iron tube ol the 

same size. .... . . 

Mr. M. G. Love, Paris, as the result of a trial with small steel rods, about .4 
inch diam, and which had a tensile strength of 48 tons per sq inch,suggests that the 
comparative strength of pillars of wrought iron, cast-iron, and steel, are probably 
about as follows: At from 1)4 to 5 diams in length, steel and cast-iron ones have 
equal strengths ; and either of them is about twice as strong as wrought iron. At 10 
diams, steel is 1)4 times as strong as cast; and 2.2 times as strong as wrought iron. 
At 40 diams, steel is 4 times as strong as cast; and 2.7 its strong as wrought. But 
this needs confirmation. Now that powerful and accurate testing machines are 
coming more into use, we may hope that the doubts at present existing on such 
subjects will be set at rest. 

Mr Stoney advises that until then steel pillars should not be trusted 
with more than 1.5 the loads of wrought iron ones. 

WOODEN PILLARS. 


The strengths of pillars, as well as of beams of timber, depend much on their de¬ 
gree of seasoning. Hodgkinson found that perfectly seasoned blocks, 2 diams 
long, required, in many cases, twice as great a load to crush them as when only 
moderately dry. This should be borne in mind when building with green timber. 

In important practice, timber should uot be trusted with more than )4 to %oi its 
calculated crushing load; aud for temporary purposes, not more than ]/ & to 

Mr. Charles Slialer Smith. C. E., of St. Eouis. prepared the 
following formula for the breaking loads of either square or rectangular 
pillars or posts, of moderately seasoned white, and common yellow pine, with flat 
ends, firmly fixed, and equally loaded, based upon experiments by himself. 

It is Gordon’s formula adapted to those woods; and gives results considerably 
smaller than Hodgkinson’s, It is therefore safer. 

Call either side of the square, or the least side of the rectangle, the breadth. Then, 


Breakg load in lbs, per 1 
Rule. sq inch of area, of a > = 
pillar of W or Y pine J 


5000f 


1 + 


'sq of length in ins 
^sq of breadth in ins 


X -004 


Or in words, square the length in ins; square the breadth in ins; div the first square 
by the second one; mult the quot by .004 ; to the prod add 1; div 5000 by the sum. 

Ex. Breakg load per sq inch, of a white pine pillar 12 ins square, and 30 ft, or 360 
fus long. Here the sq of length in ins is 360 2 = 120600. The square of the breadth is 


122 = 144 ; au( j 


129000 

144 


= 000; and 900 X -004 = 3.6; and 3.6 + 1 = 4.6. 


5000 

FinaHy, — 


= 1087 lbs, the reqd breakg load per sq in. As the area of the pillar is 144 sq ins, 
the entire breakg load is 10s7 X 144 = 156528 lbs, or 69.9 tons. 

Recent experiments on wooden pillars 20 ft long, and 13 ins square, by Mr. 
Kirkaldy, of England, confirm the far greater reliability of Mr. Smith’s formula. 
Hence we present the following new set of original tables based upon it. 

For solid pillars of cast iron and of pine, whose heights range 

from 5 to 60 times their side or diam, we may say. near enough for practice, that a 
cast iron one is about 16)4 times as strong as a pine one; but no such approximate 
ratio holds good between wrought iron and pine,or between cast and wrought iron. 


t The breakiug load in lbs per sq inch in short blocks, by Mr. Smith. 








STRENGTH OF WOODEN PILLARS. 459 


Table of breaking* loads in tons of square pillars of half 
seasoned white or common yellow pine firmly fixed and 
equally loaded. By C. Shuler Smith’s formula. (Original.) 


•*— O) 

Uzj 

sc .5 


Side 

of square pine pillar, in inches. 



Height 
in feet. 

1 1 


ix 

m 

2 

2!4 

2 X 


3 

3J4 

■*x 

3 X 

* 






BREAKING 

LOAD. 







Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 


l 

1.42 

2.54 

3.99 

5.73 

7.80 

10.1 

12.8 

15.7 

18.9 

22.3 

26.1 

30.1 

34.5 

1 


1.17 

2.22 

3.59 

5.26 

7.25 

9.6 

12.2 

15.1 

18.3 

21.7 

25.4 

29.2 

33.7 

x 


.97 

1.93 

3.19 

4.80 

6.74 

9.0 

11.6 

14.5 

17.6 

21.0 

24.7 

28.6 

33.0 

A 

% 

.81 

1.66 

2.81 

4.35 

6.19 

8.4 

10.9 

13.7 

16.8 

20.2 

23.9 

27.8 ' 

32.1 

X 

2 

.08 

1.44 

2.48 

3.92 

5.66 

7.8 

10.2 

12.9 

15.9 

19.3 

23.0 

26.9 

31.2 

2 

x 

.57 

1-24 

2.19 

3.53 

5.17 

7.2 

9.6 

12.3 

15.2 

18.5 

22.0 

25.8 

30.1 

X 

)4 

.49 

1.07 

1.93 

3.16 

4.70 

6.7 

8.9 

11.5 

14.3 

17.6 

21.1 

24.9 

29.1 

X 

% 

.42 

.93 

1.71 

2.85 

4.29 

6.2 

8.3 

10.8 

13.5 

16.7 

20.1 

23.8 

28.0 

X 

3 

.36 

.82 

1.52 

2 55 

3 89 

5.6 

7.6 

10.0 

12 7 

15.8 

19.2 

22.9 

27 0 

3 

% 

.28 

.63 

1.21 

2.07 

3.23 

4.8 

6.6 

8.8 

11.3 

14.2 

17.4 

20.9 

24.8 

X 

4 

.22 

.50 

.98 

1.70 

2.70 

4.0 

5.7 

i.i 

9.9 

12.7 

15.7 

19.0 

22.7 

4 


.18 

•40 

.81 

1.42 

2.28 

3.4 

4.9 

6.7 

8.8 

11.4 

14.1 

17.2 

20.7 

X 

5 

.15 

.34 

.68 

1.19 

1.94 

3.0 

4.2 

5.8 

7.7 

10.0 

12.6 

15.5 

18.8 

5 

X 

.12 

.28 

.57 

1.02 

1.67 

2 6 

3.7 

5.1 

6.8 

8.9 

11.3 

14.0 

17.1 

X 

6 

.10 

.24 

.49 

.86 

1.44 

2.3 

3.3 

4.6 

6.1 

8.0 

10.2 

12.7 

15.6 

6 

53 

.09 

.21 

.43 

.74 

1.26 

2.0 

2.9 

4.1 

5.4 

7.2 

9.2 

11.6 

14.2 

X 

7 

.08 

.18 

.37 

.66 

1.11 

1.8 

2 6 

3 6 

4.9 

6.5 

8.3 

10.5 

12.9 

7 

53 

.07 

.16 

.33 

.59 

.98 

1.6 

2.3 

3.3 

4.4 

5.9 

7.6 

9.6 

11.8 

X 

8 

.06 

.14 

.29 

.52 

.87 

1.4 

2.0 

2.9 

3.9 

5.2 

6.8 

8.7 

10.8 

8 

x 

.05 

.12 

.26 

.47 

.78 

1.2 

1.8 

2.6 

3.5 

4.8 

6.2 

7.9 

9.9 

X 

9 

.05 

.11 

.23 

.42 

.71 

1.1 

1.6 

2.3 

3.2 

4.3 

5.6 

7.2 

9.1 

9 

x 


.10 

.21 

.37 

.64 

1.0 

1.5 

2.1 

2.9 

3.9 

5.1 

6.6 

8 4 

X 

10 


.09 

.19 

.34 

.58 

.93 

1.4 

2.0 

2.7 

3.6 

4.7 

6.1 

7.8 

10 

X 



.17 

.31 


.86 

1.3 

1.8 

2.5 

3.4 

4 4 

5 7 

7.2 

XL 

11 



.16 

.28 

.48 

.79 

1.2 

1 1.7 

2.3 

3.1 

4.1 

5.3 

6 7 

11 

A 



.14 

.26 

.44 

.72 

1.1 

1.5 

2.1 

2.9 

3.8 

4.9 

6.2 


12 



.13 

.24 

.41 


1.0 

1.4 

2.0 

2.7 

3.4 

4.5 

5.8 

12 

13 




.21 



.84 

1.2 

1.7 

2.3 

3 1 

4.0 

5.0 

13 

14 




.18 

.31 

.46 

.70 

1.0 

1.4 

2.0 

2.7 

3.5 

4.4 

14 

15 





.27 

.41 

.63 

.91 

1.2 

1.7 

2.4 

3.1 

3.9 

15 

]« 





.24 

.37 

.57 

.78 

1.1 

1.5 

2.1 

2.7 

3.5 

16 

17 






.50 

.70 

1.0 

1.4 

1.9 

2.4 

3.1 

17 

18 







.45 

.66 

.92 

1.3 

1.7 

2.2 

2.8 

18 

20 








.76 

1.0 

1.4 

1.8 

2.3 

20 
















Height 
in feet. 


Side of square pine pillar, in inches 



Height 
in feet. 

*X 1 

*X I 

*X 1 

5 1 

ax 1 

ax 

ax 

6 

6X 

1 ex 

1 e% 

1 7 

IX 






BREAKING 

LOAD. 







Tons. 

Tons. 

Tons. 

Tons 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 


2 

35.8 

40.6 

45.7 

51.1 

56.8 

62.8 

69.0 

75.5 

82.3 

89.4 

96.8 

104.5 

112.4 

2 

3 

31.4 

36.1 

41.1 

46.2 

51.8 

57.7 

63.8 

70.2 

76.9 

84.0 

91.3 

98.9 

106.7 

3 

4 

26 8 

31.2 

35.9 

40.8 

46.1 

51.7 

57.6 

63.8 

70.4 

77.3 

84.5 

92.1 

99.9 

4 

5 

22.6 

26.5 

30.8 

35.4 

40.5 

45.8 

51.5 

57.4 

63.7 

70.3 

77.2 

84.5 

92.1 

5 

6 

18.9 

22.5 

26.4 

30.5 

35.2 

40.1 

45.4 

51.0 

57.0 

63.3 

69.9 

76.8 

84.1 

6 

7 

15.8 

19.0 

22.6 

26.2 

30.5 

35.0 

39.9 

45.0 

50.5 

56.5 

62.8 

69.4 

76.3 

7 

8 

13.3 

16.1 

19.2 

22.5 

26.3 

30.4 

34.9 

39.7 

45.9 

50.4 

56.2 

62.4 

69.0 

8 

9 

11.3 

13.7 

16-5 

19.5 

22.9 

26.6 

30.7 

35.0 

39.9 

44.8 

50.2 

56.0 

62.1 

9 

10 

9.7 

11.8 

14.2 

16.9 

19.9 

23.2 

26.9 

30.9 

35.2 

39.9 

44.9 

50.3 

56.0 

10 

11 

8.3 

10.2 

12.4 

14.8 

17.5 

20.4 

23.8 

27.4 

31.3 

35.6 

40.2 

45.1 

50.4 

11 

12 

7.2 

8.8 

10.7 

12.9 

15.4 

18.0 

21.1 

24.3 

27.9 

31.8 

36.0 

40.6 

45.5 

12 

13 

6.2 

7.7 

9.4 

11.4 

13.6 

16.0 

18.8 

21.7 

24.9 

28.5 

32.4 

36.6 

41.1 

13 

14 

5.5 

6 8 

8.3 

10.1 

12.1 

14.2 

16.7 

19.4 

22.4 

25.7 

29.2 

33.1 

37.3 

14 

15 

4.8 

6.0 

7.4 

9.0 

10.8 

12.7 

15.0 

17.5 

20.2 

23.2 

26.4 

30.0 

33.9 

15 

16 

4.4 

5.4 

6.7 

8.1 

9?8 

11.5 

13.6 

15.8 

18.3 

21.0 

24.0 

27.3 

30.8 

16 

17 

4.0 

4.9 

6.1 

7.3 

8.8 

10.4 

12.3 

14.3 

16.6 

19.1 

21.9 

24.9 

28.3 

17 

18 

3.6 

4.4 

5.5 

6.6 

8.0 

9.4 

11.2 

13.0 

15.1 

17.4 

19.9 

22.7 

25.8 

18 

19 

3.3 

4.0 

5.0 

6 0 

7.3 

8.6 

10 2 

11.9 

13.8 

16.0 

18.3 

20.9 

23.7 

19 

20 

3.0 

3.7 

4.6 

5.5 

6.6 

7.8 

9.3 

10.9 

12.6 

14.6 

16.8 

19.2 

21.8 

20 

22 

2.5 

3.0 

3.8 

4.6 

5.6 

6.6 

7.9 

9.2 

10.7 

12.4 

14.3 

16.3 

18 6 

22 

24 

2.1 

2 6 

3.2 

3.9 

4.7 

5.6 

6.7 

7.9 

9.1 

10.6 

12.2 

14.1 

16.0 

24 

26 

1.8 

2.2 

2.8 

3.4 

4.1 

4.9 

5.8 

6.8 

7.9 

9.2 

10.6 

12.2 

13.9 

26 

28 

1.5 

1.9 

2.4 

2.9 

3.5 

4.2 

5.1 

59 

6.9 

8.0 

9.3 

10.7 

12.2 

28 

30 

1 3 

1.7 

2.1 

2.6 

3.1 

3.7 

4.4 

5.2 

6.1 

7.1 

8.2 

9.4 

10.8 

30 

32 

1.2 

1.5 

1.9 

2.3 

2.7 

3.2 

3.9 

4.6 

5.4 

6.3 

7.3 

8.4 

9.6 

32 

34 

1.1 

1.3 

1.7 

2.0 

2.4 

2.9 

3.5 

4.1 

4.8 

5.6 

6.5 

7.5 

8.6 

34 

36 

1.0 

1.2 

1.5 

1.8 

2.2 

2.6 

3.1 

3.7 

4 3 

5 0 

5.8 

6.7 

7.7 

36 

38 

.9 

1.1 

1.3 

1.6 

2.0 

2 4 

2.8 

3.3 

3.9 

4.5 

5.3 

6.1 

7.0 

38 

40 

.8 

1.0 

1.2 

1.5 

1 8 

2.1 

2.6 

3.0 

3.5 

4.1 

4.8 

55 

6.3 

40 


Continued <>u next |i:tgo. 




























































460 


STRENGTH OF WOODEN PILLARS 


Table of breaking: loads in tons of square pine pillars, with 
flat ends firmly fixed, and equally loaded. (Continued.) 


Original. 


[ Height 
| in feet. 


Side of square pine pillar, in inches 

• 


Height 

in feet. 

7 * 

1 1 % 

1 8 

8 H 

£ 

oo 

1 8* 

I 9 1 

0 % 

9M 

9H 

1 io 

1 10* | 10* 






BREAKING 

LOAD. 







Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 


2 

120.6 

129.1 

137.9 

147.0 

156.3 

165.9 

175.8 

186-0 

196.4 

207.2 

218.2 

229.5 

241.0 

2 

4 

107.9 

116-3 

125.0 

133.9 

143.0 

152.6 

162.4 

172.5 

182.8 

193.4 

204.2 

215.3 

226.6 

4 

6 

91.7 

99.7 

108.0 

116.5 

125.3 

134.5 

143.5 

153.0 

162.8 

173.5 

184.4 

195.6 

207.0 

6 

8 

75.9 

83.2 

90.8 

98.7 

106.8 

115.5 

124.4 

133.6 

143.0 

153.0 

163.2 

173.7 

184.4 

8 

10 

62.0 

68.5 

75.3 

82.4 

89.7 

97.7 

105.8 

114.3 

123.0 

132.3 

141.8 

151.6 

161.6 

10 

12 

50.7 

56.5 

62.5 

68.7 

75.1 

82.2 

89.6 

97.2 

105.0 

113.5 

122.2 

131.2 

140.5 

12 

14 

41.8 

46.7 

51 9 

57.3 

62.9 

69.2 

75.7 

82.4 

89.4 

97.1 

105.0 

113.2 

121.6 

14 

16 

34.7 

38.9 

43.4 

48.1 

53.0 

58.5 

64.3 

70.3 

76.5 

83.3 

90.4 

97.7 

105.3 

16 

18 

29.1 

32.7 

36.6 

40.7 

45.0 

• 49.8 

51.9 

60.1 

65.6 

71.7 

78.0 

84.6 

91.4 

18 

20 

24.6 

27.7 

31.1 

34.7 

38.5 

42.7 

47.2 

51.8 

56.7 

62.0 

67.6 

73.5 

79.6 

20 

23 

19.6 

22.1 

24.8 

27.7 

30.9 

34.4 

38.0 

41.9 

46.0 

50.5 

55.2 

60.2 

65.4 

23 

26 

15.8 

17.8 

20.1 

22.5 

25.2 

28.1 

31.1 

34.4 

37 9 

41.6 

45.6 

49.8 

54.3 

26 

29 

13.1 

14.8 

16.7 

18.7 

20.9 

23.4 

25.9 

28.6 

31.6 

34.9 

38.2 

41.9 

45.6 

29 

32 

10.9 

12.3 

13.9 

15.7 

17.6 

19.7 

21.8 

24.2 

26.7 

29.5 

32.3 

35.5 

38.7 

32 

35 

9 3 

10.6 

11.9 

13.4 

15.0 

16.8 

18.7 

20.7 

22.8 

25.2 

27.7 

30.5 

33.3 

35 

38 

8.0 

9.1 

10.2 

11.5 

12.9 

14.6 

16.3 

18.0 

19.7 

21.8 

23.9 

26.3 

28 8 

38 

41 

6.9 

7.9 

8.9 

10.0 

11.2 

12.6 

14.1 

15.6 

17.2 

19.1 

21.0 

23.1 

25.2 

41 

44 

6.0 

6.9 

7.8 

8.8 

9.8 

11.0 

12.3 

13.6 

15.0 

16.7 

18.4 

20.2 

22.1 

44 

50 

4.7 

5.4 

6.1 

6.9 

7.7 

8.8 

10.0 

11.3 

12.6 1 

13.7 

14.9 1 

16.2 

17.5 

50 


3- 

to % 

Side of square pine pillar, in inches. 

£ 

bd Tj 

M.2 

10% 

11 | 11* | 

HJf 1 

11% 

12 | 

1 io% 

11 

11* 

11% I U% 

1 12 

£.2 



BREAKING LOAD. 



BREAKING LOAD. 




Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 



Tons. 

Tons. 

Tous. 

Tons. 

Tons. 

Tons. 


4 

238.9 

251.0 

263.2 

275.9 

288.8 

802.1 



26.5 

28.6 

31.1 

33.8 

36.7 

39 9 

42 

6 

2 1 8.8 

230.6 

242.7 

255.2 

267.9 

281.0 



24.2 

26.4 

28.7 

31.2 

33 9 

3« 8 

44 

8 

195.5 

207.0 

218.5 

230.4 

242.5 

255.1 



22.4 

24.3 

26.4 

28.7 

31.2 

33 9 

46 

10 

172.2 

183.0 

194.1 

205.6 

217.3 

229.5 



20.7 

22.6 

24.5 

26.7 

28.9 

3| 4 

48 

12 

150.3 

160.3 

170.7 

181.5 

192.5 

204.0 



19.2 

20.9 

22.7 

24.7 

26.8 

29 2 

50 

14 

130.5 

139.8 

149 4 

159.4 

169.6 

180.2 



17.8 

19.5 

21.1 

23.0 

25.0 

27 2 

52 

16 

113.2 

121.7 

130 4 

139.6 

149.0 

158.8 



16.6 

18.2 

19.7 

21.5 

23.3 

25 4 

54 

18 

98.7 

106.2 

114.1 

122.5 

131.0 

140.0 



15.5 

17.0 

18 4 

20.1 

21.8 

23 7 

56 

20 

86.2 

92.9 

100.0 

107.6 

115.4 

123.6 



14.5 

15.9 

17.2 

18.8 

20.4 

22 ? 

58 

22 

‘75.6 

81.7 

88.1 

95.0 

102.0 

109.5 



13.6 

14.9 

16.2 

17.7 

19.2 

20 9 

60 

24 

66.7 

72.2 

77.9 

84.1 

90.5 

97.3 



11.8 

12.9 

13.9 

15.2 


18 0 


26 

59.1 

64.0 

69.2 

74.9 

80 6 

86.8 



10.1 

11.1 

12.0 

13.2 

14.3 

15 6 

70 

28 

52.6 

57.1 

61.8 

66.9 

72.1 

77.7 



8.9 

9.8 

10.6 

11.6 

12.5 

13 7 

75 

30 

47.0 

51.1 

55.3 

60.0 

64.7 

69.9 



7 8 

8.6 

9.4 

10.2 

11.1 

12 1 


32 

42.1 

46.0 

49.9 

54.0 

58.4 

63.0 



7.0 

7.7 

8.4 

9.1 

9 9 

107 

ft*; 

34 

38.2 

41.5 

45.0 

48.8 

52.8 

57.1 



6.2 

6.8 

7.4 

8.1 

8 8 

9 6 


36 

34.6 

37.7 

40.9 

44.3 

48.0 

51.8 



5.6 

6.1 

6.7 

7.3 

8 0 

8 7 


38 

31.5 

34.2 

37.2 

40.4 

43.9 

47.4 



5.1 

5.6 

6.1 

6.6 

7 2 

7 ft 


40 

28.8 

31.3 

34.0 

37.0 

40.1 

43.5 



3.5 

3.9 

4.2 

4.6 

5^0 

5.5 

120 


Continued on next page. 


Roinarks.JUi- Kirkaldy found for Ri^a and Dantzic firs, 

20 1 1 and , ,^J ns square, (or 18^ sides high,) 148 and 138 tons total; or .87G 
ap ( l -817 tons, (19 60 and 1829 lbs,) per sq inch. Mr Smith’s rule gives for common 
pine, 160 tons total; or .947 ton, or 2121 lbs, per sq inch. Hodgkinson would give 
for Riga about 297 tons total. s 

. Each of Mr Kirkaldy’s 20-ft pillars shortened about .6 of an inch total - or 03 
inch per ft; or Uof an inch in 4 ft 2 ins, under a mean of 1900 tbs per sq inch' 
The writer has known 8 unbraced pillars of hemlock, tolerably seasoned, 
nC. ins square, and 42 ft high, to be gradually loaded each with 32 tons, or 71680 
lbs total; (or .2 - __2 ton, or 498 lbs per sq inch) without appreciable yielding. As¬ 
suming their strength and stillness to be about as for Mr Smith’s pine, (as in all 

hn[ wS S, K e l ?h0 " ? h / him T W f 39 ‘ 9 tons total - With these same data, 
but with Hodgkinson s formula, they should yield at 69.3 tons; and vrith 
Hodgkinson s own data, for seasoned red deal, at 91.6 tons. See Remarks, p 462 . 




































































STRENGTH OF WOODEN PILLARS 


461 


Table of breaking loads in tons of square pillars of half- 
seasoned white or common yellow pine, with flat ends 
firmly fixed, and equally loaded. By C. Shaler Smith’s formula. 
(Continued.) 


As this table was partly made by interpolation, the last figure is not always pre¬ 
cisely correct. 


Original. 


+-> • 

r T m <D 

2 ^ 

Side of square pine pillar in inches. 

jC ** 

£ 

E5.2 

13 

14 

1 15 

1 16 

1 11 

1 18 

19 

20 | 

21 

22 

23 

24 

K.S 






BREAKING LOAD. 







Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 


4 

358 

418 

482 

552 

625 

703 

786 

872 

964 

1060 

1161 

1265 

4 

6 

335 

394 

456 

526 

599 

676 

760 

847 

938 

1033 

1134 

1236 

6 

8 

308 

367 

429 

500 

572 

649 

732 

818 

910 

1005 

1106 

1208 

8 

10 

281 

339 

400 

466 

537 

612 

694 

780 

870 

964 

1064 

1166 

10 

12 

252 

307 

365 

432 

502 

576 

656 

740 

829 

922 

1022 

1124 

12 

14 

225 

277 

333 

397 

464 

536 

614 

696 

784 

876 

973 

1074 

14 

16 

201 

250 

303 

363 

428 

497 

573 

652 

739 

829 

925 

1024 

16 

18 

179 

224 

274 

331 

392 

458 

531 

608 

692 

780 

873 

972 

18 

20 

160 

201 

248 

301 

359 

422 

492 

566 

647 

732 

822 

919 

20 

22 

143 

182 

224 

274 

329 

388 

455 

526 

604 

686 

773 

866 

22 

24 

127 

163 

203 

249 

301 

357 

421 

488 

563 

642 

726 

816 

24 

26 

115 

148 

184 

226 

275 

328 

389 

453 

523 

599 

680 

767 

26 

28 

103 

133 

167 

206 

252 

302 

359 

420 

490 

560 

638 

721 

28 

30 

93 

121 

152 

189 

231 

278 

332 

389 

453 

522 

597 

677 

30 

32 

84 

109 

138 

173 

212 

256 

307 

361 

421 

487 

558 

635 

32 

34 

76 

99 

126 

159 

196 

237 

284 

335 

392 

455 

523 

597 

34 

36 

69 

91 

116 

146 

180 

219 

264 

312 

366 

426 

490 

560 

36 

38 

63 

84 

107 

134 

166 

203 

245 

290 

341 

397 

458 

525 

38 

40 

58 

77 

99 

124 

154 

188 

227 

270 

318 

372 

429 

494 

40 

42 

54 

71 

91 

115 

143 

175 

212 

253 

298 

349 

403 

465 

42 

44 

50 

66 

84 

107 

133 

163 

198 

236 

280 

328 

380 

438 

44 

46 

46 

61 

78 

99 

123 

152 

185 

221 

263 

308 

358 

413 

46 

48 

43 

57 

73 

92 

115 

142 

173 

207 

247 

290 

337 

389 

48 

50 

40 

53 

68 

86 

107 

133 

162 

194 

231 

272 

317 

367 

50 

52 

37 

50 

64 

81 

101 

124 

152 

182 

217 

256 

300 

347 

52 

54 

35 

47 

60 

76 

95 

117 

144 

172 

205 

242 

283 

328 

54 

56 

33 

44 

56 

71 

89 

110 

135 

162 

193 

228 

267 

310 

56 

58 

31 

41 

52 

67 

84 

103 

127 

153 

182 

215 

253 

294 

58 

60 

29 

38 

49 

63 

79 

98 

120 

144 

172 

204 

240 

280 

60 

65 

25 

33 

43 

55 

69 

86 

105 

126 

151 

179 

211 

246 

65 

70 

22 

29 

37 

48 

60 

74 

92 

111 

134 

159 

187 

218 

70 

75 

19 

25 

33 

42 

53 

66 

82 

98 

118 

141 

166 

195 

75 

80 

16 

22 

29 

37 

46 

58 

72 

87 

105 

125 

148 

174 

80 

85 

14 

19 

26 

33 

41 

52 

65 

78 

94 

112 

132 

156 

85 

90 

13 

17 

23 

30 

37 

46 

58 

70 

85 

102 

120 

141 

90 

95 

12 

16 

21 

27 

33 

42 

53 

64 

77 

93 

108 

127 

95 

100 

11 

14 

19 

24 

30 

38 

48 

58 

70 

84 

99 

117 

100 

110 

10 

12 

16 

20 

26 

33 

40 

48 

58 

70 

82 

97 

110 

120 

9 

11 

14 

17 

22 

28 

34 

41 

49 

60 

71 

83 

120 

130 

7 

9 

12 

14 

18 

23 

29 

36 

43 

52 

61 

72 

130 

140 

6 

8 

10 

12 

16 

20 

25 

31 

37 

44 

53 

62 

140 

150 

5 

7 

9 

11 

14 

18 

22 

27 

32 

38 

46 

54 

150 

160 

5 

6 

8 

10 

13 

16 

20 

24 

29 

34 

41 

48 

160 

170 

4 

5 

7 

9 

11 

14 

17 

21 

25 

30 

36 

43 

170 

180 

4 

5 

6 

8 

10 

12 

15 

19 

22 

27 

32 

38 

180 

190 

3 

4 

5 

7 

9 

11 

14 

17 

20 

24 

29 

34 

190 

200 

3 

4 

5 

6 

8 

10 

12 

15 

18 

22 

26 

31 

200 

































462 


STRENGTH OF WOODEN PILLARS 


Table of breaking loads in tons, or in lbs, per sq inch of 
cross section of half seasoned square pine pillars, whose 
heights are measured by one of their sides. 


Height 
in sides. 

BR LOAD PI 

JR 8Q IN. 

Height | 

in sides. 

BR LOAD PI 

ER SQ IN. 

Height | 

in sides.j 

BR LD PER 

SQ IN. 


Height 

in sides. 

BR LD PER 

SQ IN. 

1 

Tons. 

2.2232 

Lbs. 

4980 

26 

Tons. 

.6027 

Lbs. 

1350 

51 

Tons. 

.1960 

Lbs. 

439 


76 

Tons. 

.0924 

Lbs. 

207 

2 

2.1969 

4921 

27 

.5697 

1276 

52 

.1888 

423 


77 

.0902 

202 

* 3 

2.1544 

4826 

28 

.5398 

1209 

53 

.1826 

409 


78 

.0879 

197 

4 

2.097 S 

4699 

29 

.5116 

1146 

54 

.1763 

395 


79 

.0862 

193 

5 

2.0290 

4545 

30 

.4853 

1087 

55 

.1705 

382 


80 

.0839 

188 

6 

1.9513 

4371 

31 

.4607 

1032 

56 

.1647 

369 


81 

.0821 

184 

7 

1.8665 

4181 

32 

.4379 

981 

57 

.1598 

358 


82 

.0799 

179 

8 

1.7772 

3981 

33 

.4165 

933 

58 

.1545 

346 


83 

.0781 

175 

9 

1.6857 

3776 

34 

.3969 

889 

59 

.1496 

335 


84 

.0763 

171 

10 

1.5942 

3571 

35 

.3781 

847 

60 

.1451 

325 


85 

.0746 

167 

11 

1.5040 

3369 

36 

.3599 

809 

61 

.1406 

315 


86 

.0728 

163 

12 

1.4165 

3173 

37 

.3447 

772 

62 

.1362 

305 


87 

.0714 

160 

13 

1.3317 

2983 

3S 

.3295 

738 

63 

.1321 

296 


88 

.0696 

156 

14 

1.2513 

2803 

39 

.3152 

706 

64 

.1286 

288 


89 

.0683 

153 

15 

1.1745 

2681 

40 

.3018 

676 

65 

.1250 

280 


90 

.0670 

150 

16 

1.1027 

2470 

41 

.2889 

647 

66 

.1210 

271 


91 

.0656 

147 

17 

1.0353 

2319 

42 

.2772 

621 

67 

.1179 

264 


92 

.0638 

143 

18 

.9723 

2178 

43 

.2661 

596 

68 

.1147 

257 


93 

.0625 

140 

19 

.9134 

20 16 

44 

.2554 

572 

69 

.1112 

249 


94 

.0616 

138 

20 

.8585 

1923 

45 

.2455 

550 

70 

.1085 

243 


95 

.0603 

135 

21 

.8076 

1809 

46 

.2357 

528 

71 

.1054 

236 


96 

.0589 

132 

22 

.7603 

1703 

47 

.2268 

508 

72 

.1027 

230 


97 

.0576 

129 

23 

.7165 

1605 

48 

.2183 

489 

73 

.1000 

224 


98 

.0567 

127 

24 

.6755 

1513 

49 

.2100 

472 

74 

.0973 

218 


99 

.0554 

124 

25 

.6380 

1429 

50 

.2031 

455 

76 

.0951 

213 


100 

.0545 

122 


Remark.—Gordon and llodgkinson compared. The difference 

between them is greater in wooden pillars than in hollow cast iron ones. More¬ 
over, in the latter, Gordon is sometimes greater, sometimes less, than Hodgkin- 
son, as seen per lower table. Mr. Smith’s assumed strength and stiffness of pine 
may safely be taken at about one-fourth less than Hodgkinson’s for his red deal; 
and with this assumption , Uodgkinson’s rule would make the strength of Smith’s 

K ine (in all our tables for wooden pillars) greater, to the extent shown 
y the multipliers in the following table. The truth is probably between 
the two. See Remark, p. 460. 


Ht in 
sides. 

Multr. J 

Ht in 
sides. 

Multr. 


Ht in 
sides. 

Multr. 


Ht in 
sides. 

Multr. 

Ht in 
sides. 

Multr. 

5 

1.04 

12.5 

1.23 


20 

1.44 


35 

1.72 | 

50 

1.67 

7.5 

1.09 

15 

1.30 


25 

1.54 


40 

1.76 

60 

1.63 

10 

1.15 1 

17.5 

1.37 


30 

1.64 


45 

1.71 1 

80 

1.59 


Gordon’s and llodgkinson’s hollow cylindrical cast iron 

pillars compared. 

The thickness is usually from ^ to Vg of the outer diameter; and for these 
limits, the column G, (Gordon), and H, (Hodgkinson), show the proportions of 
the breaking loads. 

Thickness = ^of the outer diameter. 


Ht in 
Diams. 

G . 

H. 

Ht in 
Diams. 

G . 

H. 

Ht in 
Diams. 

G . 

H. 


Ht in 
Diams. 

G . 

H. 

Ht in 
Diams. 

G . 

H. 

5 

1 

1.25 

10 

1 

1.11 

20 

1 

.97 


40 

1 

.90 

70 

1 

.90 

8 

1 

1.12 

15 

1 

1.02 

30 

1 

.94 


50 

1 

.88 

100 

1 

.97 



Thickness = 

% of the outer diameter. 



5 

1 

1.23 j 

10 

1 

1.08 1 

20 

1 

.92 


40 

1 

.81 

70 

' i 

| .82 

8 

1 

1.10 | 

15 

1 

.98 I 

30 

1 

.88 


50 

1 

.80 

100 

! i 

1 .88 

























































































STRENGTH OF MATERIALS 


463 


Art. 4. Ultimate average tensile or cohesive strength of 

Timber, 

Being tlie least weights in pounds which, if attached to the lower end of a vert rod 
one inc h square, firmly upheld at its upper end, would break it by tearing it apart. 
For large timbers we recommend to reduce these constants to part. 


The strengths in all these tables may T , 
readily be one-third part more or less per 
than our averages. 8q. inch. 


Aider .. 

Ash, English.. 

“ American (author) abt 

Birch.. 

** Amer’n black. 

Bay-tree... 

Beech, English. 

Bamboo . 

Box . 

Cedar, Bermuda. 

“ Guadaloupe . 

Chestnut. 

horse. 

Cyprus. 

Elder. 

Elm.. 


14000 

1G000 

16500 

15000 

7000 

12000 

11500 

6000 

20000 

7600 

9500 

13000 

10000 

6000 

10000 

6000 


“ Canada. 

Fir, or Spruce. 

Hawthorn. 

Hazel. 

Holly .. 

Hornbeam. 

Hickory, Amer’n. 

Lignum Vitae, Amer’n. 

Lancewood. 

Larch, Scotch. 

Locust. 

Maple. 


13000 

10000 

10000 

18000 

16000 

20000 

11000 

11000 

23000 

7000 

18000 

10000 


Lbs per 
sq. inch. 


Mahogany, Honduras. 

“ Spanish.... 

Mangrove, white, Bermuda.... 

Mulberry. 

Oak, Amer’n white. 

“ “ basket. 

“ “ red. 

“ Dantzic, seasoned . - 

“ Riga . 

“ English. 

“ live, Amer’n. 

Pear. 

Pine, Amer’n, white, red, ( 
and Pitch, Memel, Riga... J 

Plane. 

Plum. 

Poplar. 

Quince. 

Spruce, or Fir. 

Sycamore. 

Teak. 

Walnut. 

Yew. 


8000 

16000 

10000 

12000 


10000 


10000 

10000 

11000 

11000 

7000 

7000 

10000 

12000 

15000 

8000 

8000 


Across the grain. Oak.. 

“ “ “ Poplar. 

« “ “ Larch,900 to 

“ “ “ Fir, & Pines 


2300 

1800 

1700 

550 


These are averages. The strengths vary much with the age of the tree; the 
locality of its growth; whether the piece is from the center, or from the outer por¬ 
tions of the tree; the degree of seasoning; straightness of grain; knots, &c, &c. Also, 
inasmuch as the constants are deduced from experiments with good specimens of 
small size, whereas large beams are almost invariably more or less defective from 
knots, crookedness of fibre, &c, it is advisable in practice to reduce these constants 
as recommended above. 




































































464 


STRENGTH OF MATERIALS. 


Art. 5. Average ultimate tensile or cohesive strength of 

Metals, per square inch.* 


The ultimate tensile or pulling load per square inch of any 
material is frequently called its constant, coefficient, or modulus of 
tension, or of tensile strength. 


Antimony, cast. 

Bismuth, cast. 

Brass, cast 8 to 13 tons, say 18000 to 29000 ft> . . 

“ wire, unannealed or hard, 80000. Annealed. 

Bronze, phosphor wire, hard, 150000. Annealed. 

Copper, cast 18000 to 30000. 

“ bolts, 28000 "to *38000*.".*..’.. 

“ wire (annealed 16 tons); uuannealed. 

Gold, cast. 

“ wire, 25000 to 30000. 

Gun metal of copper and tin, 23000 to 55000. 

“ “ cast iron, U. S. ordnance, 36000 to 40000. 

Iron, cast, English.13400 to 22400.. 

“ “ ordinary pig..13000 to 16000. 

American cast iron averages one-fourth more than the above. 
Average cast iron, when sound, stretches about .00018; or 1 part 
in 5555 of its length ; or % inch in 57.9 ft. for every ton of ten¬ 
sile strain per sq inch, up to its elastic limit, which is at about 
% its break-strain. The extent of stretching, however, varies 
much with the quality of the iron ; as in wrought-iron. 

Cast, malleable, annealed 18 to 25 tons. 

Iron, wrought, rolled bars, 40000 to 75000, the last exceptional... 

“ “ “ “ ordinary average. . 

“ “ “ “ good “ . 

“ “ “ “ superior... 

“ “ “ “ best American, (exceptional). 

“ “ “ “ Low Moor, English, average. 

“ “ “ plates for boilers, &c, 40000 to 60000. 

“ “ Englisli rivet iron 55000 to 60000 


wire, annealed.30000 to 60000. 


unannealed, or hard...50000 to 100000. 

“ “ “ ropes, per sq inch of section of rope .. 

“ “ large forgings, 30000 to 40000. 

In important practice, good bar iron should not be trusted per¬ 
manently with more than about 5 tons per sq inch ; which will 
stretch it about Y inch ’ n from 20 to 25 ft. 

Good bar iron stretches about 1 part in 12000 of its length ; or 
about 1 inch in 1000 ft; or Y inch in 125 ft, for every ton of 
tensile strain per sq inch of section, up to its elastic limit. 
This limit usually ranges between 8 and 13 tons per sq inch, or 
about half the breaking strain, according to quality. The 
ultimate stretching of rolled bars is from 5 to 30 per ct of the 
original length; usually 15 to 20 per cent. Plates and angle 
iron 3 to 17 per cent. Heating, even up to 500° Fah, does not 
weaken bar iron or steel. For stretch by heat see p 212. 

Lead, cast, 1700 to 2400.by author... 

“ wire, 1200 to 1600. Pipe 1600 to 1700.“ 

Platinum wire, annealed, 32000. Unannealed. 

Steel, plates, range, 60000 to 103000. 

“ “ of Hussey, Wells & Co, Pittsburg, Pa, 91500 to 97400 

“ “ Bessemer. 

“ Bessemer tool. 

“ wire, annealed 30 to 50 tons. Unan, 50 to 90 tons. 


Pounds 

per 

sq.inch. 

Tons 
per 
sq. in. 

1000 

.45 

3200 

1.4 

23500 

10.5 

49000 

22 

63000 

28.1 

24000 

10.7 

30000 

13.4 

33000 

14.7 

60000 

26.8 

20000 

8.9 

27500 

12.3 

39000 

17.4 

38000 

17 

17900 

8 

14500 

6.47 

48160 

21.5 

57500 

25.7 

44800 

20 

50400 

22.5 

60000 

26.8 

76100 

34 

60000 

26.8 

50000 

22.3 

57500 

25.7 

45000 

20.1 

75000 

33.5 

38000 

16.8 

35000 

15.6 

2050 

.92 

1650 

.74 

56000 

25 

81500 

36.4 

94450 

42.2 

98600 

44 

112000 

50 

156800 

70 


*I,arg;e bars of metal bear less per sq inch than small ones. In cast iron 

ones 1, 2 and 3 ins sq, the strengths per sq inch were about as 1, .85 and .66; and wrought iron prob¬ 
ably averages about the same. See top ot' next page. 

Iron burs re-rolled cold have tensile strength increased 25 to 50 per ct, with no increase of 
density. They are said to lose this strength if reheated. 




















































STRENGTH OF MATERIALS. 


465 


Average ultimate tensile, or cohesive strength of Metals 

per square inch. (Continued.) 


Capt. James B. Eads found that forged steel bolts for the St. Louis bridge, 5% 
ins diaiu, and 22 to 36 ft. long, broke short with only 30000 tbs per sq inch ; while 
bolts of but % inch section, cut from, the large ones, in no instance broke with 
100000 lbs per sq inch, but stretched considerably. 

Steel, cast, Bessemer ingots, average. 

“ best American Bessemer ingots . 

“ “ rolled and hammered, 120000 

to 130000. 

“ homogeneous, Cammell & Co, England, No 1. 

“ “ “ “ “ No 2. 

“ “ “ “ “ No 3. 

“ puddled bars, rolled and hammered, 65000 to 135000.... 

Steel. Experiments by Lieut. W. S. Shock,U. S. N., at Washington, 
on steel from the Black Diamond Steel-Works, Pittsburg, 
Pa. All the pieces were cut from the same bar, three 
pieces for each exp. They were turned down to a diam 
of .62 of an inch at the intended point of fracture, by a 
groove, in shape of a circular segment, with a chord of 
about l inch : 

“ the bar in its original condition, 109500 to 131900. 

“ heated to light cherry-red, then plunged into oil of 8*2° 

Fall, 201300 to 227500 ."T. 

“ heated to light cherry-red, then plunged into water of 79° 
Fah. Then tempered on a heated plate, 152500 to 176100.. 
heated to light cherry-red, then plunged into water of 79° 

Fah,1327U0 to 150500 . 

Tempering in oil usually increases the strength from 40 to 
80 per cent. 

“ chrome, made at Brooklyn, N. Y., and tested at West Point 
Foundry, N.Y., (specific gr 7.816 to 7.956,) 163000 to 199000. 

Average of 12 specimens . 

“ made from very pure Swedish iron, but containing differ¬ 
ent proportions of carbon. The bars were 21% ins long, 
with 14 ins of this length turned down to a uniform di¬ 
am of 1 inch. The breaking wts, however, in the table, 
are per sq inch : 

.33 per ct, stretched 1.37 ins. 


Mark No. 
“ No. 


carbon 

U 


No 
No. 
No. 
No. 
No. 10 
No. 12 
No. 15 
No. 20 


43 

.48 

.53 

.58 

.63 

.74 

.84 

1.00 

1.25 


137 
1.25 
1.12 
0.81 
1.00 
0.69 
1.12 
1.00 
0 62 


With more than about 1.5 per ct of carbon the tensile strength of 
steel diminishes. A bar of the above No. 15, which broke at 
60 tons per sq inch, when turned down for 14 ins of its length ; 
broke with 79% tons per sq inch when turned down at one 
point only. This is owing to the fact that the last could not 
stretch as much as the first, and therefore its diam could 
not be diminished as much before breaking. All its fibres 
pulled more unitedly. It will be observed that the steel of 
greatest strength stretched the least before breaking. This 
stronger steel would break under a suddenly applied force, or 
impulse, more easily than a weaker one would ; because the 
weaker one, by its stretching, gradually breaks the force of the 
impulse, on the same principle as a spring. Hence the steel, 
iron, &c, which is strongest against a gradually applied force 
or strain, may be unfit for uses where the strain comes upon it 
suddenly. The average ultimate tensile strength of steel is about 
twice that of wrought iron. Its deflection as a beam within the elastic 
limits is about 4 that of wrought, or % that of cast iron. Its average stretch 

31 


Pounds 

per 

sq. inch. 

Tons, 
per 
sq. in. 

63000 

281 

86600 

38.6 

125000 

55.8 

58240 

26 

71680 

32 

76160 

34 

100000 

44.6 


120700 
214400 
161300 
141600 


180000 


68100 

76160 

84000 

95200 

92960 

100800 

101920 

123200 

134400 

154560 


53.9 


95.7 

73.3 


63.2 


80 


30.4 
34 

37.5 

42.5 

41.5 
45 

45.5 
55 
60 
69 



































466 


STRENGTH OF MATERIALS, 


ia about .1 inch In 111 ft for every ton per sq inch of load, up to its elastic limit, which generally 
ranges at between and % of its breaking strength ; the latter being for the harder, stronger, and 
less stretchy kinds. A uniform bar of rolled steel, gradually loaded, will stretch from -yP to ^ 
of its length before breaking; or from of an inch to 2.4 ins per foot, according to quality. The 
mean of these is nearly of the length, or 1 j^inch to a foot. When steel, especially if hard, haB 
to be heated to softness in order to give it a required shape, it is thereby weakened. 


Average ultimate tensile or eoliesive strength of Metals 
per square inch. (Continued.) 



Pounds 

per 

sq. inch. 

Tons 
per 
sq. in. 

Silver, cast. 

41000 

18.3 

Tin, English block. 

4600 

2 

“ wire. 

7000 

3350 

3 1 

Zinc, cast...3000 to 3700 ; (the last by author). 

1.5 



Art. 6. Average ultimate tensile or cohesive strength of 

various materials. 


The strengths in all these 

Pounds 

Tons 


Pounds 

Tons 

part more or less than our 

per 

per 


per 

per 

averages. 

sq. inch. 

sq. ft. 


sq. inch. 

sq. ft. 

Brick, 40 to 400 . 

220 

14.1 

Marble,strong,wh Italy.* 

1034 

66.5 

Caen stone, 100 to 200 . 

150 

9.7 

“ Champlain,varie- 



Cement, hydraulic. Port- 



gated * . 

1666 

107.1 

laud, pure, 7 days 



“ Glenn’s F’lls.N.Y. 



in water . 

300 

19.3 

blk,* 750tol034. 

892 

57.4 

“ 6 months old. 

450 

28 9 

“ Montg’y co, Pa, 



“ 1 year old . 

550 

35.4 

PT,TV * 

1175 

75 6 

Common hyd cements 



o 1 ‘ V . 

“ “ white*... 

734 

47.2 

average 1-6 as much. 



“ Lee,Mass,white.* 

875 

56.3 

The last, neat, adhere 



“ Manchester, Vt,* 



to brick and stone with 



550 to 800 . 

675 

43.4 

from 15 to 50 lbs when 



“ Tennessee, varie- 



only 1 month old . 

32 

2 

gated * . 

1034 

66 5 

At end of 1 year, 3 



Oolites, 100 to 200 . 

150 

97 

times as much . 

96 

6 

Plaster of Paris, well set. 

70 

4.5 

See “Cement,” p 675, &c 



Rope, Manilla, best. 

12000 

771 

Glass, 2500 to 9000 (p 432) 

5750 

385.7 

“ hemp, best. 

15000 

965 

Glue holds wood together 



Sandstone, Ohio*. 

105 

6.75 

with from 300 to 800... 

550 

35 

“ Pictou, N. S * 

434 

27.9 

Horn, ox. 

9000 

579 

“ Conn, red.*.... 

590 

37.9 

Ivory.. 

16000 

1029 


2475 

159.1 

Leather belts, 1500 to 



“ Peach bot’m,* 3025 

5000. Good. 

3000 

193 

to 4600 

381 l> 

94R 1 

Mortar, common, 6 mos 



Stone, Ransome’s artif.... 

300 

19.3 

old, 10 to 20. 

15 

.96 

Whalebone. 

7600 

489 


To find the diam in ins of a round rod to bear safely a given pull 

in lbs. 


p,j am / given pull X coef of s afety 

in ins — ult tensile strength III7 

v of material in lbs per sq inch x 

Iron is weakened by extreme cold. 

The belief (originating with Styffof Sweden,) is gaining ground that iron and 
steel are not rendered more brittle by intense cold , but that the great number of 


* By the author’s trials with one of Riehle’s testing machines. Sections 

broken 1% sq inches. 
























































STRENGTH OF MATERIALS. 


467 


breakages of rails, wheels, axles, &c, in winter, is owing to the more seveie blows 
incident to the frozen and unyielding nature of the eartli at that period of the year. 
But Sandberg’s experiments show conclusively that although these metals may per¬ 
haps bear as much steady force, gradually applied, in winter as in summer, yet their 
resistance to impulse ,, or sudden force, is not more than % or X / A as great in severe 
cold: which renders them less flexible and less stretchy. It is probable that this 
fact does not receive as much attention as it should, in proportioning iron bridges, &c. 

Some experiments with good wrought iron showed that even at ’23° Fah, or only 
9° colder than freezing point, there was a loss of strength of from 2% to 4 pei 
cent. 





468 


RIVETS AND RIVETING 


RIVETS AND RIVETING. 


R, Fig 9 3, p 469, shows the usual shapes of rivets as sold.* 

The weight** in the following table of course include the head; but the lengths, as usual, 
are taken “ uuder the head; ” or are those of the shanks only. In practice, discrepancies of 5 or C 


per ct in wt may be expected. 

From Carnegie Bros. & Co’s 


“Useful Information,’’ by C. L. Strobel, C E. 


Length 



Diameters of Rivets In inches. 



of Shank. 
Ins. 

% 

X 

% 

% 

Vs 

1 

IX 

IX 




Weight of 100 Rivets, In pounds. 



V 

3.0 

8.5 







% 

3.8 

9.9 

17.3 






1 

4.6 

11.2 

19.4 

25.6 

38.9 




Va. 

5.4 

12.6 

21.5 

28.7 

43.1 

65.3 

91.5 

123 

<% 

6.2 

13.9 

23.7 

31.8 

47.3 

70.7 

98.4 

133 

% 

6.9 

15.3 

25.8 

34.9 

51.4 

76.2 

105 

142 

2 

7.7 

16.6 

27.9 

37.9 

55.6 

81.6 

112 

150 

y 

8.5 

18.0 

30.0 

41.0 

59.8 

87.1 

119 

159 

8 

9.2 

19.4 

32.2 

44.1 

64.0 

92.5 

126 

167 

74 

10.0 

20.7 

34.3 

47.1 

68.1 

98.0 

133 

176 

3 

10.8 

22.1 

36.4 

50.2 

72.3 

103 

140 

184 

A 

11.5 

23.5 

38.6 

53.3 

76.5 

109 

147 

193 

% 

12.3 

24.8 

40.7 

56.4 

80.7 

114 

154 

201 

% 

13.1 

26.2 

42.8 

59.4 

84.8 

120 

161 

210 

4 

13.8 

27.5 

45.0 

62.5 

89.0 

125 

167 

218 

A 

14.6 

28.9 

47.1 

65.6 

93.2 

131 

174 

227 

y 2 

15.4 

30.3 

49.2 

68.6 

97.4 

136 

181 

236 

% 

16.2 

31.6 

51.4 

71.7 

102 

142 

188 

244 

5 

16.9 

33.0 

53.5 

74.8 

106 

147 

195 

253 


17.7 

34.4 

55.6 

77.8 

110 

153 

202 

261 

g 

18.4 

35.7 

57.7 

80.9 

114 

158 

209 

270 

19.2 

37.1 

59.9 

84.0 

118 

163 

216 

278 

6 

20.0 

38.5 

62.0 

87.0 

122 

169 

223 

287 

y 2 

21.5 

41.2 

66.3 

93.2 

131 

180 

236 

304 

7 

23.0 

43.9 

70.5 

99.3 

139 

191 

250 

321 

y 2 

24.6 

46.6 

74.8 

106 

147 

202 

264 

338 

8 

26.1 

49.4 

79.0 

112 

156 

213 

273 

355 

9 

29.2 

54.8 

87.6 

124 

173 

234 

306 

389 

10 

32.2 

60.3 

96.1 

136 

189 

256 

333 

423 

11 

35.3 

65.7 

105 

148 

206 

278 

361 

457 

12 

38.4 

71.2 

113 

161 

223 

300 

388 

491 


The diam of rivets for bridge work is from y to 1 inch; usually % to 
and for plates more than .5 inch thick, it is about 1.5 times the thickness; 
and for thinner ones about twice; but these proportions are not closely adhered 
to. The common form of rivets as sold is shown at R, Figs 3, a head 
and the shank in one piece; and S shows the same when after being heated 
white hot it is inserted into its hole, and a second head (conical) formed on it by 
rapid hand-riveting as it cools. When longer than about (> ins they 
are cooled near the middle before being inserted, lest their contraction in cooling 
should split off their heads. The hemispherical heads often seen, called snap 
heads, are formed by a machine. The two heads alone require about 
as much iron as 3 diams length of shank. Length of a head = about 1 
diam of shank ; and its width about 2 diams of shank. 


Riveting of Steam and Water Tight Joints. 

Joints for boilers and water-tight cisterns are usually proportioned about 
as per the following table by Fairbairn ; and are made as shown either by Fig 1 
or Fig 2. Fig 1 is called a single-riveted, and Fig 2 a double-riveted’ 
lap-joint. The dist a a , or c c, is the lap. 

Mr Fairbairn considers the strength of the single-riveted lap-joint to be about 
.56; and that of the double-riveted, about .7 that of one of the full unholed 


* Price in Philada, 1886, about 3 y 2 cts per lb. 









































RIVETS AND RIVETING. 


469 


plates, when both joints are proportioned as in his following table. But some 

later experimenters consider about 

^ a - 7 ^, " ' ‘ 

V H- , . 3- f x— 1 — -'— L 


■M7“ 


■xr 


a 



/ c 

o 

o 

° -c 


o 

O : / 

• \ 

■ 

* i 


Fig 1. Fig 2. 

proportions include friction (Art 4), without which they would be about .4 and .5. 


5 and .6 as nearer the correct aver¬ 
age. Experiments on the subject 
are quite conflicting; and it is 
plain that no one set of propor¬ 
tions can precisely suit all the dif¬ 
ferent qualities of plate and rivet 
iron. With fair qualities of both, 
there is every reason to rely upon 
.5 and .6 (or about one-seventh 
part less than Fairbairn’s assump¬ 
tion) as safe for practice. These 


Fairbairn’s table for proportioning' the riveting; for steam 

and water-tight lap-joints. 


Thickness of 

Diameter of 

Length of shank 

From center to 

Lap in single 

Lap in double 

each plate. 

rivets. 

before driving. 

center of rivets. 

riveting. 

riveting. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

3-16 

% 

A 

M 


2 h 

% 

5-16 

A 

k 

$ 

V 4 

m 


VA 

‘ 6 -h 

% 

% 

P 

m 

2 

3% 

12 

13-16 

2 

2 V s 

3% 


15-16 

2 A 

3 

2 : K 

4/b 

7A 

VA 

3/t 

3j4 

5^ 


Riveting of iron girders, bridges. <&c. 


i msm 


1 N 


KC 




e'-K? —x7 


FT 


-ST" 


10 



Figs 3. 



} |o o o 


{ jo o o 

M j 

\ 

) I O 0 o 




— ^^() 


P \, ■ - T7.- P 

° V D V"V—vy \/ 

fc> 



Art. 1. The subject of riveting is abstruse, and involved in 
much uncertainty; and experimental results are very discrepant. We here pro¬ 
pose merely to confine ourselves to what is considered the best joint; and for 
safety we shall omit friction; see Art 4. In girder and bridge work the lap- 
joints above described are seldom used. Instead of them, the plates p , I igs_3, to 
be joined, are butted up square against each other, thus forming a butt-joint, 
i i, Fig D; and are united by either a single covering-plate, cover, 
wrapper, lisli-plsite, or welt e e , Fig K; or the best of all by two of them, 
as at A, ot o o,o o, Fig B. In what follows, the term plate never includes the 
covers. The single cover, like the lap-joint, allows both plates and cover to bend 
under a strong pull, somewhat as at W, thus weakening them materially; whereas 
the double cover o o, o o, Fig B, keeps the pull directly along the axis of the plates, 
thus avoiding this bending tendency. It also brings the rivets into double shear, 
thus doubling their strength. When there is but one cover, it should be at least 
as thick as a plate; and when there are two, experience shows that each had bet¬ 
ter be about two-thirds as thick as a plate, although theory requires each to be 
but half as thick as a plate. 



































































470 


RIVETS AND RIVETING. 


The length w w of covers across the joint is equal to that of the joint. 

Butts require twice as many rivets as laps, because in the lap each 
rivet passes through both the joined plates; and in the butt through only one. 

The rivets ami plate on one side only (right or lett) of the joint- 
line i i of any properly proportioned butt-joint D, represent the full strength 
of the joint, inasmuch as those on one side pul) in one direction, against those on 
the other side, which pull in the opposite direction. Therefore in designing such 
joints we need keep in mind only those on one side, as is done in what follows. 
Thus a single, double, or triple-riveted butt-joint D implies one, two, or three 
rows of rivets on each side of the joint-line i i, and parallel to it. In a prop¬ 
erly proportioned lap the strength is as all the rivets, because one-half of them 
do not pull against the other half, but one end of every rivet pulls in one direc¬ 
tion, and its other end in the opposite direction. 

The net iron, net plate, or net joint, is that which is left between 
the rivet holes, and outside of the two outer ones, all on a straight line drawn 
through the centers of the holes of one row. Its width and area are called the net 
ones of the joint. That between other rows does not increase the strength. 

In Figs 3, N, and K, the rivets are plainly exposed only to single shear; 
that is, the opposing pulls of the two plates tend to shear each rivet across only 
one circular section; whereas in Fig B, with two covers, or in Fig A, each rivet 
is exposed to double shear, one just above and one just below the joined 
plates. 

Art. 2. Bridge-joints are not required to be steam or water¬ 
tight like those of boilers or cisterns; and, therefore, by increasing the breadth 
of the overlap, or the length of the covers, the rivets may be placed in several 
rows behind each other, as the 3 rows of 3 rivets each in M and D, instead of only 
one row of 9 rivets, as in L. By this means, without losing any of the strength of 
the 9 rivets, or of the net iron, we may narrow the width of the plate to an ex¬ 
tent equal to the combined diams (6 in this case) of the holes thus dispensed with 
in the. one row. Moreover, by using more than one row we lessen the weakening 
effect shown at W. This mode of placing the rivets directly behind each other in 
several rows, as at M, and at the left-hand half of Fig D, constitutes Mr Fair- 
bairn’s chain riveting; but the joint will be somewhat stronger if the rivets 
are placed in zigzaging order, as in the right-hand haif of Fig D. 

The dist apart of the rows from cen to cen should not be less 
than 2 diams. It is questionable to what extent this increase in the number of 
rows may be carried without an appreciable loss of strength in the rivets conse¬ 
quent upon the impossibility of quite equalizing the strains on the separate rows. 
But it is probable that if we do not exceed 2 or 3 rows in laps, or the same num¬ 
ber on each side of the joint-line in butts, we may in practice assume that each 
row, and each rivet, is nearly equally strained. 

Rivet-holes are usually of about one-sixteenth inch greater diam than the 
original rivet, so as to allow the hot rivet to be easily inserted. The subsequent 
hammering swells the diam of the rivet until it tills the hole. We may either 
take this increased diam of rivet into consideration, as we have done, in calcula¬ 
ting its shearing and crippling strength, as explained farther on, or with reference 
to increased safety we may omit it. Drilled rivet-holes are said to be better 
than punched ones, as the drilling does not injure the iron around them; but on 
the other hand their sharper edges are said to shear the rivets more readily. 
Hence, such edges are sometimes reamed otf. Both these points are, however, 
disputed; and both modes are in common use. 

The dist from the edge of a hole to the end of a plate or cover should 
not be less than about 1.2 diams, to prevent the rivets from tearing out the end 
of the plate; nor nearer the side edge of a plate than half the clear dist between 
two holes as given hy the Rule in Art 5. The first is rather more than Fairbairn 
directs. 

Rivet holes weaken the net iron left, between them, not only by the 
loss of the part cut out, but either by disturbing the iron around them, or perhaps 
by changing the shape of the net line of fracture, which may not then resist 
tension as well as while it was a continuous straight line. Some deny both cause 
and effect entirely, each party basing its opinion on experiments. But the mass 
of evidence seems to the writer to show that the net iron loses on an average 
about one-seventh of the strength due to the net width. With a view to safety, 
which we consider to be of paramount importance, we shall in what follows 
assume (until the question is definitely settled) that there is such a loss of 
strength in the net iron. 

Riveted joints for resisting compression should depend, not as 
might be supposed upon their butting ends, but upon either the shearing or the 
crippling strength of the rivets; for contraction or bad work may throw the 


RIVETS AND RIVETING. 


471 


pressure on the rivets. Machine riveting- is somewhat stronger than that 
done (as is assumed in our examples) by hand. The thickness of plates 
used in girders, tubular bridges, &c, is usually .25 to .5 inch; with thicker ones 

In’ p ? I)ari "? ly . 1 ' 1 large ones A packing piece, as the shaded piece 

in P is one inserted between two plates to prevent their being bent or drawn 
together by the rivets. 

Art. 3. A riveted joint may yield in three ways after being 
properly proportioned namely, by the shearing of its rivets; or by the pulling 
apart of the net plate between the rivet holes; or by the crippling (a kind of 
compression, mashing, or crumpling) of the plates by the rivets when the two are 
too forcibly pulled against each other. It also compresses the rivets themselves 
transversely at a less strain than the shearing one; and this partial 
yielding ot both plates and rivets allows the joint to stretch, and may thus 
produce injurious unlooked-for strains in other parts of a structure, considerably 
before there is any danger of actual fracture. Or in steam and water joints it may 
cause leaks, without farther inconvenience, or danger. For a long time this 
crippling had entirely escaped notice, and it was supposed that the only important 
point in designing a riveted joint was that the tensile strength of the net plate 
and the shearing strength of the rivets should be equal to each other. 

The crippling strength of a joint is as the number of rivets, in a lap 
or the number on one side of the joint-line in a butt X diam X thickness of joined 
plate. This product gives the crippled area of the joint. \Ve shall here call the 
diam X thickness of plate, the crippling area" of a rivet. If there are 2 or 
more plates (not covers) on top of each other at one joint, their united thickness 
is used for finding the crippling area. The ultimate crippling unit, 
by which the above product is to be multiplied for the actual ultimate crippling 
strength of the joint, may be safely taken at about 60000 lbs, or 26.8 tons, per so 
inch. ^ 

The (liam of a rivet in ins to resist safely a given single-shearing 
force is found thus: Mult the shearing force by the coef of safety, that is by the 
number, 3, 4, or 6, <fcc, denoting the required degree of safety. Call the product g. 
Mult the ultimate shearing strength per sq inch of the rivet-iron, by the decimal 
.7854. Call the product b. Divide g by b. Take the sq rt of the quotient. The 
shearing force and the shearing strength must both be in either lbs or tons. 

Or by a formula, 


Diam in ins = 


-v 


Shearing force X coef of safety 

Ult shearing strength per sq inch X .7854 


If the rivet is to be double-sheared, first mult only half the shearing 
force by the coef of safety. Then proceed as before. 

Or, near enough for practice, mult the diam in single shear by the decimal .7. 
The ultimate shearing unit for average rivet-iron may be taken at 
about 45000 lbs, or 20.1 tons per sq inch of circular sheared section. 


Table of ultimate single shearing strength of rivets. 

(market sizes), in single shear; at 45000 lbs or 20.1 tons per sq inch. 

This table is not to be used when as in our “ Example,” Art 5, the 
i crippling strength of the rivet governs the strength of the joint. 

If the rivet is in double shear it will have twice the strength in the 
, table. 

For the diam in double shear to equal the strength in the table, mult 
I the diam in the table by the decimal .7; near enough for practice ; strictly, .707. 


Diam. 

Ins. 

Diam. 

Ins. 

Bis. 

Tons. 

Diam. 

Ins. 

Diam. 

Ins. 

His. 

Tons. 

Diam. 

Ins. 

Diam. 

Ins. 

Bis. 

Tons. 

Vs 

.125 

552 

.246 


.562 

11183 

4.99 

1 

1.000 

35343 

15.8 

.187 

1242 

.554 

% 

.625 

13806 

6.16 


1.062 

39899 

17.8 

M 

.250 

2209 

.986 

.687 

16705 

7.46 

1 % 

1.125 

44731 

20.0 

.312 

3452 

1.54 

% 

.750 

19880 

8.88 

1.187 

49838 

22.2 

% 

.375 

4970 

2.22 

.812 

23332 

10.4 

VA 

1.250 

55224 

24.6 


.437 

6765 

3.02 

% 

.875 

27060 

12.1 

1.312 

60885 

27.2 

X A 

.500 

8836 

3.94 

.937 

31064 

13.9 


1.375 

66820 

29.8 





































472 


RIVETS AND RIVETING, 


The tensile strength of a properly proportioned Joint is 

equally as either the sectional area of the net plate (not covers) across the cen¬ 
ters of only one row of rivets; or as the shearing or the crippling (as the case 
may be) areas of all the rivets in a lap, or of all the rivets on one side of the 
joint-line in a butt. The tensile strength of fair quality of plate iron, before the 
rivet holes are made, averages about 45000 lbs, or 20.1 tons per sq inch; but we 
shall for safety assume, as stated in Art 2, that the making of the holes reduces 
the strength of the net iron that is left about one-seventh part, or to 38500 lbs, 
or 17.2 tons per sq inch. 

Rem. Even this is considerably too great for laps, or for butts 

with one cover, owing to the weakening of the iron in such by the bending shown 
at W, Figs 3. But we are not speaking of such. 

Art. 4. The friction between the plates in a lap, or between the 

plates and the covers in a butt, produced by their being pressed tightly together 
by the contraction of the rivets in cooling, adds much to the strength of a joint 
while new, perhaps as much as 1.5 to 3 tons per sq inch of circ section of all the 
rivets in a lap. or of all on one side of a single-cover butt; or 3 to 6 tons of all on 
one side of a double-cover butt. In quiet structures, this friction might continue 
to exist, either wholly or in part, for an indefinite period; but in bridges, <fcc, sub¬ 
ject to incessant and violent jarring and tremor, it is probably soon diminished, 
or entirely dissipated. Hence good authorities recommend not to rely on it, and 
it is, therefore, omitted in what follows. 

Art. 5. We now give rules for finding the number of rivets required for a 
double cover butt-joint (the only kind of which we shall treat), and their 
clear or net distance apart. This dist + one diain is the piteli of the rivets, or 
their dist from center to center. The principle of the rule will be explained 
further on, at Art 7, p 474. 

First, select a diam of rivet either equal to or greater than .85 times the 
thickness of the plate. In practice they are generally 1.5 times for plates % inch 
or more thick; and 2 for thinner than ]/^ in. 

Second, mult the greatest total pull in pounds that can come upon the entire 
joint by the coef (3, 4, or 6, &c) of safety, and call the product p. 

Third, multiply the crippling area of the rivet (that is, its diam X the thick¬ 
ness of plate) by 60000. The prod is the ult crippling strength of a rivet. Call it m. 

Fourth, divide p by m. The quotient will be the number of rivets to sustain 
the given pull with the reqd degree of safety. 

Then, the clear distance apart will be 

Number of rows X Diam X 60000 
38500 


Fifth. The clear dist from either end hole of a row to the side edge of the plate, 
should be not less than half the clear dist between two rivets in a row. 

Example. A double-cover butt-joint in .5 inch thick plate is to bear an actual 
pull of 33750 lbs, with a safety of 4; or not to break with less than 33750 X 4 = 
135000 lbs. How many rivets must it have; and how far apart must they be? 

First, Hero .85 times the thickness of the plate is .5 X -85 = .425 inch; there¬ 
fore, our rivets must not be less than .425 inch in diam; but we will take .75 inch 
diam. 

Second, The greatest pull X coef of safety = 33750 X 4 = 135000 lbs = p. 
Third, The crippling area of a rivet X 60000 = .75 X .5 X 60000 = 22500 = m. 
._ p 135000 . . , 

Jrourlh, — = ■ = 6 rivets required on each side of the joint-line. 

m 22500 1 J 

And the clear space or net width between them will be, if the 6 rivets 
are in one row : 


Diam X 60000 45000 

38500 “ 38500 


1.1688 ins. 


1 0188 

And the pitch = net space -f diam = 1.1688 + .75 = 1.9188 ins, = — — 
= 2.56 diams. -75 

In practice, to avoid troublesome decimals, we might make the net space 1.2 ins; 
and the pitch 1.95; but to show farther on the working of the rule, we adhere to 
the more exact ones. 

Fifth, The clear dist from each end hole to the side edge of the plate is half of 
1.1688 = .5844 ins. 

The entire width of net iron is equal to one clear space X number of 
rivets = 1.1688 X 6 = 7.0128 ins; and the entire width of plate is equal to one 
pitch X number of rivets, = 1.91S8 X 6 = 11.5128 ius. 







RIVETS AND RIVETING. 


473 


The area of cross section of unholed plate is 11.5128 X .5 = 5.7564 sq ins; its ten¬ 
sile strength before the holes are made is 5.7564 X 45000 = 259038 tbs. 

135000 

The strength of onr joint, omitting friction, is therefore . = .52 of that of the 

original unholed plate. 259038 

If the 6 rivets are in 2 rows of 3 rivets each, the elear (list be¬ 
tween two rivets in one row will be twice as great as before, or twice 1.1688 
= 2.3376 ins. Pi tell = 2.3376 + .75 = 3.0876 ins = 3.0876 h- .75 = 4.12 diams. 
tllear dist front end hole to side edge of plate == half of 2.3376 = 1.1688. 

'Entire width of net iron = 2.3376 X 3 — 7.0128 ins. Entire width 
of plate = 3.0876 X 3 = 9.2628 ins. Area of cross section of unholed 
plate=9.262Sx. 5=4.6314 sq ins. Ultimate tensile strength, unholed 
= 4.6314 x 45000 = 208413 lbs. Ult strength of riveted joint, omitting 


friction = 


oaqiio *65 of that of the unholed plate. 

ZUo41o 


Thus we see that the arrangement with two rows gives the same strength as one 
row, with a less total width and area of plate. It of course requires longer covers. 

If the 6 rivets are in 3 rows of 2 rivets each, the area of cross 
section of the unholed plate is 4.2565 sq ins. Its tensile strength, 

191542 Ebs. Strength of riveted joint = = °f that of the unholed plate. 

The entire width of net iron (.70128 ins); its area (.70128 X .5 — .35064 sq ins); 
and its ultimate tensile strength (.35064 X 38500 = 135000 lbs), are the same in each 
case. The last is the required breaking strength of the joint, as in the beginning 
of our example; and is equal to the combined crippling strength of the six rivets. 


Art. 6. The, distance apart of the rows, from center to center of 
rivets, should not be less than two diameters of a rivet-hole. 

Rem. 1. With our constants for tension, shearing, and compression, the 
rivets will not yield first by shearing- in a double-cover butt (and 
of course in double shear), except when the diam is either equal to or less than 
.85 of the thickness of the plate, which will rarely happen. At .85 the crippling 
and shearing strength of a rivet are equal when using our assumed coeffs of crip¬ 
pling, shearing, and tension. 

Rem. 2. Our example was chosen to illustrate the rule. It will rarely hap¬ 
pen in practice that the rule will give a number of rivets without a fraction ; or 
that may be divided by 2 and by 3 without a remainder. In case of a fraction, it 
is plainly best to call it a whole rivet; although the joint thereby becomes a trifle 
stronger than necessary. Or rivets of a slightly dirt diam may be used. If the 
number of rivets comes out say 7 or 9, we may make 2 rows of 3 and 4, or of 4 and 
5, &c. Moreover, the width of the plate is frequently fixed beforehand by some 
requirement of the structure, and we must arrange the rivets to suit, taking care 
in all cases to maintain the calculated area of net iron in one row, &c. 


Rem. 3. We have (as we at first said we should do) confined ourselves to the 

simple butt-joint with 2 covers, and with the 
rivets in either 1, or in 2 or more parallel rows 
on each side of the joint-line; this being the 
strongest and the one in most common use in 
engineering structures. Necessity at times 
calls for less simple arrangements, for which 
we cannot afford space, and the strength of 
which is not so readily calculated. These 
sometimes yield results which appear strange 
to the uninitiated; thus, this lap-joint breaks 
across the net iron of one plate, along either c c or o o, where there is most of it, and 
where, therefore, it might be supposed to be the strongest. 

Rem. 4. The following table shows approximately the comparative 
strengths of the common forms of joints when properly proportioned; varying 
with quality of sheets, and of rivets: 



The original unholed plate. 

Double-riveted butt with two covers. 
Double-riveted butt with one cover.. 
Single-riveted butt with one cover... 

Double-riveted lap. 

Single-riveted lap. 


With 

Without 

friction. 

friction. 

1.00 

1.00 

.80 

.64 

.65 

.52 

.50 

.40 

.65 

.52 

.50 

.40 
















474 


RIVETS AND RIVETING. 


Rem. 5. The above tabular strengths for the lap-joints will be approx¬ 
imately attained by adopting the following proportions, according as the joint is 
double- or single-riveted. 




Double rlv. zigzag. 

Single rlv. 



Id thicknesses. 

Id diams. 

In thickuesses. 

In diams. 

Calling thickness of plate. 

1. 

.6 

1. 

.6 

Then make diam of rivet.. 

1.67 

1.0 

1.67 

1.0 

it 44 

breadth of lap. 

9.0 

5.4 

5.67 

3.4 

44 41 

pitch from cen to cen. 

7.0 

4.2 

4.5 

2.7 

41 44 

(list from end of plate to 






edge of holes. 

2.0 

1.2 

2.0 

1.2 

44 (4 

dist apart of rows from 






cen to cen. 

3.33 

2.0 



Rem. <1. 

If two or more plates on top of each other, as the 


four in A B or M II, are to be jointed together so as to act as one plate of the 
thickness c c, the diams of the rivets, and the thickness of the covers c c, e e will 
depend upon whether the junctions of the plates are all in one line with each 
other as at c c, in A B, or whether they break joint with each other as at 0, 1, 2, 3 
in M II. 



It is plain that the two covers c c by means of their connecting rivets convey 
from A to B, across the joint c c, all the strength that partly compensates for the 
severance of the four plates at that joint; whereas the two covers ee, e e } and 
their rivets in like manner convey from n of one single plate, to o of the adjoining 
one, across the joint between those two letters, only the strength that partly com¬ 
pensates for the severance of that single plate; and so with the joints at 1, 2, and 
3. Therefore the covers c c. and their rivets, must be four times as strong as those 
at any one of the four joints 0, 1, 2, 3. The first, c c, are to be regarded as joining 
two solid plates A and B, each of the fourfold thickness cc; and the others as 
joining two of the single thickness. The covers c c will, therefore, each be about 
two-thirds of the thickness c c; and the others each about, two-thirds as thick as 
a single plate. Thus, suppose each of the 4 plates in A B or M II to be % inch thick , 
making cc 3 ins. Then each cover, c, is % of 3 ins, or 2 ins thick ; or the two covers, 
cc, together 4 ins, which is thus the effective thickness of the joint, cc. But each 
cover, e e, is only % oi% inch, or ^ inch thick; and the effective thickness of joint 
at either 0, 1, 2, or 3, is that of the 3 unbroken plates plus that of the 2 covers, or 
(3 X %) + (2 X 'A) = 3i< ins. 

Art. 7. Principle of the Rule in Art 5. With our constants for 

shearing (45000 lbs per square inch) and for crippling (60000 lbs per square inch), and 
with diameter of rivet equal to, or greater than, .85 times the thickness of the plate, 
as by our rule, the crippling strength of a double cover butt joint will be equal to, or 
less than, its shearing strength. Therefore, to avoid waste of material, either in the 
plate or in the rivets, we must make 

Tensile strength of net plate n . .. . A _ 

across one row of rivets Crippling strength of all the rivets. Or, 

Total net 

width of v Thickliess v Tension _ Crippling area .. Crippling .. Total number 
plate X of P late X «n>t ~ of one rivet X unit X of rivets. 

Now, by Art 3, the crippling area of a rivet is = diam of rivet X thickness of 
plate. We take the crippling unit at 60000 lbs; and the teusion unit at 38500 lbs. 
Therefore (transposing) we must make 

Diam of w Thickness 


Total net width 
of plate 


= rivet 


X 


of plate 


X 60000 X 


Total number 
of rivets 


Thickness of plate X 38500 





































RIVETS AND RIVETING. 475 


By making the clear (list between each end rivet of a row and the side edge of 
plate = halt the clear dist between two rivets in a row; and calling the sum of the 
two end dists one space, we have 

Number of spaces _ Number of rivets . 

in a row — in a row. 60 tUat 

The clear dist between two rivets in a row, which is 


Total net width of plate 
Number of spaces in a row is a * so = 


Total net width of plate 
Number of rivets in a row 


Diam of .. Thickness .. cnnnn Total number 
rivet x of plate x wwu x of rivets 


Thickness .. Number of rivets 

of plate X d8&U0 X in a row. 


But 


Total number ot rivets 

.r;-:--—:—-—:- = N umber ot rows. 

N umber ot rivets in a row 


Therefore, omitting “thickness of plate,” common to both numerator and denom¬ 
inator, we have, as in rule in Art 5, 

Clear dist Diam of rivet X 60000 X Number of rows 
apart = 38500 


But if the diameter of the rivets is less than .85 times the 
thickness Of the plates, the shearing strength of a double-cover butt joint 
(with our assumed constants for shearing and crippling) is less than its crippling 
strength. In such cases, for the clear dist between two rivets in a row, say 


Clear dist = 


Circular area of a rivet X Shearing unit 
Thickness of plate X Tension unit 


X 2 


Rem. 1. Butt joints in double shear, or with 2 covers, being the 
only ones here considered, and inasmuch as rivets may always be used with a diam 
greater than .85 of the thickness of the plate, we may in practice always use the 
Buie in Art 5 for such joints; and, therefore, we gave it alone. 

Bern. 2. When using: these rules for other kinds of joint, 
such as laps, or butts with single covers, remember that the rivets in such are in 
single shear; and, therefore, we can use Rule in Art 5 (for crippling) only when 
the diam is either 1.7 or more times the thickness of plate. If less, use 
Rule above for shearing:; all on the assumption that our foregoing coefs of 
crippling and shearing are used. 


But the coef for tension must be changed for each kind of these 
! other joints, to allow for the weakening effects of the bending shown at W, Figs 
; 3, as deduced approximately from experiment. The writer believes that the fol- 
[ lowing tension units will give safe approximate results without friction. For 
double-cover butts, double-riveted, 38500 lbs per sq inch, as adopted above. 
For double-riveted laps, or one-cover butts, 28000. For sing;le-ri veted 
laps, or one-cover butts, 24000. But, as before remarked, no great certainty is 
attainable in riveting. 

Rem. 3. A joint may fail by crippling: without the facts being 
known or even suspected, .for it does not imply that anything breaks, but 
merely that the joint has stretched ; and this might not be detected even on 
a slight inspection of it. Still it might, and probably often has sufficed to endanger, 
and even destroy both bridges and roofs by generating strains where none were 
provided for. 












476 


STRENGTH OF MATERIALS. 


Art. 7. Breaking by shearing. Let 

abed , Fig 1, represent a beam, with its ends 
resting on supports SS; with a load ?, so heavy 
as to break it by forcing its entire central part, 
oo g g, away from the two end parts a dg and be o ; 
so that while the two latter remain in their 
places, the central part slides out; or is, as it 
were, punched clean out from between them. 

This peculiar mode of fracture is called shearing, 
or delrusion. Thq force required to produce it, 
or the resistance which the beam opposes to such 
a force, may practically be assumed to be in pro¬ 
portion to the area of the sheared section. Thus, 
since the area of cross section of a beam 1 ft sq 
is 4 times as great as that of a beam 6 ins sq, the former will present 4 times as great 
a resistance to shearing; or will, in other words, require 4 times as great a load, or 
pres, to shear it across. In Fi? 1 the total sheared area is equal to twice the trans¬ 
verse area of the beam. See Eye-bars and Pins, page 612. Bridge chords are ex¬ 
posed to great shearing force where they rest ou the abuts, but it becomes less 
toward the center of the span; and so with every equally loaded beam. See p532. 

We have very few experimental data on this subject. 

The shearing strength of white pine, spruce, 
and hemlock, parallel to the fibres, by the author, 250 to 500 lbs 
per sq inch ; oak 400 to 700; and is of use in estimating the 
resistance along the line c c, Fig. 2, at the end of a tie-beam; 
or at the head of a queen-post, <fec. 

Aeross the fibres the writer found for spruce about 
3250 lbs; white pine and hemlock 2500; yellow pine 4300 to 
5600 ; white oak 4400. 

Wrought iron is stated at 35000 to 55000 lbs per sq . , . , . .. 

inch; cast iron 20000 to 30000; steel 45000 to 75000 lbs; _jP bfl ^ 

copper 33000. 

The shearing strength of steel and wrought iron is about 
part less than the tensile. The punching of rivet-holes in Fig 1 . 2^* 

iron or steel plates, is an example of shearing. The rivets in 

tubular bridges are frequently sheared in two, in time, by the motion of the platen 
through which they are driven. In punching holes, the area of section is evidently 
found by mult the circumf of the hole by the thickness of the plate in which it in 
punched. If a piece of material be supported as shown in Fig 2b^, its resistance tc 
shearing will be 3 times as great as in Fig 1, where it is sheared across in 2 places 
only; whereas in Fig 2%, shearing would have to occur at 6 places, as per the C 
dotted lines. 




Art. 8. Breaking by torsion, or twisting. Let n, Fig 3, be a verl 

cylindrical rod of any material, 1 inch diam, the lower 
end of which is immovably fixed; and let c be a lever 
whose leverage a b, measd from the axis of 

the cylindrical rod, is 1 ft. Suppose that with a spring 
balance attached to the end b of the lever, we apply force 
horizontally, and around the axis of the rod as a center, 
until the rod breaks by being twisted. Then if we mult 
together the leverage a b (1 foot) and the amount of force 
shown by the spring balance in lbs, and div the prod by 
tbe cube of the diam of the rod in ins the quot will be a 
certain number of foot-pounds ; and will be what is called 



Fig. 3. 




the constant, or coefficient for torsion , for all cylindrical bars of that material. If w< 
use a square bar, we shall get the coef for square bars; and so w ith any other shape 
So that if with any other bar, or shaft, we mult the cube of its diam in inches bj 





































STRENGTH OF MATERIALS. 477 


said constant, and div the prod by the leverage in feet, the qnot will be the force in 
lbs which will twist the bar in two. In shape of formulas, 


1 Leverage y Brkg force 
in feet in lbs 

L Cube of diam in ins 


Constant. 


eubeof diam x Constnnt Bre.kg 

And-=- t —f 

Leverage in feet in lbs. 


Also 


Cube of diam 
in ins 


X Constant 


Brkg force in lbs 


Leverage 
in feet 


And 


Leverage v Brkg force 
in feet ^ in lbs 

Constant 


Cube of diam 
in ins. 


The constant for solid cylinders of average cast iron is about 600; 
and for wrought iron 800. For puddled steel about 700; cast steel 1000 to 1700. 
Wrought copper 400. All may vary one fourth part of these more or less, 
ij For woods, rough averages. W pine or spruce 20 to 25 ft-lbs. Y pine 35. 
, r Ash 40. W oak 50. Locust 75. Hickory 85. 

j. To find, by the last formula, the diam of a rod to have a safety of 3, 4, 5, &c, 

.. against a given twisting force in lbs, first mult said force by 3, 4, 5, &c, as the case 
a may be, and use the prod as the breaking force. The diam will then be the safe one. 

Any angle described by the force at b, when made to revolve around the axis of 
the rod as a center, during the twisting process, is called the angle of torsion. The 
length of the twisted rod or shaft does not affect the amount of force reqd to produce 
rupture; but the longer it is, the greater will be the angle of torsion ; or in other 
words, the greater will be the dist through which the force must revolve around the 
axis before fracture takes place. On the other hand, a long shaft will twist through 
a given angle under a less force than a short one of same diam and material; and will 
more readily bend under torsion. Authorities say that a working shaft should not 
twist more than 1°. We should not expose it to more than .1 of its ult strain. 

If we know the force in lbs per sq inch reqd for shearing any material, see pre¬ 
ceding Art, then the force required to break a cylinder of it by torsion, is 
Torsion One-half the shearing v v Cube of rad of 

, force _ f° rce P er S( 1 inch * ' ° X cylinder in ins 

in lbs Leverage in inches. 


That of a square shaft is about times that of a round one whose diam 
i is equal to a side of the square; or about i less than that of a round one of the 
' same transverse area. For any solid rectangular shaft 


Breaking 
Torsional force = 
in lbs 


One-third of the shearing ^ The square of w The square of 
force, in lbs per sq in -*• one side * the other side. 

Square root of the sum of the ^ Leverage 

above two squares * in inches. 


Hollow shafts resist torsion better than solid ones of the same area of 
metal. Calling the outer and inner diams in ins D and d, then 

Breaking (D*— rf4) y Constant 

Torsional force = -=-;— w 

in lbs Leverage in It X D 


Strength of wrought-iron shafting. The shafting used for the 
transmission of power to the diff parts of machine-shops, many manufacturing es¬ 
tablishments, &c, is subjected to twisting strains. It is usually made cylindrical, 
and of wrought iron. Experience shows that we may safely use the following for 
shafts of iron of good quality, bearing but little weight, and well supported at proper 
intervals, say 8 or 9 ft., by self-adjusting ball and socket hangers. 

Diam of a wrought = * / Horse-pow ^T 

iron shaft in ins. \f Number of revs * 

v per minute 


Or in words: for the diameter in inches div the number of horse-powers that are 
to be transmitted along the shaft, by the number of revs which the shaft is reqd to 
make per min. Mult the quot by 125. Take the cube root of the product. This 
cube root will be the diameter itself, at the thinnest, part, at its bearings. 

The last formula shows that the faster a shaft revolves under the same number 
of horse-powers, the less is the torsional strain upon it. This may at first seem 
strange, but less so when we reflect that a horse-power is made up of pres and dist; 
therefore, the faster it moves, the less is its pressure. Hence many horse-powers re¬ 
volving rapidly will require a less diam than a small number revolving slower in 
proportion than its number. 















47 £ 


STRENGTH OF MATERIALS, 




TRANSVERSE STRENGTH. 


Art. 9. Transverse (or across) Strength. Sometimes called Relative 
Strength. 

When either a load l, Fig 1, or any other vert force acts upon a hor beam / o 
fixed at one end, the beam becomes a lever, and with the load has a tendency to 
move or revolve about the fixed end,/as a supporting fulcrum, and in so doing 
to strain or break the beam at said fulcrum, by forces of tension and compres¬ 
sion, acting hor or lengthwise of the beam, pulling apart the upper fibres at/ and 
crushing together the lower ones. The load or other force together with the wt 
of the beam also tends to break the beam by shearing or cutting it across virtir 




STRENGTH OF MATERIALS. 


479 


cally as shown at Fig 1, p 532. But it is only the first of 

these tendencies of which we speak now. It is called the load’s Moment of 
Rupture, or Breaking Moment, or merely its mo¬ 
ment,about the fulcrum/; and is measured in foot-tons, 
or foot-lbs, inch-lbs, <fcc, by mult the load in tons or lbs, 

Ac, by its leverage, or the shortest or perp dist h e or / 

.v of its line of direction a m from the fulcrum in ft or 


ins. 




If the load instead of being’ concen¬ 
trated like l is distributed in any way along the whole 
or a part of the beam, its leverage is measured from the 
fulcrum perp to the line of direction of its cen of grav; 
which is plainly the case also with a concentrated load, 

because its line of direction also passes through its cen of grav. Before the beam 
bends, its leverage is evidently greater than afterwards, and it becomes less as the 
bending increases; but as very little bending is allowed in practical cases the 
leverage may generally be assumed not to change, but to remain as when the 
beam is hor. 

The load evidently tends also to strain, or break the beam at any point what¬ 
ever as t, Fig 1, between itself and the fulcrum /, and is assisted in so doing by the 
wt of beam between t and o. Therefore any such point l may also be assumed to 
be a fulcrum. The moment of the load will of course be less at such point than 
at /because its leverage t s will be shorter. 

In the closed beam ia e o, Fig 2, the load tends to revolve about 
the neutral axis n as the supporting fulcrum of 
its lever the beam, as shown by the dotted lines, and 
thereby to strain all the fibres from top to bottom of 
the beam at the section ine, by stretching lengthwise 
those above n, and compressing lengthwise those below 
n. The greatest strain is at the top and bottom fibres; 
and from them both ways it diminishes until at n it is 
nothing. The load also stretches or compresses the 
fibres lengthwise at every vert section along the entire 
length of the beam, more or less according to its lever¬ 
age and moment at said section ; most near the fixed 
end and least near the free end; so that the extent of 
stretch indicated by s i is the total accumulated 

stretchings that have taken place in the top fibres at every point from i to a. The 
same is the case with the stretches and compressions of the fibres anywhere be¬ 
tween i e and a o, as indicated by the varying hor dists between n i, and n s , or 
between n v and n e. The compressed fibres below n , and comprised between n v 
and n e would as it were vanish, being crushed or mashed flat against the face of 
the wall. 

A closed beam a a. Fig 3. supported at only one 

point whether at the center or not, and balanced by two either equal or un¬ 
equal loads, may plainly be regarded as two 
levers each of which is essentially in the same 
condition as Fig 2. Whether the loads are con¬ 
centrated or distributed their leverages n e.n e 
are as before to be measured from n and perp to 
the lines of direction v o, v o of their centers of 
grav as in Fig 2. Both the 5 ton loads are mani¬ 
festly upheld by the support, which of course 
reacts vert upwards against them in a vert line 
with their common cen of grav n, with a force 
of 10 tons as per the central arrow. 

Rem. 1. Each end load in Fig 3 being 5 
tons, suppose each lever n e to be 4 ft. Then the moment of each load about the 
fulcrum n would be = 5 X 4 = 20 ft-tons. Hence it might seem that over the 
support the fibres of the beam near w would have to resist a combined moment 
of 10 ft-tons. But they have actually to present a resistance of but 20 ft-tons, on 
the same principle that, if t wo men pull against each other at two ends ol a rope, 
each with a force of say 30 lbs, the strain or pull on the rope is not 00 but only <50 
lbs, because strain is the reaction (pressure or pull) against each other of two equal 
opposing forces, and is equal to only one of them. The two above equal moments 
are merely two forces acting through leverages. 

A clos(Ml beam, Fig; 4, supported Jit both ends, and 
loaded at only one point, whether at the center or not, with a concentrated load, 




















480 


STRENGTH OF MATERIALS. 


may also like Fig 3 be regarded as two levers with their common fulcrum at n 
in a vert line with the cen of grav of the load. This however is by no means 
so manifest at first sight as in Fig 3, but needs a little explanation. Let the 

beam bear 10 tons concentrated at its center, then 
evidently 5 tons of it will rest pressing dow n¬ 
wards on each end support; and each support 
will therefore press upward or react against an 
end of the beam with a force of 5 tons as per the 
arrows. Now these two 5-ton reactions of the sup¬ 
ports in Fig 4 are to be considered as taking the 
place of the two 5-ton end loads in Fig 3; w hile 
the 10-ton load in Fig 4 takes the place of the 
10-ton reaction of the support in Fig 3, and hence in this view of the case is no longer 
to be considered at all as load, but merely as a fixture for holding the common ful¬ 
crum n of the two levers in place, or in equilibrium with the upward end reactions. 
Being no longer regarded as load, it of course cannot in such cases be assumed to 
have any moment of rupture; that property being now transferred to the end 
reactions. Still, to avoid awkwardness of expression we always speak of the mo¬ 
ment of the load even in such cases, rather than of the moments of the reactions 
of th°, load. In both Figs 3 and 4 the forces at work are the same in amount, but 
plainly reversed in direction. 



Rem. If tlie load Is distributed as the 6 tons in Fig 5, instead of 
concentrated as in Fig 4, we still consider the beam as consisting of two levers 

with their common fulcrum n in a vert line with 
the cen of grav c of the load. But to find the mo¬ 
ment of the load (or more correctly, the moment 
of the reactions of the supports) about n we must 
proceed a little differently. Thus let the beam be 
3 ft span, and the load uniform, weighing 6 tons, 
and being 1 ft long. Find by rule, p 481, how 
much of this load rests on each support, (4 tons on 
a, and 2 tons on o.) The upward reactions of the 
supports will therefore also be 4 and 2 tons. Then 
first find the moment about n of either one of the reac¬ 
tions, say of the 4-ton one at a. This moment will plainly be (4 tons X 1 ft) = 4 
ft-tons. Then find the moment about n of that part (3 tons) of Ihe load that is between 
n and a, by mult the wt (3 tons) of that part by the hor dist (e n = .25 of a ft) 
between its cen of grav and n. This last moment (3 tons X 25 of a ft) = .75 of a 
ft-ton, being downward, evidently diminishes or counteracts the upward moment 
of the 4-ton reaction at a about the same fulcrum n to the same extent and is 
therefore to be subtracted from it, thus leaving 4 — .75 = 3.25 ft-ton for the mo¬ 
ment of the 6-ton load about n. 

The same result will follow if we use the 2-ton reaction of o, with the hor lever¬ 
age o n, and the part of the load between o and n. To iind the moment for any 
other point than n see pp 481 to 483. ' 


6 tons 




c 

.4- a 

^o- 



e n 


0&j 

//>. 

4 

Fig;. 5. 

2 


t 


If the beam is inclined, the moment of rupture may still be found by 
using the hor span and segments instead of the inclined ones; but the resulting 
longitudinal strains, as well as the shearing forces become changed, involving much 
complication. We confine ourselves therefore to hor beams. 

In a cantilever, Fig 29)4, a load or portion of a load causes a mo¬ 
ment or strain in every part of the beam between said load or por¬ 
tion of load and the point i of support, but at no other point. 

Thus ccauses a moment at each point between i and t; and a causes 
a moment at each point between i and s; but a causes no moment 
at any point between s and t. 

The weight of the beam itself is not here included, 
to be so, cousider it as a uniform load, and use Case 3 or Case 10, t 
and add the result to that obtained for the load. 


§ 

i 


6 i 


a 


c 


Fig 

When required 
482 and 483, 


The deflection in ordinary cases may be found by the rule on page 50G. 












STRENGTH OF MATERIALS. 


481 


General Rule for moments of rupture in hor cantilevers, no 

matter how irregularly the load or loads may be distributed. Bear in mind that 
only that part of the load which is beyond (towards the free end from) any as¬ 
sumed point tends to break the beam at that point as a fulcrum, and that it does 
so with a leverage = dist of the cen of grav of that part of the load from the point. 
The other part of the load has no moment at that point. Thus the whole load 
o x tends to break the beam at g or i with a leverage = a g or a % as the case may 
be, a being the cen of grav of the load. And so for the moment at any other 
point c, Fig 29, as a fulcrum, find the wt of all the load c x between c and the free 
end t of the beam. Also find the cen of grav 
s of that part of the load. Mult the weight 
just found by its leverage c s. 

Example 1. We use a uniform load in 
order to illustrate the rule more readily. Let 
the hor yellow pine beam it be 7 ft long; its 
breadth and its depth i e each 6 ins; the whole 
load ox 4 tons; and e the point or fulcrum at 
which the moment of the load is reqd. Then the wt of the load between c and t 
is 3 tons; and its cen of grav s is 1.5 ft or 18 ins from c. Hence the load’s moment 
at c = 3 tons X 18 ins leverage = 54 inch-tons. That is, a load of 3 tons tends 
with a leverage of 18 ins to rupture the beam at c. 


Fig:. 29. 










0 o 


a s 


x 


Example 2. Let Fig 30 be a rolled 

iron I beam cantilever of the cross-section 
shown in ins at S, projecting hor 10 ft or 120 
ius, and bearing a concentrated load of 2 tons 
at its free end. The moment of the load at the 
section i e is = 2 X 120 — 240 inch-tons. 


Fig-. 30. 



1 



Hh 



s® 

1 

2 

120" JL 




S o> 


i 


mp 

* 


m 

4 


1.25J 

i e a t 

SCO X 

n 

1.75 


General Rule for M of Rup in hor beams supported 
at each end, no matter how irregularly the load or loads may be distributed. 
Let i n, Fig 31, be such a beam of yellow pine ^ 
of 6 ft or 72 ins span, 6 ins square, and loaded 
with 3 tons. First find the cen of grav c of 
the whole load a x, and what portion (1.25 
and 1.75 tons) of said load rests on each sup¬ 
port i and n, thus, as whole spau : whole load 
: : either arm : portion at other arm. Con¬ 
sider the upward reactions thus found (1.25 and 1.75 tons) of the two supports to 
be two forces acting vert upwards against the ends of the beam at i ana n as de¬ 
noted by the arrows. Let o be any point whatever in the beam at which as a ful¬ 
crum the load’s moment is required. Assume either of the upward end forces, say 
the 1.25 tons at i, to be acting at the outer end i of a lever i o (4 ft long) of which 
o is the fulcrum. Mult this force (1.25 tons) at i by this leverage i o (48 ins). Call 
the product (60) p. Find the cen of grav (s) of the part load a o (2 tons) between 
i and o. Mult said part load by the dist o s (12 ins) of its cen of grav (s) from the 
given point or fulcrum o. Deduct the product (24) from p. The remainder (36 
inch-tons) will be the moment at o of the total load a x. The same result will 
follow if we use n and the 1.75 tons reaction, but with the load x o. 


Rem. 1. If there Is no load between i and the fulcrum point, as would 
be the case if the moment had been reqd at any point between i and a instead of 
at o, then the above p by itself is the moment. Tims e is 12 ins from i, hence the 
moment at e of the entire load a a: is 1.25 X 12 = 15 inch-tons. 


32 

































482 


STRENGTH OF MATERIALS. 


Rem. 2. Although in Fig 31 the load is regarded as having no moment at either 
end i or n of the beam, yet it produces shearing forces there equal to the upward 
reactions. See “ Shearing,” p 532. . 

Although the foregoing general rules apply to all the following cases, 
still these last will often expedite calculations. 

CANTILEVERS. 

Case 1. Concentrated load at free end. Fig 32. 

Greatest moment is at o, and = load X on. At any other point 
a it is = load X « «• Make o v = greatest moment, joiu v n; 
then a c is the moment at any point a. 




Case 2. Concentrated load at any point a. Fig 

33. Greatest moment is at o, and = load X o a. At c it is = 
load X c a. Make o v — greatest moment, join v a. Then c e is 
the moment at any point c. The load has no moment between 
a and n. 


Case 3. Uniform load throughout. Fig 34. Greatest moment is at 
o, and — whole load X Half o n. At n it is nothing At any 
point a it is = load on an X half an. Make ov = greatest mo¬ 
ment, draw the dotted parabola with its vertex at n. Then u c 
gives the moment at any point a. 

If the load is not uniform the greatest moment 

is = whole load X dist from o to its cen of j 

Case 4. Load on one part. Fig 

35. Greatest moment is at o, and = load X dist from o to cen 
of grav c of load. At any point t between the load and o, mo¬ 
ment = load X t c. At any point a in the load, moment = 
load on a s X dist a e of the cen of grav of load on a s 
from a. 




W 


X 


J i 

I 


-t- 


y 


a 


.#] n c e 

Fig.3G 


9 


8 


Case 5. Several loads, wxy , Fig 36. Find 

their centers of grav c, a, s. Greatest moment is at o, 
and = (to X c o) + (x X « «) + (.V X * o). Or first find 
the common cen of grav of all the loads, and mult its 
dist from o by the sum of the three loads. Between the 
loads the moment at g = y X ff *; at f = (y X * <) + 
(xXae); and at n = (y X s «) + {x X « «) -f (w X c n ). 


Case 6. One uniform load and one local 
one. Fig 37. Greatest moment is at o. Find that of the 
uniform one by Case 3; and that of th« local one by Case 
4, and add them together. No moment between a and n. 

REAMS SUPPORTED AT ROTH ENDS. 


Fiff 38 
8 





O 


/ 


X 


m 


a 


n 




Case 7. Concentrated 
load at center. Fig 38. 

Greatest moment is at center, and = half load X half 
span. At the supports it is nothing. Make cs= moment 
at center, join s o, $ a; then n t = moment at any point 
n. Or the moment at any point n = half load X « n, a 
being the nearest support. 

CascS. Concentrated 
load not at center. Fig 


89. Greatest moment is at the load, and is = load X e o 
X e a -r- o a. Make e s = moment at load, join s o, s a; 
then at any point c the moment is c t. Or at any point 
c, between load and o, moment = load X ae X oc -f- o a. 
At any point m, between load and a, moment = load 
XoeX«®Toa. No moment at o or a. 

























STRENGTH OF MATERIALS. 


483 


Case 9. Several concentrated loads * y z, Fig 40. By Case 8 find 

the greatest moment of each load separately, 
and for each of them draw its dotted vertical 
and two inclined lines as in this fig. Then for 
the moment at any point whatever as e, measure 
the vert dists (in this case e o, e a, e c) to the 
sloping lines, and add them together. For it 
is plain that at e we have e o for the moment 
of the load x at that point; e a for that of the 
fc load y ; and e c for that of the load z ; and so at 
any other point. Or make en==eo + ea + 
e c ; also make *• i and m h respectively equal 
to the three dists above s and m, and join j hi 
n k. Then at any point along the beam j k the 
vert dist to these upper lines gives the moment. 

Case 10. Uniform load from end to end. Fig 41. The greatest 

moment is at the center c, and is = half load X quarter 
span. At any other point e moment == half load on e o 
X e o; or to half load on e a X e o. Make cs = half load 
X quarter span, and draw a parabola o s a, then at any 
point e the moment is = e t. Or, moment at any point = 
half what it would be at that point if the whole load were 
concentrated there. 

The shearing or vertical strain at the cen¬ 
ter is zero or nothing. See Art 11, p 535. 

Rem. 1. The weight of the beam itself is usually such a load, but is frequently 
so small compared with the load that in this and other cases it may be neglected. 

Rem. 9. In the case of a uniformly distributed load like that in figure 12, page 
535, the greatest moment of rupture that can occur at any given point on the span is when 
the load covers the span from end to end; and in beams or trusses of uniform depth 
the hor strains at any given section are then also greater than under any partial 
load; so that if the chords are then strong enough in every part, they will be strong 
enough for any partial load; which is not the case with web members; any one of 
which is most strained when the longest segment reaching to it is loaded. See p 536. 

Case 11. Uniform load from a support to part way across. 

Fig 42. Find the cen of grav g of the load, and by p 481 what portion of it rests 

on each support o and x. Then by General Rule, p 481 
the mom at o or x = nothing. At n or at any point a be¬ 
tween n and x it is = portion or reaction at x X * n (or 
x a as the case may be). At any point c between n and 
o moment is = reaction at x X % c — (load on c n X 
half c n) or to reaction at o X o c — (load on o c X half 
o c). This plainly applies to unequal loads also.f 


Fig. 43. 

_, 

r (j i n a x "?' 


Fig. 41. 



Hi 


m 


jjlss 

§1 


0 

6 C 

a 



T 


JE 


To find the place of greatest moment t if the load is uni- 
Fig. 43. form say, as twice x o: n o an o : nt. When the load covers 
the whole beam it becomes Case 10. 

~PP Case 19. Uniform load reaching; lo neither 
support. Fig. 43. From either support proceed as from 


x in Fig 42, except as to greatest moment, which find by trial.# 


* On this subjeet see “ Humber’s Strains in Girders,” to whloh the writer is chiefly indebted for the 
foregoing. 

t Except that then the dist from the given point c to the cen of grav or the part-load c 6 or c n is 
not necessarily equal to half c o or half c n ; and if not, said dist must be used instead of said half 
c o or half c n. 

























484 


STRENGTH OF MATERIALS. 


The moment of rupture at any point t in a bent piece R with a load 
upou or suspended from c, is equal to the load X its leverage 1l, perp to c w. 

F This moment tends to break 

the piece R at its cross section 
at t by tearing apart the fibres 
to the right of its neutral axis, 
and bv compressing those to 
the left of it; and to this mo¬ 
ment the piece R opposes the 
moment of resistance of that 
section as in the case of a beam. 

In an arched piece as 
S loaded at any one point o, 
draw o n, o m, also o w vertical and equal by scale to the load, and complete the 
parallelogram o e w c of forces. Then will o e and o c by the same scale give two 
forces info which the load is resolved, and acting in the directions o m,on, much 
as the two strings of two bows o an, o u m. The force o e tends to break the bow 
o um at any section u with a moment = the force X its leverage u v drawn from 
the point, and perp to o m; and the force o c tends to break o an at any point in 
the same way with its leverage. The section at u or elsewhere resists, as in R. 
The weights of the pieces R and S themselves have not been taken into con¬ 
sideration. 



RESISTANCE OF BEARS. 

Having the moment of rupture of the load, it is necessary to know whether 
the Moment of Resistance, or simply the Resistance, of the beam is 
sufficient to withstand it. 

The foregoing' instructions for finding moments of rupture apply to 
horizontal beams of any form of cross-section, and whether said 
cross-section is solid, as in a common wooden beam; or open, as in a bridge-truss; 
or whether, as in the rolled I beam or plate girder, the web is Solid, but of small 
cross-section as compared with that of the flanges. 

But in treating of the action of the beam in resisting: this moment 
of rupture, we shall first consider only those beams in which each 
fibre, throughout the entire cross-section, is to be regarded as opposing: the 
moment of rupture by a horizontal or longitudinal resist¬ 
ance. This is always the case in beams of solid rectangular,cylindrical, oval, 
&c, cross-section; and (strictly speaking) in those with thin solid webs, as I beams 
and plate girders; but in plate girders it is usual, on the score of safety and in view 
of the small cross-section of the web, to neglect its share of the resistance, and 
thus to regard the girder in the same light as a truss, or “ open beam.’’ For the 
manner of resistance of such beams, see pp 528, &c. 






STRENGTH OF MATERIALS. 


485 


THEORY OF RESISTANCE OF CLOSED BEAMS. 


Scientists give the following, which, however, often differs from experiment in 
beams of composite cross-section, such as I beams, &c. See example, p 489. 

In a close*! beam, Fig 2, p 479, each of the fibres throughout the entire 
depth of the yielding section i n e opposes the breaking moment of the load by a 
Resisting' Moment or Moment of Resistance of its own. As the breaking 
moment about n of the load is made up of its gravity-force or weight mult by its 
leverage or perp distance nc from the fulcrum or neutral ax n, so the resisting 
moment about n of each separate fibre, say for instance the one at i, is made up of 
its natural longitudinal resisting force or strength mult by its leverage or perp dis¬ 
tance n i above or below the same fulcrum n; and the sum of all these separate 
moments is the moment of resistance of the cross-section, i n e, of the 
beam. A line, passing through this fulcrum, transversely of the beam, and at right 
angles to the action of the breaking force, is called the neutral axis of the 
given cross-section. 

The longitudinal resisting force or strength, in pounds per square inch, of those 
fibres which are farthest from the neutral axis, is equal to the ultimate tensile or 
compressive strength of the material in pounds per square inch, plus the assistance 
in lbs per square inch, which they receive from their natural adhesion to each 
other. This adhesion resists the longitudinal sliding of the fibres upon each other, 
without which the beam cannot break. This total resistance, in pounds per square 
inch, of the fibres farthest from the neutral axis, is called the Coefficient of 
Resistance (frequently, but less aptly, the u coefficient, constant, or mo*lu> 
Ins. of rupture”) of the material of which the beam consists, and is usually 
denoted by tfc C.” It is shown (page 489) to be = 18 times the center breaking 
load, in pounds, of a beam of the given material, 1 inch square X 1 foot span; as 
given in the table, page 493. 

The other fibres of the beam are of course capable of exerting a resistance equal to 
that of the farthest fibres; but, owing to their less distance from the neutral axis, 
they are less stretched or compressed , when the beam yields. They are therefore 
unable to put forth their entire resisting power. The length, through which any 
fibre in abending beam is stretched or compressed (ie, the extent to which it is 
lengthened or shortened), is plainly (Fig 2, p 479,) proportional to its perpendicular 
distance from the neutral axis; and, inasmuch as the longitudinal resistance which 
it actually exerts is proportional to the lengthening or shortening of it, it follows that 


Perp dist from 
neutral axis to 
farthest fibre. 


Perp dist from Coefficient of resistance 
neutral axis to , . (or longitudinal resist- 
any given fibre • • ance of farthest fibre) in 
lbs per sq in 


Longitudinal re¬ 
sistance of said 
given fibre, in lbs 
per sq in 


therefore, 


or 


Longitudinal resist¬ 
ance of given fibre, = 
in lbs per sq in 


or 

and 

Longitudinal resistance 
of given fibre, in lbs 


t:t'::C:f ; 

Coefficient of y Dist from neutral 
resistance x axis to given fibre 

Dist from neutral axis to farthest fibre. 



_ Its longitudinal resist- y Its area 
— ance in lbs per sq in in sq ins 

(F = fa) 


Coefficient of y Dist from neutral . , Area of 
resistance ^ axis to given fibre given fibre 

Dist from neutral axis to farthest fibre 


C V a 


or 


F 


t 






486 


STRENGTH OF MATERIALS. 


The moment with which a given fibre resists the rupture of the beam is plainly 
= Its longitudinal resistance X Its perp dist trom the neutral axis 


or 


r = Ff 


Coefficient of ,, Dist from neutral Area of n . , . 

X axis to given f*re X given fibre ^ »'8t from neutral 
H — X axis to given fibre 


resistance 


Dist from neutral axis to farthest fibre 


or 


r = 


C V a 


t 


Coefficient of resistance Area of ^ Square, of dist from neu- 
Dist from neutral axis to X b' iveu fibre X tral axis to S iven fibre 


farthest fibre 


or 


r =-j^t' 2 a 

I/ 


It follows from the above that the strengths of solid or “ closed ” beams are as the 
squares of their depths; although in ‘■‘■open" beams, page 528, &c, it is simply as the 
depths. 

The Moment of Resistance*of the entire cross-section of the beam is 


Coefficient of resistance 

Dist from neutral axis to 
farthest fibre 


i-The sum of 1 

X I the products for > 01 
L nil the fibres j 


( Area Square of 

of the X its dist from I I 
fibre neutral axis/J 


or, R = 



X 


f sum, for all) 
\ the fibres j 


of t* a 




This sum. or “ I,” is, for convenience, called the Moment of Inertia of 
the cross-section of the beam about its neutral axis G G. For beams of rectaugular 
cross-section, such as Figs p 487, it may be found by tlie rules on that page. For 
irregular sections, as Fig 14^ above, it cannot be found by ordinary arithmetic, but 
an approximation, sufficiently close for all practical purposes, may readily be made 
thus: Both above and below’ G G, and parallel to it, draw lilies ^?'k , l to, &c, dividing 
the section into narrow strips. If these lines are equidistant, the subsequent calcu¬ 
lations will in some cases be easier; but otherwise it is immaterial whether they 
are so or not. If they are drawn no closer together, proportiemally to the size of the. 
figure , than in Fig 14^, the approximation will be near enough for practical pur¬ 
poses. The closer they are the more accurate will be the result; but however close 
they may be, it will always be a trifle too small. Begin by finding the area in sq ins, 
of the first strip x x j k, below G G. Mult this area by the square of the dist Or to 
the cen of grav of the strip. Then proceed to the next stripj k l m\ find its area; 
and mult it by the square of the dist o s to its cen of gr s. So w ith each strip below 
G G. Add all the prods together. If the section has the same shape, size, and posi¬ 
tion above G G as below it, (as would be the case with a square, I beam, or circle,) 
mult their sum by 2. The prod will be the reqd moment. But if, as in Fig 14)^, the 
section above G G differs from the portion below it, we must div it also into strips, 
and proceed as with the lower part. The sum of all the products on both sides of 
the neutral axis will then be the moment of inertia. 





















STRENGTH OF MATERIALS. 


487 


Tile position of the hor nentral axis ft ft, may l>c found by cutting 
out a correct figure, 4 or 5 ins long, of the section, drawn on thick paper oi* tin, and 
balancing it over a straight edge. The line at which it balances is ft ft. When this 
lias been done, the dist o g, in ins, to the farthest fibre, (which may be either above 
oi- below ft ft, according to the shape of the Fig,) can be measured in ius. 

The true moment __ The approx moment , The moment of inertia of each strip 
of inertia found as above ' l m n p, &c, about its own neutral axis 

the neutral axis of each strip to be taken parallel to that of the whole figure. 

Or (wh ich amounts to the same thing) 

The moment of inertia of The sum of the moments of 

the whole figure about = inertia of the several strips 
any given neutral axis about the same neutral axis, 

In which 

Moment of inertia of Its moment of in- / y , Dist 2 from neu-, 

each strip about the neu- = ertia about its own -f- ( 1TS x tral axis of fig) 
tral axis of the whole Jig neutral axis \ aiea to that of strip / 

This affords a convenient method of finding the moment for figures, like those be¬ 
low, made up of rectangles; the moment of each rectangle about its own neutral axis 
being = its breath X its depth 3 12. 


From the above it follows that in any hollow figure, as A, B, D or G, p 495, 
or in figs 1 to 5 below 



The moment of in¬ 
ertia about any- 
given neutral axis 

Thus 


Mom of in of the en- Mom of in of 

tire fig (including the _ the missing 

missing parts) about parts about the 
the same axis same axis 


Mom of in of Mom of in of Mom of in of 

channel abed — rectangle abgh — rectangle edef 
efg h about G G about G G about G G 


The moment of inertia is plainly independent of the material oi which the beam 
consists, of the span, and of the manner in which the beam is supported or loaded; 
and is the same for all beams of a given cross-section. It follows from the foregoing 
that it is proportional to the cube of the depth of the beam. 

moments of Inertia of a few well-known figs are given below. Those of 
similar figs are to each other as their breadths X cubes of depths. 

Square. (Fourth power of side) -4- 12, whether any side or diagonal is vert. 

Parallelogram, rectangular or otherwise; neutral axis parallel to either two 
of the sides. Breadth X (cube of depth) -v- 12. The breadth must be measured 
parallel to the neutral axis; and the depth at right angles to it. 

Hollow square or rectangle. [(B x f> 3 ) — (b X <^)J s~ 12.* 

Circle. Rad 4 X .7854. Semicircle. Rad 4 x .1098. 

Ring. (Outer rad 4 — Inner rad 4 ) X .7854. 

Ellipse. Long diam vert. Half short diam X (half long diam) 3 X -7854. 

Elliptic ring. Loug diam vert. Let L, S, l , s, be half the long and half 
the short diams. Then [(S X L 3 ) — (s X £ 3 )J X .7854. 

Triangle. Base x Perp lit 3 -f-36. The base is that side which is parallel to 
the neutral axis. This does not apply to hollow triangles. 

< b > 


A 

D 

v 



D 




6 

< > 

"A ^ 

d d 


< 

A-p 

d 

_ T V 

b 

k 

JCc 




A 

d! d' <. 




b' 

Fig 3. 


< 1 / > 
Fig 4. 


Fig 5. 


1. (B.D 3 — 2 b.d 3 ) -f- 12. 

_ (5' _ b) d" 3 } 3. 5. [b.d 3 


2. (B.D 3 + 2 bd 3 ) -i- 12. 3 ;«>d 4. ( b.d 3 +- b'.d ' 3 

— (b — hr).(d — c) 3 + b'.d ' 3 — (b' — k).(d' — c') 3 ] -h 3. 


* R and b are respectively the outer and inner dimensions parallel to the neutral axis, whether 
said axis be lengthwise or crosswise of the figure. D and d are the cutter and inner dimensions per. 
per.dicular to the neutral axis. 









































488 


STRENGTH OF MATERIALS. 


From the last formula on page 486, we have 

Coefficient of Resistance 

Moment of Resistance = ^ from neutral axis to X Moment of inertia 

farthest fibre 



For beams of square or rectangular cross-section, this be¬ 
comes 

Coefficient of v Area of cross-sec- ^ depth 
III omen t of Resistance tion in sq ins in ins 

Resistance — --——- 


( R 


Ad\ 
6 > 


In rolled I beams, the moment of resistance is approximately found thus: 
all the dimensions in inches. 


Moment /Area of cross- % area of v Depth Elastic limit of 

of resist- = I section of one cross-section 1 X of X iron in lbs per sq 

ance \flauge of web / beam inch 


the area of web being = its thickness X extreme depth of beam ; and the area of 
one flange being = (area of whole beam — area of web) 2. For average rolled 
iron, the elastic limit may be taken at 22400 lbs or 10 tons; or about half the ulti¬ 
mate tensile strength. 

When a beam is upon the point of failing, its 

Moment of rupture is — its moment of resistance. 

In other words 


Load in lbs X span in ins 

in 


Co-efficient of resistance 
in lbs per sq in 


X 


Dist from neutral axis to 
farthest fibre, ins 

V m t t 

Here m is, according to circumstances, 1, 2, 4, 8, &c, as follows 


Moment of 
inertia, ins 


When the beam is firmly fixed at one end, and loaded at the other, 

“ “ “ “ uniformly, 

“ “ merely supported at both ends “ 

“ “ “ “ uniformly, 

“ “ firmly fixed “ “ at the center, 

“ “ “ “ ‘ uniformly, 


m = 1 
m = 2 
at the center, m = 4 
m = 8 

ill = 8 

ill = 16 


By some authorities the last two are given at 6 and 12. Prof De Volson Wood 
gives 12 for the liist. The term “ fixed ” in such cases is rather indefinite. 

Therefore, 


Total breaking 
load in lbs 


Mo ment of inertia X Co-efficient of resistance X m 
Dist in ins from neutral axis to farthest fibre X span in iris 


or 


w== JC« 

tl 


For the neat, or extraneous, breaking load ; if uniformly loaded deduct the weight 
of the beam itself. If supported at both ends and loaded at the center, or if fixed at 
one end and loaded at the other, deduct half the wt of the beam. 











STRENGTH OF MATERIALS. 489 


On page 493 is a table of center breaking loads in pounds, for beams 1 inch square, 
and of 1 foot span, supported at both ends. In such beams 


■y, . e . .. Depth 3 X Breadth 1 

Moment of inertia = —--—- = — 

1 mi 

vx (p 488) is 4; the dist in ins from the neutral axis to the farthest fibre is ^; and 
the span in inches is 12. The last formula on page 488 becomes, in such cases” 

Total center break- _ Coefficient of res istance X & X 4 _ Coefficient of resistance 
iug load in lbs i X 18 

( W= TS C=18W) 

or, in other words, 


Coefficient of 18 times the center breaking load of a beam 
resistance of any = of the given material, 1 inch square, 1 foot 
given material span, supported at both ends 

We thus see that the foregoing theory is identical, in principle, with the practical 
method for beams of solid cross-section, given on pages 491, &c. For beams of com¬ 
posite cross-section, such as hollow beams, I beams, &c, the theory, as already 
remarked, gives results differing from those of experiment. Thus: 

\\ hat is the ceuter breaking load of a solid cast-iron beam 4 ins square, and 6 ft, 

| or 72 ins, clear span, supported at both ends? 

4 y 43 4 V 64 256 

Here the moment of inertia is - ^ = —— = —— = 21.333. The coeff 

12 12 12 

of resistance, or 18 times our constant for cast-iron on p 493, is 2025 X 18 = 
36450. Since the beam is supported at both ends, and loaded at the middle, m is 4. 
The dist o g of the farthest fibre from the neutral axis must in a square be equal to 
% of one side; consequently it is here 2 ins. The clear span is 72 ins. Hence, 


Breakg_ 
load — 


Mom of In X Coeff of res X 
Dist o g of farthest fibre X span — 


21.333 X 36450 X 4 __ 3110352 
2 X 72 144 


= 21600 lbs = 9.64 tons. 


By our table, p 502, a beam of average cast-iron, 4 ins deep, 1 inch broad, and 6 ft 
span, breaks with 2.41 tons; consequently, four such, or one 4 ins square, would 
break with 2.41 X 4 = 9.64 tons; thus confirming the accuracy of the foregoing. 
Applied in the same way to solid cylinders, the result corresponds equally well with 
experiment and with our table on p 503. 

But for Mr Clark’s hollow squares, p 516, the formula gives 3.06 tons in¬ 
stead of the actual 2.15 ; and for his hollow cylinders 2.980 instead of 2.287. A 
true Hodgkinson beam, P 518, with top flange of 1 by 3 ins, bottom flange 
1.5 by 12 ins, vert web .75 inch thick, total depth 15 ins, clear span 20 ft, has a 
moment of inertia of 780; dist from neutral axis to upper fibre 10.7 ins, and to the 
lowest one 4.3 ins. By Hodgkinson it would yield at the lower] flange, and by his 
rule with a center load of 29.24 tons. By the formula it would be 19.8 tons. Beam 
1, p 520, actually broke with about 52 tons; by the formula it would be 40. 

The Coeff of resistance for average rolled iron is about 45000 lbs or say 
20 tons per sq inch. Cast-iron 36000 lbs or 16 tons. Good straight-grained, well- 
seasoned white pine or spruce 8100 lbs or 3.6 tons; yellow pine 9000 or 4 ; good oaks 
10000 or nearly 4.5. But as large beams are liable to defects and imperfect season¬ 
ing, not more than about two-thirds of these constants should be used in practice. 

















490 


STRENGTH OF MATERIALS. 


Comparison between models and actual structures. Many 

practical men imagine that if a model is strong, an actual bridge, roof, &c, con¬ 
structed with precisely the same proportions, must be equally strong in propor¬ 
tion to its size. This arises from their ignorance of the fact that the strength 
of similar beams, trusses, &c, increases only in proportion to the squares of their 
spans; while their weight increases as the cubes of the spans; so that a model 
5 or 10 ft long may show a great surplus of strength; while the roof or bridge of 50 
or 100 ft span, constructed like it in every respect, may break down under its own 
weight. 

We may compare the two in the following manner: Let us suppose a model 4 feet 
long of a bridge truss, its wt 0 lbs, and the extraneous center load reqd to break it 
120 lbs, or 20 times its own wt. Then its entire center breakg load, including half 
its own wt, is 120 + 3 = 123 lbs. Now suppose we are going to build a bridge truss 
of 200 ft, or 50 times the span of the model. The strength of the truss will be 50 a , 
or 2500 times that of the model; that is, it will require for its entire center breakg 
load, 50 3 X 1^3 = 2500 X 123 = 307500 lbs. Its wt, however, will be 50 3 , or 125000 
times that of the model, or 125000 X 6 = 750000 lbs; and one-half of this weight, or 
375000 lbs, must be deducted from its entire center strength, in order to find its ex¬ 
traneous center load. But in this case the half weight is greater than the entire 
center strength; consequently the truss would break under its own w r t. If, instead 
of a center load in the model, we had broken it by an equally distributed one, the 
calculation would plainly be the same, except that in the model the entire weight, 
instead of ^ of it, would be added to the extraneous load for the entire distributed 
breakg load; and in the truss, its whole weight must be deducted from its breaking 
strength, to get the extraneous distributed load. 

If (he breaking load of a model is 2, 3 or 4, &c, times as great as its 
weight, then a similar structure 2, 3 or 4, &c, times as large in every particular will 
break under its own weight. 


STRENGTH OF MATERIALS. 


491 


PRACTICAL. METHODS FOR FINDING STRENGTHS OF 

BEAMS. 


Fig - . 4. 



B 


"\7l 



To find constants. In 

beams of the same material, and 
exactly alike, except in their 
breadths, n d, the strengths vary 
in the same proportion as those 
breadths; that is, if one is 2, 3, or 
10 times broader than the other, its strength will be 2, 3, or 10 times as great. If 
they are alike, except in their clear lengths or spans, a a, between the points of sup¬ 
port, their strengths will be inversely as those lengths; that is, if one is 2, 3, or 10 
times longer than the other, it will be but %, or Ay part as strong. If they are 
alike, except in point of depth, o d , measured vert, their strengths will be directly as 
the squares of their depths; that is, if one is 2, 3, or 10 times as deep as the other, 
it will also be 4, 9, or 100 times as strong; or in other words, will require 4, 9, or 100 
times as great a load to break it. See Art. 11. It must be remembered that we are 
now speaking only of strength, or resistance to brealcing; and not of stiffness, or resist¬ 
ance to bending, or deflecting. Stiffness follows laws very dilF from 
those of strength. See Art 26, &c. 

Now, if we combine all the three foregoing elements of size, namely, length, 
breadth, and depth, we have the fact, that the strength of any beam, of any size, of 

breadth X the square of its depth. 

any given material, is in proportion to its - ^ ^ - There¬ 

fore, if we find by actual trial, what center load will break any beam of known size; 
, , . , . breadth X sq of depth 

and then find what is the proportion between its -—----, and its 

length 

brealig load, said proportion (or, more strictly speaking, ratio) will also be that 
which any similar beam has to its breakg load, and will therefore serve to calcu¬ 
late the breakg load of any other similar beam of the same material. For instance, 
if we take any piece of average good white pine, say 6 ins broad, 10 ins deep, and 12 
. , , . breadth X sq of its depth . , 6 X 160 

.feet clear span, we find that its-is equal to-——=50. 

length 12 

And if we gradually load this at its center until it breaks, we shall find that the 
breakg load, including half the wt of the clear span of the beam itself, amounts to 

breadth y sq of depth 

22500 lbs. Therefore, the proportion between the * n in3 _ i n i ns and the 


length in feet 

the same as 1 to 450; that is, the breakg 
weight, may be found by mult its 


i breakg load, is as 50 to 22500; which is 
| load of the beam, including half its own 
breadth y sq of depth 

i>i iQ s _injns_ by 459 , And in this same manner may be found the total 

length in feet 

center breakg load of any rectangular beam of average quality of white pine. For 
tlfe neat load one-half the wt of the clear span must be deducted. It is self-evident 
that the weight of the beam assists to break it, as well as the neat load; and the 
extent to which it does so, is the same as if one-half of its unsupported wt were 
concentrated at its center. Hence the rule. On this principle the rule in Art 12 is 
thus found for any material, is called its coef lor een 
or its constant for the same, as it does not vary with the 
If we take a piece, all of whose dimensions are 1, as 1 

breadth y sq of depth 

deep and 1 ft span; then the in ins x in ins will be 


based. The ratio 

breakg - loads. 

size of the beam. 


inch wide, 1 inch 
1 X sq of 1 


length in feet 

= 1 ; and the breakg load (including half its own weight) of such a piece 

is, at once, the constant reqd. See Remarks 1 to 4. In an average piece of white 
pine, this load will be found to be about 450 fibs ; or the same as the constant obtained 
from the large beam. So the student may find them for himself; 
and if he uses materials not included in our table, Art 10, it will be well to supply 
the deficiency, by inserting his own results. 

The foregoing directions for finding coefficients may be more briefly expressed 




















492 


STRENGTH OF MATERIALS. 


by a formula. After finding the neat center breakg load, by experiment, add to It 
one-half the wt of the clear span of the beam, for a total center load; then the 

Span v Total load 

Coef for breakg ) __ in teet in lbs , , . i 

strength J Breadth . Square of depth 

in ins * in inches. 

Ill ft cylinder, as the breadth and the depth are each equal to the diam, 

It is plain that B X D a , amounts to the same thing as diam*; and it is always so 

expressed. 

Rem. 1. The variation in strength of equal beams of the same material is so great, 
that it is necessary to experiment with several pieces, in order to find an average for 
a constant. The loads given in the preceding tables are also constants, but for crush- | 
ing and tension. They are averages of the strengths of the materials, derived from j 
experiment. The actual strength of any particular specimen, if of superior quality, . 
may be considerably greater than the average; or on the other hand, if of very poor j 
quality, it may fall as much below it. We should always keep this in mind when 
referring to any table of constants; and if we have doubts as to the quality of the j 
piece of material which we are about to employ, we should make a corresponding ] 
deduction from the constant in the table. 

Rent. 2. If, instead of pine, we had experimented with oak, iron, stone, 

the process for finding the constant would have been precisely the same. If in¬ 
stead of a square beam, we use cylindrical, or triangular ones, or any other 
shape, such as hollow cylinders, H, T, or U beams, Ac, we shall in the same way 
establish constants for either larger or smaller beams of those shapes, and of precisely 
the same proportions in every part. See Remark, p 509. Or if, instead of support¬ 
ing the beam at both ends, we secure it firmly at one end, and load it at the other 
end until it breaks, we shall obtain the constant /or beams fixed at one end, and loaded 
at the other , &c. Remember that the constants are for loads at 
rest. If they are liable to jars, jolts, vibrations, Ac, a large margin must be 
left for safety. Moreover, the constants given in tables are generally deduced from i 
small specimens free from important defects; whereas large beams of any kind of 
material usually contain irregularities, which diminish their strength ; and on this 
account larger allowances for safety should be made as the dimensions of the beam 
increase. 

Rem. 3. It is not necessary that b and d he taken in ins, and 

lengths in ft. They may all be in ins, ft, yds, or any other measure, but since in 
every-day practice we usually speak of the breadths and depths of beams in ins, and 
of their lengths in ft, it becomes more convenient so to consider them. If other meas¬ 
ures be used, the constant will of course be diff; but it will still be such that if the 
same measure be used for calculating the strength of another beam, the final result 
will be the same as before. In like manner, the loads may all be taken in tons, &c, 
instead of lbs; but in giving the rule, it must be stated what measures have been ;i 
employed. 

Rem. 4. There are peculiarities in some materials, which les¬ 
sen the reliability of constants derived from experimenting with small pieces. 
Thus, a large beam of cast iron will break with a less load in proportion than a small 
one; because, in the interior of thick masses of that material, more time to cookis 
required than in the outer surfaces; in consequence of which, there is a want of uni¬ 
formity in the arrangement of the particles of iron, and this conduces to weakness. 
All we can do in such cases, is to exercise judgment and caution in making sufficient 
allowance for safety. 


Art. 10. Table of constants or coefficients for the quiescent 
breaking loads of rectangular beams, supported horizon¬ 
tally at both ends, and loaded at the center; being the average qui- , 
escent breaking loads in lbs (including one-half the weight of the beams themselves) j 
for beams 1 inch square, and 1 foot clear length between the supports. For safety 
in practice, not more than about ^ to ^ of these constants should be employed; de- ! 
pending upon the importance of the structure, its temporary or permanent charac¬ 
ter, and the degree of vibration to which it will be exposed. Thus a roof will prob¬ 
ably be as safe at as a bridge at x / & . Even with a perfectly safe load, a beam may 
bend too much. See Art 26. 

If any of these coefficients be mult by .589 (or say .6) it will give that for a cylin¬ 
drical beam whose diam - side of the square. Or if mult by .71 it will give that 
for a square beam with its diagonal vertical. Any of these constants 
■nay vary one-third part either more or less. For any beam, 


fen. Breakg 
load in lbs. 


Breadth (ins) X Square of depth (ins) 
clear span in feet 


X Constant. 










STRENGTH OF MATERIALS 


493 


One Third part of any of these constants (except those for wrought 
iron and steel), may bo taken in ordinary practice as about the average constant 
for the greatest center load within the elastic limit. The loads 
here given for wrought iron and steel, are already the greatest within elastic limits. 


Transverse Strengths, in lbs. See explanation, Art 10, p 492. 


WOODS. 

Ash, English . w 

“ Amer White (Author). B 

“ Swamp. * 

Black. © 

Arbor Vitae, Amer. 0 

Balsam, Canada. 

Beech, m 

“ Amer . zz 

.® » 

Birch, Amer Black... 

ig “ Amer Yellow.3 3 

» Cclar, Bermuda .?r 5 

“ Guadaloupe... .g -• 

» “ Amer White,. \ 

i.' or Arbor Vitae J 

;r ™ 


"l 

j Chestnut .§ o 

i Elm, Amer White.° ? 

“ Bock, Canada.. ® 


CT3 




~s r* 

B.g 
s "■ 

.§*§ 


-i 

o. 

•b 

5 

a 


Hemlock 
Hickory, Amer.. 

“ “ Bitter nut. 

Iron Wood, Canada. 

Locust . 

Lignum Vitae . 

Larch .— 

Mahogany . 

Mangrove, White. 

“ Black. 

Maple, Black. 

I “ Soft.. Z 

Oak, English. 

“ Amer White (by Author) 

“ “ Red, Black, Basket... 

“ Live . 

Pine, Amer White...(by Author) 
“ “ Yellow. “ “ 

“ “ Pitch. .. “ “ 

“ Georgia.. 

Poplar . 

Poem. . 

Spruce .(by Author) 

“ Black. 

Sycamore .. 

Tamarack .. 

Teak . 

Walnut . 

Willow . 


I 


METALS. 


Brass . 


it 

a 

u 


a 

a 

u 


common pig - 
castings from pig 


hies.. 


ins thick. 


Wrought iron does not break 


650 

but at about the average of 2250 
lbs its elas limit is reached. 


650 

Steel, hammered or rolled; elas 


400 

destroyed by 3000 to 7000.. 

5000 

600 

Under heavy loads hard steel 


250 

snaps like cast iron, and soft 
steel bends like wrought iron. 


350 


850 

550 

STONES, ETC. 

Blue stone flagging, Hudson River 

125 

850 

Brick, common, 10 to 30. .average 

20 

400 

“ good Amer pressed, 30 to 

40 

600 

50.average 

250 

450 

Caen Stone . 

25 

Cement, Hydraulic, English Port¬ 
land. artificial, 


650 

7 days in water 

30 

800 

1 year in water 

50 

500 

** “ Portland, King- 


800 

ston, N. Y., 7 


800 

days in water. 

30 

600 

“ “ Saylor’s Port., 7 


700 

days in water. 

26 

650 

“ “ Common U. S. 


400 

cements, 7 dys 


750 

in water. 

5 

650 

The following hydraulic ce- 


550 

ments were made into prisms, in 


750 

vertical moulds, under apressure 


750 

of 32 lbs per sq inch, and were 


550 

kept in sea water for 1 year. 


600 

Portland Cement, English, pure, 


850 

1 year old... 

64 

600 

Roman Cement, Scotch, pure. 

23 

450 

American Cements, pure, av about 

25 

500 

Granite, 50 to 150.average 

100 

550 

“ Quincy. 

100 

850 

Glass, Millville, N. Jersey, thick 


550 

flooring .... (by Author). 

170 

700 

Mortar, of lime alone, 60 days old 
“ 1 measure of slacked lime 

10 

450 


550 

in powder, 1 sand 

8 

500 

“ 1 measure of slacked lime 


400 

in powder, 2 sand . 

7 

750 

Marble, Italian, White (Author) 

116 

550 

“ Manchester, Vt, “ 

95 

350 

“ East Dorset, Yt, “ 

111 


“ Lee, Mass, “ “ 

86 


“ Montg’y Co, Pa, Gray “ 

103 

850 

“ “ “ Clouded “ 

142 

2100 

“ Rutland,Vt, Grav 

70 

2000 

“ Glenn’sFalls,N.Y.Black “ 

155 

2300 

“ Baltimore, Md, white, 


coarse.“ 

102 

2025 

Oolites 20 to 50. 

35 

Sandstones, 20 to 70 ..average 

45 

1800 ? 

“ Red of Connecticut and 

45 

2250 

New Jersey. 


Slate, laid on its bed, 200 to 450, av 

325 























































































494 


STRENGTH OF MATERIALS, 


Art. 11. General facts respecting the breakg loads of a uni- 

rm beam of any form of section. Calling the breakg load, 

When the beam is firmly fixed at one end, and loaded at the other . 1 

Then when so fixed, and uniformly loaded, it will be. 2 

When merely supported at both ends, and loaded at the center. 4 

“ “ “ “ “ and uniformly loaded ... 8 

Firmly fixed, or tightly confined at both ends, and loaded at the center... 8 
“ “ “ “ “ “ and uniformly loaded. 16 


By some authorities the last two are given at 6 and 12. The term “fixed” in such 
cases is rather indefinite Prof. De Volson Wood gives 12 for the last. 

Rem. 1. When one beam of any form of section is 2, 3, or 4, &c, times as long, broad, 
and deep, as another, its weight will be 8, 27, or 64, &c, times as great, or as the cubes 
of the linear dimensions; but its breaking load will be only 4, 9, or 16, &c, times as 
great; that is, the strengths of similar beams are to each other as the squares of 
their respective linear dimensions. But in these breakg loads are included the 
weights of the beams themselves ; one-half of which must be deducted when the 
load is all at the middle of the beam; or the whole of it when equally distributed. 
When beams are of the moderate dimensions usually employed in buildings, their 
own weight is usually so small in comparison with their loads, that at times it may 
safely be neglected; but as they become longer, since their weight increases much 
more rapidly than their strength, it at hist constitutes too important an item to be 
overlooked; for the beam may become so great as to break under its own weight; i 
although proportioned precisely like a small one which may safely bear many times J 
its own weight. 

When a square beam is supported on its edge, instead of on a 

side (or in other words has its diag vert), it will bear but about as great a 

breakg load. 

The deflections or bendings of beams (see p505) are directly in 
proportion to the load, and to the cube of the length; and inversely to the breadth, 
and to the cube of the depth, all while within limits of elasticity. 

Rem. 2. It is very important to remember that a beam will bear a much greater 
load placed upon it in small amounts at a time; or (in case it is all applied together) 
if its pres be allowed to come upon the beam very gradually, than if it be placed 
upon it suddenly, even without any jarring, and without any previous momentum. 
When applied in small amounts, or gradually, the bending takes place slowly and 
with slight momentum; but when applied all at once, the great load descends im¬ 
mediately and rapidly to the full extent of the bending due to the load itself, and in 
so doing acquires a momentum which carries it still further; thus producing a strain 
which authorities maintain to be just twice as great as in the former case. A heavy 
train coming very rapidly upon a bridge, presents a condition intermediate between 
the two. The tables of constants always suppose the load to be applied very grad¬ 
ually, but as this is frequently not the case in practice, an allowance must be made 
accordingly. 


Art. 12. To find the quiescent breakg load of a hor square, 
or rectangular beam B. p 491, of any material, supported 
at both ends, and loaded at the centre. 

Rule. Mult together the square of its depth d o in ins, its breadth n d in ins, and 
the coef from the table p. 493. Div the prod by the clear length a a between the 
supports, in ft. The quot will be the reqd breakg load in lbs: including, however, 
half the wt of that portion of the beam which lies between the supports, and which 
must be deducted in order to get the neat load. 

Ex. What will be the center breakg load of a beam of Connecticut red sandstone, 
15 ins wide, 10 ins deep, and 12 ft long between its supports? Here 
Depth 51 ^ Breadth v Constant 
in ins in ins * 


Clear length in ft 
load reqd. 


100 X 15X45 
12 


67500 

“Ti 


— = 5625 lbs; the breakg 


But we must deduct one-half the wt of the clear length of the beam. Now abeam 
of red sandstone, of 15 ins, by 10 ins, by 12 ft, contains 21600 cub ins == 12^4 cub ft; i 
and a cub ft of red sandstone weighs about 140 tbs; therefore the beam weighs j 
12.5 X 140 = 1750 tbs; one-half of which, or 875 tbs, must be taken from the 5625 1 
lbs of breakg load, leaving 4750 tbs as the actual extraneous, or neat breakg load. 

Item. 1. If cylindrical, first find the breakg load of a square beam, 
of which each side is equal to thediamof the cylinder. Mult this load by the dec .6, 
or, more correctly, .589. Hence a square one is 1.7 times as strong as a cylindrical 
one. 















STRENGTH OF MATERIALS. 


495 


If ova.1, or elliptic, first find the load for a rectangular beam, whose sides 
are respectively equal to the two diams, and mult it by .6. 

a *triangular, and its base (whether up or down) hor, 
first find the breakg load for a rectangular beam, whose breadth is equal to the 
base; and its depth equal to the perp height of the triangle; and take one-third 
of the result as an approximation. When the edge is down, the ends must rest in 
triangular notches in the supports; otherwise, they will be crushed when loaded. 
See Art 16. 

For beams of sucli sections as A to G, the following rude rules of 
thumb will often be preferred to more intricate ones, being sufficiently approxi¬ 
mate for ordinary purposes, and for any material. See near end of Art 33, p 516. 

l or the closed Figs A, B, O, G, (each one supposed to be of equal 
thickness throughout,) first find the load for a solid beam of the same size aud shape. 




'/X'/'/A 


111 


TOP 


Then find that of a beam of the size of the hollow part. Subtract the last from 1 lie 
first. 

For C, (its top and bottom being of equal size,) first find for a rectangular beam 
a a a a. Then for two beams corresponding to the two hollows v v. Subtract these 
last from the first. ■ 

For E or F, find for three separate beams r r, i i, t n, and add them together. 

For angle and T iron, see p 525. 

For U and other shapes in common use, we may use the formula, p 488, 
or experiment with a model made of the given material, and thus find the necessary 
constant, as directed, p 491. See Remark, p 509 ; also Art 33, p 516. 

For I beams, see Art 37, and for Hodgkinson beams, see Art 

35. 

Rem. 2. In this case we may remove *4 part of the SIDE 

material of the solid square, or rectangular beam, without 
diminishing its breaking strength, although it will bend 
more. The width may remain uniform, and the depth be re¬ 
duced either at top or bottom, as shown by the dotted lines 
at m, Fig 5, strictly two parabolas with bases at load, but the 
straight ones are best in practice. Or the depth may remain 71 

uniform, and the breadth be reduced, as shown by the dots at 
n, which is a top view of the beam. Theoretically, the dotted Fig. 5. 

lines in n might meet at the ends of the beam ; but in prac¬ 
tice this would not generally leave sufficient material at the ends for the beam to 
rest upon securely. 

Such reductions of beams are rarely made when they are 
of wood; but in iron ones much expense may be saved 
thereby. 

Rem. .3. Load at one end of the beam. 

Fig 6. the other end fixed, imagine the load to 
be at the center, and calculate it by the foregoing rule. C U 

Then div the result by 4. 

In this case the lower side of the beam may be cut away 
in the form dr a parabola, as shown by the dots. To draw 
this curve, see page 153. Or the depth may be left uni¬ 
form, and the sides be cut away, as shown by the dots at t , which is a top view. 

Art. 1.3. When the load Is equally distributed along the 


=3| 

d! 


1 


1111 


fi 


m 

W 

■ mm 


Fig. 6. 




is 

entire clear length of a horizontal beam, 
supported at both ends, as in Fig 7, instead of be¬ 
ing all applied at the center, assume it to be at the center, 
and proceed precisely as in the foregoing rule, Art 12. 

Then mult the load by 2. But in this case the wt of the 
entire clear length of the beam is to be deducted for the 
ueat load. 

Ex. What will be the equally distributed breakg load of the beam of sandstone 
in the last example ? Here the center breakg load has already been found to be 



.. 


I 

r 


Fig. 7. 
































































496 


STRENGTH OF MATERIALS. 



5625 lbs; and 5G25 X 2 = 11250 lbs, the reqd distributed load. From this subtract 
the wt of the entire 12 feet clear length of beam, or 1750 lbs; and the rem, 9500 lbs. 
is the neat extraneous breakg load. About -A- part of this, or 950 lbs, is quito as 
much as should be trusted upon so variable and treacherous a material as red sand, 
stone. 

Rem. 1. A beam requires twice as much breaking load, equally distributed , as it 
will at its center. In this case the breakg strength of the 
beam ivill not be diminished if the top be cut away in the form 
of a true semi-ellipse, as shown by the dots in Fig 7. Or if the 
depth must be kept uniform, the sides may be trimmed to two 
parabolas o c o, o to, Fig 8. The mode of drawing these figs 
to any span and height will be found under “Mensuration,” 
pp 150, 153; but in practice circular segments will answer. 

Item. 2. Toad uniformly distributed along 

THE ENTIRE CLEAR LENGTH, y g , Fig 9, OF A HOR RECTANGULAR 
beam, firmly fixed at one end only, assume it to be at the cen¬ 
ter. as ill Art 12,and calculate it by the rule in that Art. Then 
div the result by 2. From the quot deduct entire wt of beam 
for neat load. 

In this case, theoretically, we may cut off one-half the pro¬ 
jecting part y o of the beam, as by the dotted line y o, without 
diminishing its breakg strength. But in practice it will rarely 
be advisable to reduce it to a mere thin edge at o. Or the depth 
c s of the beam may be left uniform, and the sides bo cut away, 
as shown by the two semi-parabolas a c, a c, at t, which is the 
top of the beam. If a c, a c, be even made straight, instead of 
parabolas, it is plain that there would still be a considerable 
saving of expense, if the beam is of iron. 

Art. 14. When the entire breakg load is applied at any 
point o, Fig lO, not at the center: first find by Art 12, what would be 

the center breakg load; including half the weight 
of the beam. Then ; making c the center of the 
span, and having the lengths ao, og, ac and eg; 

extraneous /said total \ half the 

concentrated / center s acXcg\ _ weight 

breaking — l breaking ^ ao y^ 0 g) of the 
load at o \ load / beam 


Fig. 9. 



□ 


_ 

a o 

c y 

it 

■ 



I 


O 


Fig. 10. 


This rule is not exact if the load 

rests upon the beam for a short distance on either or on both sides of the point o, 
but only when it all rests at that very point alone; if it does not the load may be in¬ 
creased. See p 483. 

Rf.m. 1. As a given load approaches either support, its breaking moment decreases; 
but its tendency to shear the beam between itself and the nearest support increases. 
See Rem, p 533. 

Rf.m. 2. This beam will bear to be reduced, as at m or n. Fig 5 ; except, that in¬ 
stead of reducing from its center, as in Fig 5, we must do so from where the load is 
applied. 



Art. 15. When the beam, instead of 
being lior, is inclined, as in Fig 11, in 

any of the foregoing cases, the lior dist o y must 
be taken as its span, instead of the actual clear 
length o c ; and s o, s y instead of a o and a c. This 
applies also to beams fixed at one end,#nd whether 
the inclination is upward or downward from the 
fixed end. 


Fig. 11. 


Note. The quantity of material in inclined beams may be 
reduced, in the came manner as in hor ones. 


Art. 16. Triangular beams of wood, according to Barlow’s experi¬ 
ments with pine, require about % greater breakg loads with the base up, than when 




it is down. Or with the base down, about j less than when up. Tredgold considers 
them about equally strong in either position; and that to find the center breakg 
load, we may first calculate it by Art 12, as if the beam were a rectangular one with 
the same base and perp height as the triangle ; and take %of the result Hence, 
the triangle is not an economical shape for a beam; for with only X / A the strength of 
a rectangular one, it has half as much material. 

Hodgkinsou, with cast-iron triangular beams, base up, 


























STRENGTH OF MATERIALS, 


497 


mada the breakg loads equal to % of those of rectangular bars, as in wood. Rennie’s 
experiments give about the same proportion, with the base up ; but with the b;.se 
down, he made the strength nearly twice as great, or about j 6 g- that of a rectangular 
beam of the same width and vertical height. The comparative strengths in the two 
positions will vary in diff materials, inasmuch as it is affected by tho comparative 
resistances which any given material presents to tension and compression. Within 
the limit of elasticity the beam will be equally strong, whether the edge or base be 
up; and will bend equally in either case; so also with the Ilodgkinson, or any other 
form of beam. 

Art. 17. To find the side of a square lior beam supported at 
both ends, and reqd to break under a given quiescent center 
load. 

Rule. Mult the clear bearing in ft, by the given breakg load in pounds. Div tho 
prod by the corresponding constant p 493. Take the cube root of the quot. This 
cube root will be the reqd depth or breadth of the beam, approximately, in ins. 
When the size of the beam is so great that its wt must be taken into consideration, 
increase either its breadth,as directed,in Art 20; or its depth, as per Art 21, or,first 
find the approx side as before. Then calculate the wt of a sq beam having that side. 
Add half this wt to the given cen load, and with this increased cen load, repeat the 
whole calculation. The resulting side will be the reqd one very approx, but still a 
mere trifle too small. 

The breakg, or the safe load of a square beam, if mult by .6, will give that of a 
cylinder, whose diam is equal to a side of the square one. 

Art. 18. When the beam is reqd to bear its center load 
safely, mult the given safe load by the number of times it is exceeded by the 
breakg load. Then find, by Art 17, the side of a square beam to break under this in¬ 
creased load. The beam thus found will evidently be approximately the safe one for 
the actual load: exclusive, however, of the wt of the beam. When this must be in¬ 
cluded, increase the dimensions as directed in Art 17. 

If the load is equally distributed, first div it by 2, then proceed precisely as before. 

Art.. 19. When the beam is cylindrical, and reqd to break 
under its center load, to find its diam, mult the load by 1.7,and by Art 17 
find the side of a square beam, to break under this increased load. The side thus 
found will also be approximately the reqd diam. See Rems 1 and 2. 

If to be borne safely, first mult it by the number of times it is to be ex¬ 
ceeded by the breakg load. Then mult the prod by 1.7, and proceed precisely as 
before. See Rems 1 and 2. 

Rem. 1. In neither case, however, is the wt of the beam itself included. When this 
is necessary, first find the approximate diam as before. Then calculate the wt of a 
beam having this diam Add half this wt to the given cen load, in either case; and 
with this increased center load, repeat the whole calculation. The resulting diam 
will be the required one very approximately, but still a mere trifle too small. 

Rem. 2. If live load is equally distributed, first take one-half of 
it as being a center load, and with this proceed precisely as before. 

Art. 20. To find tlie breadth of a lior rectangular beam, 
supported at both ends, to break under a given quiescent 
center load ; mult the center load in lbs by the span in feet. Mult the square 
of the depth in ins by the constant p 493. Div the first prod by the last. Tho 
quot will be the breadth approximately. Calculate the wt of a beam having this 
breadth. Then say, as the center load is to half this wt, so is the breadth found, to 
a new breadth to be added to it. It will still be somewhat too small, owing to the 
neglect of the wt of the breadth last added. This may readily be found, and its 
corresponding breadth added. 

Rem. 1. If Ike loud is to be borne safely, (without any regard to the 
amount of deflection,) first mult it by the number of times it is exceeded by the 

breakg load. .. 

Rem. 2. If in either case equally distributed, take half of it as it a 
center load, and proceed precisely as before. 

Art. 21. To find the depth, when the breadth is given, mult 
the load in fbs by the span in feet. Mult the breadth in ins by the constant p 493. 
Div the first prod by the last; take the sq rt of the quot for an approximate 
depth. Calculate the wt of a beam having the depth just found ; add half of it to the 
given center load, and with this new load repeat the whole calculation; for a more 
approximate depth, but still somewhat too small, owing to the neglect of the wt of 
the depth last added. We may find this, and repeat the whole calculation, or we 
may merely increase the breadth by Art 20. .... , ,, 

Rem. If the load is to be borne safely, or if It is equally distributed, sec Remarks, 









498 


STRENGTH OF MATERIALS. 


Art. 22. To find the safe dimensions to be given to a rec¬ 
tangular beam of given span, supported at both ends, and 
which is at the same time exposed both to a transverse 
strain and to a longitudinal tensile or pulling one, or a 
longitudinal compressive one. The writer is unable to suggest any 
better rules than the following, which are at least safe. Namely, when the longi¬ 
tudinal strain is tensile, find separately the safe dimensions as if for a beam alone; 
and as if for a tie alone; and add the two resulting areas together. When the longi¬ 
tudinal strain is compressive, find separately the 6 afe dimensions as if for abeam 
alone; and as if for a pillar alone ; and add the two resulting areas together. 

Example 1. A wrought iron rectangular beam of 10 ft span is to sustain 
with a safety of 6 , an equally distributed transverse load of 100000 lbs; and a pulling 
strain of 200000 lbs. Of what size must it be? 

Here the distributed load of 100000 lbs is equal to a safe center one of 50000 lbs; 
or to a breaking center one of 50000 X 0 = 300000 lbs. 

Now first we may assume for the beam some probable approx depth, say 12 ins. 
Then we find by Art 20 that its breadth as a beam alone will be 

Breakg load in lbs X span in ft 300000X 10 3000000 0 . 

sq of depth in ins X coef, p 493 144 X -500 360000 

Again, a bar to bear a pull of 200000 lbs with a safety of 6 , should not break with 
less than 1200000 lbs; therefore since fair bar iron breaks with about 50000 lbs per 
sq inch, we have 1200000 = 50000 = 24 sq ins as the area of bar for the pull alone. 
We may add all of this to the width of the beam, making it 24=12 = 2 ins wider; 
or 10.33 ins wide in all. Or we may add it all to the depth, thus makiug the beam 
24 8.33 = 2.88 ins deeper, or 14.88 ins in all. Or part may be added to the breadth, 

and part to the depth. 

Example 2. A wrought iron rectangular beam of 10 ft span, is to sustain 
with a safety of 6 , an equally distributed transverse load of 100000 lbs, and a com¬ 
pressive strain of 200000 lbs. Of what size must it be? 

Here first assuming some probable approx depth, say 12 ins, we find as before that 
its breadth as a beam only will be 8.33 ins. As to the compressive force, it is plain 
that a pillar for sustaining it should be a hollow one with its sides as wide as possi¬ 
ble; and this is to be effected by placing it around the outside of our beam. The 
pillar will therefore have sides of about 8.33 and 12 ins wide; and its breaking load 
must be 200000X6 = 1200000 lbs, or say 536 tons. Now the length of the pillar 
measured by its narrowest side is 120-f-8.33 = 14.4 sides; and by table, p. 444, we 
find that a hollow square wrought iron pillar 14.4 sides long, breaks with 15.5 tons 
per sq inch of its metal area. Hence we require 536 15.5 = 34.6 sq ins metal area 

for our pillar. Now the circumf of the pillar is 8.33X2 + 12X2 = 40.66 ins. 
Hence its thickness must be 34.6 = 40.66= .85 of an inch. Hence both the breadth 
and the depth of the beam must each be increased twice that much or 1.7 inch ; thus 
making it 10.03 ins broad and 13.7 ins deep. 

It is plain that our pillar is thicker than necessary, because 
the tabular widths are supposed to be by outside measure, whereas our width of 8.33 
ins is inside measure. The final outer width of 10.03 ins would make the pillar only 
12 sides long; at which it would require 15.7 instead of 15.5 tons per sq inch to 
break it. Other considerations too abstruse to be explained here, combine to make 
the resulting dimensions in both examples somewhat in excess. 


. 

1*1 .it i 

th u : 
























STRENGTH OF MATERIALS. 


499 


Art. 23. Table of safe quiescent loads for horizontal rec- 
lang'iiiar beams of white pine or spruce, one inch broad, 
supported at both ends, and loaded at the center; together 
with their deflections under said loads. 


The safe load is here one-sixth of the breaking load. 

For the neat loads, deduct % the \vt of the beam itself. The deflections , 
however, are the actual ones; the wts of the beams having been introduced in cal¬ 
culating them, by the rule in Art 27. 


Loads applied suddenly will double the deflections in the table; as 
when, tor instance, if a load is held by hand, just touching a beam, the hold should be 
suddenly loosed. 

Caution. Iuasmuch as this table w T as based upon well seasoned, straight 
grained pieces, free from knots, and other defects, we must not in practice take 
morethan about two-thirds of the loads in the table for a safety of 6 in ordinary 
bmlding timber of fair quality, and with these reduced loads should not reduce 
the deflections. 

Observe also that our table is for safe center loads, but it is plain that 
in practice we cannot always apply the term in its utmost strictness; otherwise 
the load would have to be sustained by a mere knife-edge, at the very center of 
the beam. Now, in the instance Rem. p. 600, if we attempted to sustain the center 
load of 6075 lbs upon such a knife-edge, it would at once cut the beam in two. If 
we even applied it along 3 or 4 ins of the length, it would cut into it, and we should 
not have a safety of 6 against crushing the top of the beam until as iii the case of 
the ends we distributed the load along full 46 ins of length, or about 32 ins for a 
safety of 4. 


The safe load is here % of the breakg one; and the last at450 lbs at the 
center of a beam 1 inch square, and 1 foot clear length between its supports. For 
mere temporary purposes, part may be added to the loads in the table, thus mak¬ 
ing them equal to the % of the breakg load. But in important structures, subject 
to vibration, part should be deducted from the tabular loads, thus reducing 
them to y of the breaking load. This is especially necessary if the timber is not 
well seasoned. 

With the safe loads in this table a beam may bend too 
much for many practical purposes. When this is the case, we may, by reducing 
the loads, reduce the deflections in nearly the same proportion; or see table, p 512. 

All the loads in the Table are superabundantly safe against shearing-. 
Against crushing- at the ends, Ac, see “Cautions” below the Table, 
p 500. Original. 


Depth 

Span 4 ft. 

Span 6 ft 

f-lpan 8 ft 

Span 10 ft 

Span 12 ft 

Span 14 ft 

Span 16 ft 

Wt. of 

in ft rtf 

beam. 

load 

def. 

load 

def. 

lorn 

def 

loud 

def. 

load 

def. 

!":id 

def. 

load 

def. 

beam. 

Ins. 

lbs. 

ins. 

lbs. 

ins. 

lbs. 

ins. 

lbs. 

ins. 

lbs. 

ins. 

lbs. 

ins. 

lbs. 

ins. 

lbs. 

1 

19 

.39 

13 

.92 

10 

1.8 

8 

3.0 

6 

4.4 





2 

2 

75 

.22 

50 

.45 

38 

.82 

30 

1.3 

25 

1.9 

21 

2.7 

19 

3.7 

4 

3 

170 

.13 

114 

.30 

85 

.53 

67 

.84 

57 

1.3 

48 

1.7 

42 

2.3 

C 

4 

300 

.10 

200 

.22 

150 

.39 

120 

.63 

100 

.92 

86 

1.3 

75 

1.7 

8 

5 

469 

.08 

312 

.18 

234 

.31 

187 

.50 

156 

.72 

134 

1.0 

117 

1.3 

10 

6 

675 

.06 

450 

.15 

337 

.26 

270 

.41 

225 

.60 

193 

.83 

168 

1.1 

12 

7 

919 

.06 

612 

.12 

460 

.22 

367 

.35 

306 

.51 

262 

.70 

230 

.93 

14 

8 

1200 

.05 

800 

.11 

600 

.19 

480 

.31 

400 

.45 

343 

.61 

300 

.81 

16 

9 

1520 

.04 

1014 

.10 

760 

.17 

607 

.27 

507 

.40 

434 

.54 

380 

.72 

18 

10 

1875 

.04 

1250 

.09 

937 

.16 

750 

.24 

625 

.35 

536 

.49 

468 

.64 

20 

11 

2270 

.04 

1514 

.08 

1135 

.14 

907 

.22 

757 

.32 

648 

.44 

567 

.58 

22 

12 

2700 

.03 

1800 

.07 

1350 

.13 

1080 

.20 

900 

.29 

772 

.40 

675 

.53 

24 

14 

3675 

.03 

2450 

.06 

1837 

.11 

1470 

.17 

1225 

.25 

1050 

.34 

918 

.45 

28 

16 

4800 

.02 

3200 

.05 

2400 

.10 

1920 

.15 

1600 

.22 

1372 

.30 

1200 

.40 

32 

18 

6075 

.02 

4050 

.05 

3037 

.09 

2430 

.14 

2025 

.20 

1736 

.27 

1513 

.35 

36 

20 

7500 

.02 

5000 

.04 

3750 

.08 

3000 

.12 

2500 

.18 

2145 

.24 

1875 

.31 

40 

22 

9075 

.02 

6050 

.04 

4537 

.07 

3630 

.11 

3025 

.16 

2693 

.22 

2268 

.29 

44 

24 

10800 

.02 

7200 

.04 

5400 

.06 

4320 

.10 

3600 

.15 

3088 

.20 

|2700 

.26 

48 


(Continued on next page.) 




































500 


STRENGTH OF MATERIALS 


Table, continued. (Original.) 


Depth 

Span 18 ft. 

Span 20 ft. 

Span 25 ft. 

Span 30 ft. 

Span 35 ft. 

Span 40 ft. 

Wt. of 
10 ft of 

beam. 

load 

def. 

load 

def. 

load 

def. 

load 

def. 

load 

def. 

load 

def. 

beam. 

Ins. 

lbs. 

ins. 

lbs. 

ins. 

lbs. 

ins. 

lbs. 

ius. 

lbs. 

ins. 

lbs. 

ins. 

lbs. 

6 

150 

1.4 

135 

1.8 

108 

2.9 

90 

4.5 

77 

6.5 

67 

9.2 

12 

7 

204 

1.2 

184 

1.5 

147 

2.5 

122 

3.9 

105 

5 8 

92 

7.6 

14 

8 

267 

1.0 

240 

1.3 

192 

2.1 

160 

3.2 

137 

4.6 

120 

6.4 

16 

9 

338 

.92 

304 

1.2 

243 

1.9 

202 

2.8 

174 

4.0 

152 

5.5 

18 

10 

417 

.82 

375 

1.0 

300 

1.7 

250 

2.5 

214 

3.5 

188 

4.9 

20 

11 

505 

.74 

454 

.93 

363 

1.5 

302 

2.2 

259 

3.2 

227 

4.3 

22 

12 

600 

.68 

540 

.85 

432 

1.4 

360 

2.0 

308 

2.9 

270 

3.9 

24 

14 

817 

.58 

735 

.72 

588 

1.2 

1 490 

1.7 

420 

2.4 

367 

3 2 

28 

16 

1067 

.50 

960 

.63 

768 

1.0 

640 

1.5 

548 

2.1 

480 

2.8 

32 

18 

1350 

.45 

1215 

.56 

972 

.90 

810 

1.3 

694 

1.8 

607 

2.5 

36 

20 

1666 

.40 

1500 

.50 

1200 

.79 

1000 

1.2 

857 

1.6 

750 

2.2 

40 

22 

2017 

.37 

1815 

.45 

1452 

.72 

1210 

1.1 

1037 

1.5 

907 

2.0 

44 

24 

2400 

.33 

2160 

.41 

1728 

.65 

1440 

.96 

1234 

1.3 

1080 

1.8 

48 

26 

2817 

.31 

2526 

.38 

2018 

.60 

1684 

.88 

1449 

1.2 

1263 

1.6 

52 

28 

3267 

.28 

2940 

.35 

2352 

.55 

1960 

.81 

1680 

1 1 

1470 

1.5 

56 

30 

3750 

.26 

3375 

.33 

2700 

.50 

2250 

.76 

1928 

1.1 

1687 

1.4 

60 

32 

4267 

.25 

3840 

.30 

3072 

.45 

2560 

.71 

2194 

1.0 

1920 

1.3 

64 

34 

4817 

.23 

4335 

.29 

3468 

.44 

2890 

.67 

2477 

.92 

2167 

1.2 

68 

36 

5400 

.22 

4860 

.27 

3888 

.43 

3240 

.63 

2777 

.86 

2430 

1.1 

72 


White oak, and best Southern pitcb pine will bear loads ^ 

greater. 

For cast iron, mult the loads in the table by 4.5; and for wrought by 

5.3. For these new loads, mult the def's by .4 lor cast; and by .3 for wrought. 

If tile load is equally distributed over the span, it may be twice as 
great as the center one, and the dels will be 1*4 times those in the table. If the 
loads iu the table be equally distributed aloug the whole beam, the defs will 
be but five-eighths as great as those in the table. See Art 26. p 505. When more 
accuracy is reqd, half the wt of the beam itself must be deducted from the center 
load; and the whole of it from an equally distributed load. The wt of the beam, in 
the last column, supposes the wood to be but moderately seasoned, and therefore to 
weigh 28.8 lbs per cub ft. 

Uses of the foregoing table. Ex. 1. W r hat must be the breadth 
of a hor rect beam of wh pine, 18 ins deep, supported at both ends, and of 20 ft clear 
length between its supports, to bear safely a load of 5 tons, or 11200 lbs at its center? 
Here, opposite the depth of 18 ins in the table, and in the column of 20feet lengths, 

we find that a beam 1 inch thick will bear 1215 lbs; consequently, -VIP? = 9.22 ins, 

the reqd breadth; for the strength is in the same proportion as the breadth. 

Ex. 2. What will be the safe load at the center of a joist of white pine, 18 ft long, 
3 ins broad, and 12 ins deep? Here, iu the col for 18 ft. and opposite 12 ins in depth, 
we find the safe load for a breadth of 1 inch to be 600 lbs; consequently, 600 X 3 = 
1800 lbs, the load reqd. 

Rem. Cautions in the use of the above table. For instance, in 
placing very heavy loads upon short, but deep and strong beams, we must take care 
that the beams rest for a sufficient dist on their supports to prevent all danger from 
crushing at the ends. Thus, if we placo a load of 6075 lbs at the center of a beam 
of 4 feet span, 18 ins deen. and only 1 inch thick, each end of the beam sustains a 

607 9 

vert crushing force of —~ = 3037 lbs, and that siilowise of the grain, in 

which position average white pine, spruce, and hemlock crush under about 800 
lbs per sq inch, and do not have a safety of 6 until the pressure is reduced to about 
133 lbs per sq inch. Therefore our beam, in order to have a safety of 6 against 
crushing at its ends, must rest on each support 3037. -s- 133 = 23 sq ins; or for a 
safety of 4 nearly 16 sq ins. When a pressure is equally distributed side- 
wise (that is, at right angles to the general direction of the fibres) over the entire 
pressed surface of a block or beam (to ensure which, the opposite surface must be 
supported throughout its entire length) the resulting compression might readily 
escape detection unless actually measured. But when a considerable pressure is 
applied to only a port ion of the surface, as of caps and sills where in contact with 
the heads and feet of posts, or at the ends of loaded joists or girders, the com¬ 
pression becomes evident to the eye, because the pressed parts sink below the 
impressed ones, in consequence of the bending or breaking of the adjacent fibres 
What in the first case (especially if slight) would be called compression, would 



























STRENGTH OF MATERIALS. 


501 


in the second be called crushing 1 ; even when neither might be so great as 
to be unsafe. 

Owing to the resistance which said adjacent fibres oppose to being bent or 
broken, it is plain that a given pressure per sq inch, or per sq foot. &c„ 
will cause somewhat less compression or crushing when applied to only a part of 
a surface, than when to the whole of it. 

The writer has seen 40 half seasoned hemlock posts, each 12 ins square, 
footing at intervals of 5 ft from center to center, upon similar 12 X 12 inch hem¬ 
lock sills, to which they were tenoned, and which rested throughout their entire 
length on stone steps. Each post was gradually loaded with 32 tons, or equal to 
say 500 lbs per sq inch; and their feet all crushed into the sills from % to Vg.inch. 
Their heads crushed into the caps to the same extent. In practice the pres¬ 
sure at the heads and feet of posts is rarely, if ever, perfectly equable; and the 
same remark applies to the ends of loaded joists, girders, &c., in which a slight 
bending will throw an excess of pressure upon the inner edges of their supports. 

See other cautions on p. 499. 


STONE BEAMS. 

Table of safe quiescent extraneous loads for beams of good 
building granite one inch broad, supported at both ends, and loaded at the 
center; assuming the safe load to be one-tenth of the breaking one; and the latter 
to be 100 lbs for a beam 1 inch square, and 1 foot clear span. The half weight of 
the beams themselves is here already deducted by the rule in Art 12. at 170 

lbs per cub ft. 


00 

a 

•H 




CLEAR 

SPANS IN 

PEET. 




a 

JS 

1 

2 

3 

4 

5 

6 

7 

8 

10 

12 

15 

20 

§* 

Q 

Safe center load i in pounds. 

i 

10 

5 











2 

40 

20 

13 

10 









3 

90 


29 

21 

17 








4 

160 

79 

52 

39 

31 

26 

21 






5 

250 

124 

82 

61 

48 

40 

34 






6 

360 

179 

119 

89 

70 

58 

48 

42 

32 




7 

490 

244 

162 

120 

96 

79 

67 

58 

45 

36 

27 

16 

8 

639 

319 

212 

158 

126 

104 

88 

76 

59 

47 

36 

22 

10 

999 

499 

331 

248 

197 

163 

139 

120 

94 

76 

58 

38 

12 

1439 

718 

478 

357 

284 

236 

201 

174 

137 

111 

85 

58 

14 

1959 

978 

650 

487 

388 

322 

274 

238 

188 

153 

118 

81 

16 

2559 

1278 

&50 

636 

507 

421 

359 

312 

246 

201 

157 

109 

18 

3239 

1818 

1077 

806 

643 

534 

455 

396 

313 

257 

200 

141 

20 

3999 

1998 

1329 

995 

794 

660 

563 

490 

388 

319 

249 

176 

22 

4839 

2417 

1609 

1205 

961 

800 

682 

594 

470 

387 

303 

216 

24 

5758 

2877 

1916 

1434 

1145 

951 

813 

708 

562 

463 

362 

260 

27 

7288 

3642 

2425 

1815 

1450 

1205 

1030 

898 

713 

588 

462 

332 

80 

8998 

4496 

2995 

2243 

1791 

1489 

1273 

1110 

882 

728 

573 

415 

83 

10888 

5441 

3624 

2714 

2168 

1803 

1542 

1345 

1069 

883 

696 

505 

86 

12958 

6476 

4314 

3231 

2581 

2147 

1836 

1603 

1275 

1054 

832 

606 


If uniformly distributed over the clear span, the safe extraneous 
loads will he twice as great as those in the table. 

For good slate on bed the safe loads may be taken at about 3 times; for 
good sandstone on bed at about one-half; and for good marble or 
limestone on bed at about the same as those iu the table. See table, p 493. 





































502 


























































































































STRENGTH OF MATERIALS, 


503 
















































































































































504 


STRENGTH OF MATERIALS. 


LIMIT OF ELASTICITY IN BEAMS. 

The limit of elasticity of a beam of any particular form, 
or material, is determined by experiment with a similar beam, as in the case 
of constants for breakg loads. <tc. Thus, load a beam at the center, by the careful 
gradual addition of small equal loads; carefully note down the def that takes place 
within some mins (the more the better) after each load has been applied; in order to 
ascertain when the defs begin to increase more rapidly than the loads; for when this 
takes place, the load for elastic limit has been reached. 

It is not the defs of the whole beam that are to be noted, but those of its 
clear span only. Several beams should be tried to get an average 
constant; tor even in rolled iron beams of the same pattern, and same iron, there is 
a very appreciable diff of strengths and defs. 

Then, to get the constant, so as to apply it to similar beams, using the total load 
applied during the equal defs, including Yi wt of beam, 

Constant for greatest center __Span i n ft X 'lolal load in l bs_ 

loads within elas limit Breadth in ins X Square of depth in ins 






STRENGTH OF MATERIALS. 


505 


Rem. Of course, in practice, it is frequently difficult to ascertain with precision 
when, or under what load, the defs actually do begin to increase more rapidly than 
the successive loads. For although theoretically the defs are equal for equal loads, 
until the elastic limit is reached, yet in practice ., they are only nearly equal, up to 
that point. This is owing to the fact that no material composing a beam is perfectly 
uniform throughout in texture and strength; so that instead of perfect equality of 
defs, we shall have an alternation of larger and smaller ones. Therefore, some judg¬ 
ment is reqd to determine the final point; in doing which, it is better, in case of 
doubt, to lean to the side of safety. 

Rem. Within the limits of elasticity, a beam of irregular shape, such as a T, or a 
Hodgkinson beam, a triangle, &c, will bend to the same extent, whether its top or 
its bottom be uppermost. 

Rem. It is assumed always that the load is not subject to jars or vibrations. These 
would increase the deflections. 

Constants for greatest center loads within limits of elastici¬ 
ty, may be had near enough for common practice by taking one-third of the break- 
I ing constants in the table on p 493 ; except those for rolled iron and steel.* 

Then, y of the 

Greatest center load Breadth v Square of depth v constant on 
within elas limit of — in ins in ins * page 493.* 

rectangular beam in lbs Span in feet 

Art. 26. Inflections, or bendings, of beams, under their 
loads. The foregoing relates to the strength of beams, or their resistance to 
' breakg; the following to their stiffness , or resistance to bendg. The two follow very 
! ditf laws. 

It is with the defs within the elastic limit that the engineer is chiefly interested. 
They then are directly as the load and as the cube of the span; and inversely as 
the breadth, and as the cube of the depth; and this, with the following, applies 
j not only to all rectangular beams, but to all others of whatever cross section, 
1 provided the sections are similar. 

! 

With same span, breadth, and load, the deflections within elas limits 
are in all cases inversely as the cubes of the depths. Hence the depths are 
! inversely as the cube roots of the deflections. 

With same span, breadth, and deflection, the depths are directly 
as the cube roots of the loads. Hence the loads for equal deflections are as 
the cubes of the depths. 


If the deflection of a beam supported at both ends and loaded at the 

center be called. 

Then that of the same beam, with the same load uniformly dis¬ 
tributed will be.. 

Firmly fixed at both ends, and loaded at the center, by Mosely. 

“ “ “ “ “ “ uniformly loaded. 

Fixed at one end. and loaded at the other. 

“ “ “ uniformly loaded. 


1 . 

.625, or % 
.2, or \ 
.125, or % 
16. 

6 . 


The extent to which a beam may bend under even a perfectly safe load, may be 
too great for many purposes in every-day practice. Tredgold and others assume, 
that in order not to be observed, or that it may not cause the plaster of ceilings 
to crack, &c, a beam should not deflect at its center more than the part of 
its span, or ^ of an inch per ft. Thus, if its span be 20 ft, it should not bend 
i more than fgths, or % of an inch, which is also ^jjth of 20 ft. For such cases see 
Art 29. 


Shafts of wheels in machinery should not deflect more than half of this, nor a 
bridge more than say of its span, or inch per foot, under its heaviest load. 

We shall allude first to defs within the limits of safety, or of the elasticity of the 
beam; and afterward to those not exceeding of the span. After the elastic 
limit is passed, the defs increase irregularly,and more rapidly than before; and the 
beam becomes unsafe. As a general rule, the elasticity of a wooden beam is not 
injured for practical purposes, if the quiescent load does not exceed about % of the 
breaking one. 


* Except for wrought iron ami steel ; for which take the whole con¬ 
stant. 
















506 


STRENGTH OF MATERIALS. 


The Constant for Deflection for any given material, within the limit of 
elasticity, is the deflection, in inches, of a beam of that material, 1 inch square, and 
1 foot span, supported at both ends, and loaded at the center with a total weight 
(including five-eighths of the weight of the clear span of the beam) of one pound. 
Such constants may, like those of transverse strength, (see p 491,) he readily found 
by experiment. Thus, at the center of any rectangular beam, placed hor upon sup¬ 
ports at each end, place any load that is within its elastic limit, and measure the 
resulting def in ins. Mult the wt of the span of the beam by .625, add the prod to 
the neat load, for a total load. Mult together the total load in lbs. and the cube of 
the span in feet. Also mult together the breadth in ins, and the cube of the depth 
in ins. Div the first prod by the last one Div the def by the quot. Tho last quot 
will be the reqd constant for any rectangular beam of the same kind and quality of 
material, whether wood, metal, stoue, &c. That is, 


The constant for 
def withiu 
elastic, or safe 
limits of team 


} 


Def in ins 
divided by 


{ 


Total load .. Cube of span 
in lbs * in feet 

Breadth .. Cube of depth 
in ins * in ins. 


We add to the experimental neat load, the .625. or % of the wt of the clear span 
of the beam itself, because the wt of the beam equally distributed throughout its 
span, also aids in producing the def; and it does so to the same extent that % of it 
would do, if collected at the center of an imaginary beam having the same strength 
throughout as the real one, but having itself no weight. Therefore, in applying the 
constants for def to beams intended for actual use, we must not omit to add % of 
the wt of the span, to the intended center load, for an equivalent total center load, 
before making the calculations for def. The weights of similar beams (that is, 
beams -proportioned exactly alike in every part, but of diff sizes) increase so much 
more rapidly than their clear spans, that although a small one may safely bear a 
load of many times its own wt, a much larger one will break down without any 
load. Having by experiment found the constant of def for any given material, the 
def of any similar beam of the same material, whether larger or smaller, and loaded 
at the center, may be found thus: 


Def 

within safe _ 
or elastic 
limit in ins 


Total equivalent 
center load in lbs 


Cube of span 
in ft 


Breadth 
in ins 


Cube of depth 
in ins. 


Constant 


The def of a beam of any form whatever of cross section, 

if within the limit of elasticity, may be found approximately thus, 


Def in 
ins. 


load v Cube of span v f . 

ill tbs ^ in incViua ® 


in inches 


mod of elas v moment of inertia 
p 434 A in ins, p 486 


Coef d. Beams supported at both ends; center load.02083 

“ “ “ “ uniform “ .01302 

“ fixed at one end ; loaded at the other.33333 

“ “ “ “ ; load uniform.12500 


This formula gives the def produced by the load only. To find that arising from 
the weight of the beam itself, consider said wt as a uniform load; then find the re¬ 
sulting def by the same formula, and add it to that of the load. 


In a rectangular beam supported hor at both ends and loaded at the 
middle within elas limit, the def in ins will be 


Def in ins — 


/ Load , .625 wt of clear span\ v cube of clear span 
yin tbs ^ of beam in lbs / * in ins 

4 X coef elas X breadth in ins X cube of depth in ins 


And the center load in lbs, (including .625 wt of clear span of beam) 
required to produce any given def in ins within elas limit of such a beam will be 

4 X coef elas V breadth y cube of depth v given def 
__ A in in s A in ins A in ins 

cube of clear span in ins. 


Toad 

in tbs 


















STRENGTH OF MATERIALS. 


507 


Tabic of constants for tlic deflections, within the safe, or 
elastic limits, of hor rectangular beams, supported at both ends and loaded 
at the center. The timbers are supposed to be well seasoned; if not, the constant 
should be increased. 


White oak .00023 * 

Best southern pitch pine,) r»w>- * 

and white ash ./ ' 

Hickory .00016* 


White pine. 

Ordinary yellow pine. 

Spruce ... 

Good straight-grained hemlock. 
Ordinary oaks. 


.00032* 


Cast iron .000018 to .000036.Mean .000027* 

Bar iron .000012 to .000024.Mean .000018 

Steel, rolled .000010 to .000020.Mean .000015 

Full and reliable experiments on the strength and deflections of the various steels 

are much needed. 


It is evident that the stiffer the material is, the smaller will be its constant for bind¬ 
ing. All these constants vary somewhat with the quality of the metal. The defsalsoof 
timber of the same kind, vary so much with the degree of seasoning, the age of the 
tree, the part it is cut from, &c, that the writer considers it mere affectation to pre¬ 
tend to assign constants for practical use, more nearly approximate than he has here 
done. They are averages deduced from his own experiments on good pieces, well 
seasoned; and the loads were allowed to remain on for months, instead of minutes, 
as usual. Every structure is more or less exposed to vibrations and jars, which in 
time increase the deflections. In several instances, our experimental timbers bore 
their breakg loads for months before they actually gave way. And in all kinds, less 
than of the breakg load produced in a few months a permanent set, or def. 


The following are deduced from single experiments only. An allowance is made 
for the weight of the beam. 


Rolled iron beams proportioned exactly as the 7-inch Phoenix beam, .00003031 

“ “ “ “ 30 lbs, 9 inch, “ “ .0000321+ 

“ “ “ “ 50 “ heavy 9 inch “ “ .0000264 

“ “ “ “ 41% lbs, 12 inch “ “ 0000313+ 

“ “ “ “ 51% lbs, 15 inch “ “ 0000365+ 

“ “ “ 66% lbs, 15 inch “ “ .0000438 


Art. 27. To find the def in inches, of a hor rectangular 
beam, supported at both ends, and loaded at its center, with 
any jgiven load within its elasticity; mult the weight of the clear 
beam itself, in lbs, by the decimal .625. Add the prod to the given center load in lbs. 
Call the sum the total load. Mult together this total load, the cube of the span in 
ft, and the constant from the upper table. Also mult together the breadth in 
ins, and the cube of the depth in ins. Div the first prod by the last one. 

Ex. What will be the def of such a beam of average white pine, 9 ins broad, 1‘2 
ins deep, 21 feet clear span, and weighing 450 lbs; with a neat center load of 1218.75 
lbs? 

Here first, 450 X -625 = 281.25 lbs. And 281.25 -+- 1218 75 = 1500 lbs total load. 


Hence, 

1500 X 21 s X -Const. _ 1500 X 0261 X .00032 
' 9 X 12 3 — 9 X 1728 


4445.2 

, * * ■ = .286 inch; reqd def. 

15552 


Rem 1. When the load is all at one point not at the center, 

o Fig 10 p 496, mult together the two dists n a, o g, from the load to the points of 
support. Mult the prod by 4. Div the result by the clear span. Use the quot as if 
it were the span, in the last rule. The wt of the beam is not here taken into account; 
it will of course somewhat increase the def. 


* Averages near enough for ordinary practice by the writer’s own trials. Call- 

ingr *hc average elastic def of a steel beam, 1 , that of a similar 

average wrought one will be 1.2 ; and that of a cast one 1.8. If that of an average cast beam be 1, 
that of a wrought one will be .67 ; and that of a steel oue .56. If that of a wrought one be 1, cast will 

be 1.5; and steel .83. , „ ,. 

t We believe that these four beams have the same proportions, as nearly as the process of making 
them will admit of; so that .000033 may be taken as a near enough average for all four. As before 
remarked, extreme accuracy must never be expected in such matters. ' r ”" ,h “ 

tical beam will often give differences greater than this. 


Two halves of the same iden- 



























508 


STRENGTH OF MATERIALS, 


Rem. 2. When the neat load is equally distributed alone: 

the span, instead of all being at the center, then for an equivalent total center 
load, add together the neat load, and the entire vvt of the clear span of beam ; and 
mult the sum by the dec .625. With the resulting equivalent center load, proceed 
precisely as in the foregoing example. 

Ex. The def of the foregoing beam of white pine, 9 ins broad, 12 ins deep, 21 
feet span, weighing 450 lbs, and bearing an equally distributed load of 1218.75 lbs? 

Here first 450 + 1218.75 = 1668.75. And 1668.75 X -625 = 1042.97 lbs = equivalent 
center load. Hence 


1042.97 X 21 3 X -00032 3090.862 . ... 

-—-P- = , , ■ - = .1987 ins, reqd def. 

9 X 12* 15552 ’ 1 

Rem. 3. With an equally distributed load, including the wt of the 
beam, the def is only %, or the .625 part as great as it would be if the same total 
load, including the entire wt of the beam, were all applied at the center. 

Rem. 4. If the beam in any of these, or the following cases, is inclined. Fig. 
11, p 496, use the hor dist o y , instead of the actual span o c. 

Art. 28. Rule 1. To find the neat center load which will (to¬ 
gether with the wt of'the beam itself; produce any given def 
within the elastic limit of the beam; find the cube of the clear length 
in teet; mult this cube by the constant from the table on p 507. Also mult the 
breadth in ins, by the cube of the depth in ins. Div the first prod by the last one. 
l)iv the given def in ins, by the qnot, for the total reqd load in tbs. Mult the wt of 
the clear length of the beam in lbs by .625, and deduct the prod from the load so ob¬ 
tained, for the neat load. By formula, 

Cube of length v Constant 
in feet ^ " ^ n7 


Total load, 

including = Deflect -i_ 
wt of beam 111 1118 


in p507 


Breadth 
in ins 


X 


Cube ot depth 
in ins. 


Ex. What center load in lbs will (together with the wt of the beam itself) pro¬ 
duce a def of .286 of an inch, in a beam of white pine, 21 ft span, 9 ins broad, 12 ins 
deep, and which weighs 45u lbs? See table, p 499- 

Cube of 21 . Const. Breadth. Cube of 12. 

Here 9261 X -00032 = 2.9635. And 9 X 1728 = 15552. 

Ami rjm~ - 000190S - A,,d “ 1500 *“• 

For the neat load we must deduct .625 of the wt of the beam ; or 450 lbs X .625 => 
281.25 lbs: so that the neat load is 1500 — 281.25 = 1218.75 lbs, as in Ex 1, Art 27. 

If the load is uniformly distributed, use precisely the same rule for get¬ 
ting the total load. Then mult this load by 1.6. Deduct the entire wt of the clear 
length of beam. 

Ex. What equally distributed load will deflect the foregoing beam .1987 ins? 
Here, proceeding as before, the only diff is that instead of .286 def, we have .1987 

def to be div by .0001906. And —= 1042.5 lbs, as the equivalent center load. 

And 1042.5 X 1.6 = 1668 lbs for the total distributed load, including the entire wt 
of the beam, or 450 Jbs. Hence 1668 — 450 = 1218 lbs, the neat distributed load reqd; 
agreeing with the preceding example within % of a ft>; the diff being owing to a 
neglect of small decimals in the calculation. 

Rule 2. Tlie length, depth, neat center load, and def being: 
given, to tind the breadth. 


Neat cen load v Cube of length v Constant 

jjj ^ foot '' in A rt ‘7A 


in feet 


in Art 26 


Cube of depth 


X 


Def 
in ins 


Breadth 
in ins 
approx. 


in ms 

Or sufficient for the neat load alone. 

Now calculate the wt of a beam with the breadth already found. Mult this wt by 
.625, then say, as 

Breadth .625 of the Additional 

first * * weight of * breadth 
found the beam reqd. 

Add these two breadths together, and their sum will be the total breadth reqd, more 
approximately; but still somewhat too small, inasmuch as it provides only for the 


Neat center 
load 








STRENGTH OF MATERIALS, 


509 


wt of the beam of the breadth first found, and not for that having the additional 
breadth. This may readily be calculated and added. 

Rule 3. The length, breadth, neat center load, and def, 
being- given, to find the depth. 


Neat cen load v Cube of length v Constant 
in lbs in feet in Art 26 

Breadth .. Def 
in ins in ins 


Cube of depth 
in ins 
approx. 


Take the cube root of this for the depth itself, approximately. Rem. This, 
like the breadth given by the preceding formula, is too small, inasmuch as it does 
not allow for the wt of the beam. Therefore, when greater accuracy is required, 
proceed thus: Calculate the wt of a beam having the depth just found. Mult this 
wt by .625. Add the prod to the neat center load. Consider the sum as a new neat 
center load; and using it instead of the neat center load first given, go through the 
whole calculation again, to obtain a new cube of depth. The cube root of this will 
be more nearly correct; but still a trifle too small, for the same reason as in the fore¬ 
going case. 




i 9 


I* T 1 ff|*f ( 


Rem. In experimenting for constants of 

any kind, with beams of irregular cross-sections, this, for in¬ 
stance, it is quite immaterial which breadths and depths are 
tneasd; thus, for the breadth we may take ab, Im , cd. or oe, 

: &c ; and for the depth, either nc,lv, m d, b o, &c. It is only 
i necessary to state what parts actually have been taken, so 
that the corresponding ones may be measd in any other beam 
which is to be calculated by the constant derived from the 
experiment. This remark applies to all constants involving 
the breadth and the depth. The constant itself will of course vary according to 
which dimensions are taken in the experiment; but the results derived from it when 
applied to other beams of similar forms, will not be affected thereby, if the corre¬ 
sponding parts be measd in both cases* 

* We mar even take any einyle. oblique measurement, as ab, Im, nc. ad, &c. and call it both the 
breadth and the depth, this applies to rectangular, or to any other shaped beams. 












510 


STRENGTH OF MATERIALS. 


Art. 29. It is often required that the deflection shall not ex¬ 
ceed a certain amount. In such cases, if we ca 1 the greatest altowa >ie 
deflection in inches per foot of span, « I>,”* the formula p o08 tor total ceil- 
ter load (including .625 of the weight of the clear span ol the beam) becomes 


load* in lbs 


D X span in feet X dep ths j n inches X breadth in inches 
Span 3 in feet X constant, p 507 

D X depth 3 in inches X brea dth i n inches 


Span 2 in feet X constant, p 507 

total load found /weight of clear \ 

— as above — ^ span of beam * u 

For the total uniformly distributed load 


Extraneous center load 

in lbs 


1,6 X D X depth 3 in inches X breadth in i nches 
load* in lbs = Span 2 in feet X constant, p 507 


Extraneous uniformly total distributed load 
distributed load in lbs found as above 


weight of clear 
span of beam 


To find the breadth required: 

Breadth in inches extraneous center load in lbs X span 2 in ft X constant p507 
approximately = D X depth 3 in inches 


For a closer approximation, add to the breadth so found 


said breadth X 


.625 X weight of clear span of beam 
center load in lbs 


The sum will still be a mere trifle too small. 

If the load is uniformly distributed, use distributed load X -625, 

instead of center load, in the first formula for breadth. For a closer approxima¬ 
tion, add to the breadth so found, 


said breadth X 


weight of clear span of beam 
uniform load 


The sum will still be a mere trifle too small. 

To find the depth required 


Depth 

in inches 
approximately 



extraneous center v span 2 in v 
load in lbs feet * 

D X breadth in inches 


constant, 
p 507 


Then calculate the wt of the entire clear span of a beam having this depth, 
mult it by .625, and add the prod to the neat center load. Consider the sum as 
a new neat center load; and using it instead of the one first given, go through 
the whole calculation again, for a new depth. This will be the reqd depth more 
approx, but a little too small. 

If the neat load is uniformly distributed, first mult it by .625. Use 
the prod as a center load, and by the foregoing formula find first the approx 
depth. Then calculate the wt of the entire clear length of a beam having that 
depth. Mult this wt by .625, and add it to the prod used as a center load. Con¬ 
sider the sum as a new center load ; and using it instead of the one first used, 
go through the whole calculation again, for a new depth. This will be the reqd 
depth, approx, but a mere trifle too small. 

To find the side required for a square beam 


Side of square 
beam in inches 
approximately 



extraneous center v span 2 in .. constant, 
load in lbs * feet * p 507 

D " 


The fourth root is = the square root of the square root. 


* When D = ^ (i e when the greatest allowable deflection is = span) see tables pp 512, 























STRENGTH OF MATERIALS. 


511 


For a closer approximation, find the weight of a square beam with the side 
just found. Multiply it by .625, and add the product to the extraneous center 
load. Employ the formula again, using this increased load instead of extrane¬ 
ous center load. The side thus obtained will still be a mere trifle too small. 

To find the diameter required for a solid cylindrical 
beam. 


niainetPf 4 I 1 >7 w extraneous center .. span 2 in constant 
in inches = \j * IoaJ in lbs X >Vet X P 507 
approximately \ D 

The fourth root is = the square root of the square root. 

For a closer approximation, find the weight of the clear span of a beam with 
the diameter just found. Multiply said weight by .625. Add the product to the 
original given center load. Then repeat the formula, using the sum last ob¬ 
tained, instead of extraneous center load. The resulting diameter will still be 
a t rifle too small. 

The stiffness of a cylinder is to that of a square heam, whose breadth 
and depth are each equal to the diam of the cylinder, as .589 to 1; or that of the 
square one is to that of the cylinder as 1 to .589, or as 1.698 to 1; in practice we 
may use .6 and 1.7. Hence, the cylinder will bend 1.7 times as much as a square 
one, under the same load. 

When, in any of the foregoing cases, the beam is inclined, Fig 11, 
p 496, take the horizontal distance oy for the span, instead of oc. 







494 


STRENGTH OF MATERIALS, 


Art. 11. ftencral facfN respecting- the breakg loads of a uni- 

rm beam of any form of section. Calling the breakg load, 

When the beam is firmly fixed at one end, and loaded at the other . 1 

Then when so fixed, and uniformly loaded, it will be. 2 

When merely supported at both ends, and loaded at the center. 4 

“ “ “ “ “ and uniformly loaded ... 8 

Firmly fixed, or tightly confined at both ends, and loaded at the center... 8 
“ “ “ *• “ “ and uniformly loaded. 16 


By some authorities the last two are given at 6 and 12. The term “ fixed” in such 
cases is rather indefinite Prof. De Volson Wood gives 12 for the last. 

Rem. 1. When one beam of any form of section is 2, 3, or 4, &c, times as long, broad, 
and deep, as another, its weight will be 8, 27, or 64, &c, times as great, or as the cubes 
of the linear dimensions; but its breaking load will be only 4, 9, or 16, &c, times as 
great; that is, the strengths of similar beams are to each other as the squares of 
their respective linear dimensions. But in these breakg loads are included the 
weights of the beams themselves ; one-half of which must be deducted when the 
load is all at the middle of the beam; or the whole of it when equally distributed. 
When beams are of the moderate dimensions usually employed in buildings, their 
own weight is usually so small in comparison with their loads, that at times it may 
safely be neglected; but as they become longer, since their weight increases much 
more rapidly than their strength, it at last constitutes too important an item to be 
overlooked; for the beam may become so great as to break under its own weight; 
although proportioned precisely like a small one which may safely bear many times 
its own weight. 

When a square beam Is supporter! oil its edge, instead of on a 

side (or in other words has its diag vert), it will bear but about Toths as great a 
breakg load. 

The deflections or bendings of beams (see p505) are directly in 

proportion to the load, and to the cube of the length; and inversely to the breadth, 
and to the cube of the depth, all while within limits of elasticity. 

Rem. 2. It is very important to remember that a beam will bear a much greater 
load placed upon it in small amounts at a time; or (in case it is all applied together) 
if its pres be allowed to come upon the beam very gradually, than if it be placed 
upon it suddenly, even without any jarring, and without any previous momentum. 
When applied in small amounts, or gradually, the bending takes place slowly and 
with slight momentum; but when applied all at once, the great load descends im¬ 
mediately and rapidly to the full extent of the bending due to the load itself, and in 
so doing acquires a momentum which carries it still further; thus producing a strain 
which authorities maintain to be just twice as great as in the former case. A heavy 
train coming very rapidly upon a bridge, presents a condition intermediate between 
the two. The tables of constants always suppose the load to be applied very grad¬ 
ually, but as this is frequently not the case in practice, an allowance must be made 
accordingly. 


Art, 12. To find the quiescent- breakg load of a lior square, 
or rectangular beam 11. p 491, of any material, supported 
at both ends, and loaded at the centre. 

Rule. Mult together the square of its depth d o in ins, its breadth n d in ins, and 
the coef from the table p. 493. I)iv the prod by the clear length a a between the 
supports, in ft. The quot will be the reqd breakg load in lbs; including, however, 
half the wt of that portion of the beam which lies between the supports, and which 
must be deducted in order to get the neat load. 

Ex. What will be the center breakg load of a beam of Connecticut red sandstone, 
lft ins wide, 10 ins deep, and 12 ft long between its supports? Here 
Depth* .. Breadth w Constant 
in ins in ins * 


Clear length in ft 
load reqd. 


100 X 15X45 
12 


^ = 5625 
12 


lbs; the breakg 


But we must deduct one-half the wt of the clear length of the beam. Now abeam 
af red sandstone, of 15 ins, by 10 ins, by 12 ft, contains 21600 cub ins = 12*4 cub ft; 
and a cub ft of red sandstone weighs about 140 lbs; therefore the beam weighs 
12.5 X 140 = 1750 lbs; one-half of which, or 875 lbs, must be taken from the 5625 
lbs of breakg load, leaving 4750 lbs as the actual extraneous, or neat breakg load. 

Rein. 1. If cylindrical, first find the breakg load of a square beam, 
of which each side is equal to thediam of the cylinder. Mult this load by the dec .6, 
or, more correctly, .589. Hence a square one is 1.7 times as strong as a cylindrical 
one. 









STRENGTH OF MATERIALS. 


495 


If oval, or elliptic, first find the load for a rectangular beam, whose sides 
are respectively equal to the two diams, and mult it by .6. 

tf^NSular, and its base (whether up or down) hor, 
first find the breakg load tor a rectangular beam, whose breadth is equal to the 
I b 2 s 6; and Jts depth equal to the perp height of the triangle; and take one-third 
fit,the result as an approximation. When the edge is down, the ends must rest in 
triangular notches in the supports; otherwise, they will be crushed when loaded 
See Art 16. 

For beams of such sections as A to G, the following rude rules of 
thumb will often be preferred to more intricate ones, being sufficiently approxi¬ 
mate for ordinary purposes, and for any material. See near end of Art 33, p 516. 

*«»• closet! Fig’s A, B, B, G, (each one supposed to be of equai 

thickness throughout,) first find the load for a solid beam of the same size and shape. 




§ 






Then find that of a beam of the size of the hollow part. Subtract the last from the 
first. 

For C, (its top and bottom being of equal size,) first find for a rectangular beam 
a a a a. Then for two beams corresponding to the two hollows v v. Subtract these 
last from the first. ■ 

For E or F, find for three separate beams r r, i i, t n, and add them together. 

For angle and T iron, see p 525. 

For U and other shapes in common use, we may use the formula, p 488, 
or experiment with a model made of the given material, and thus find the necessary 
constant, as directed, p 491. See Remark, p 509 ; also Art 33, p 516. 

For I beams, see Art 37, and for Hodghinson beams, see Art 

35. 

Rem. 2. In this case we may remove part of the SIDE 

material of the solid square, or rectangular beam, without 
diminishing its breaking strength, although it will bend 
more. The width may remain uniform, and the depth be re¬ 
duced either at top or bottom, as shown by the dotted lines 
at m, Fig 5, strictly two parabolas with bases at load, but the 
straight ones are best in practice. Or the depth may remain n 

uniform, and the breadth be reduced, as shown by the dots at 



"" 



m 

i is 



TOP 


If 


n, which is a top view of the beam. Theoretically, the dotted 


Fig. 5. 


JIM 


wm- 


1 


t 


3 


m 
% 
MB 


n 


lines in n might meet at the ends of the beam ; but in prac¬ 
tice this would not generally leave sufficient material at the ends for the beam to 
rest upon securely. 

Such reductions of beams are rarely made when they are 
of wood; but in iron ones much expense may be saved 
thereby. 

Rem. 3. Load at one end of the beam. 

Fig’ 6. the other end fixed, imagine the load to 
be at the center, and calculate it by the foregoing rule. 

Then div the result by 4. 

In this case the lower side of the beam may be cut away 
in the form dl a parabola, as shown by the dots. To draw 
this curve, see page 153. Or the depth may be left uni¬ 
form, and the sides be cut away, as shown by the dots at t , which is a top view. 

Art. 13. When the load is equally distributed along- the 
entire clear length of a horizontal beam, 
supported at both ends, as in Fig 7, instead of be¬ 
ing all applied at the center, assume it to be at the center, 
and proceed precisely as in the foregoing rule, Art 12. 

Then mult the load by 2. But in this case the wt of the 
entire clear length of the beam is to be deducted for the 
neat load. 

Ex. What will be the equally distributed breakg load of the beam of sandstone 
in the last example ? Here the center breakg load has already been found to be 


Fig. 6. 




Fig. 7. 





























































514 


STRENGTH OF MATERIALS. 


A single beam of wood under each rail, and firmly braced against 
lateral motion, will suffice for light railroad bridges of very small span. If single 
beams of sufficient depth cannot be procured, built-beams may be used ; see g, and ij. 
Figs 62, p 613. Assuming the weight of entire bridge and load at two tons per foot, 
the following dimensions may be used: 


Span in Ft. 

Size of Beam. 

Span in Ft. 

Size of Beam. 

5 

8 X 10 ins. 

15 

12 X 18 


9 X 12 “ 

17^ 

13 X 20 

10 

10 X 14 “ 

20 

14 X 22 

1 

11 X 16 “ 

22^ 

16 X 24 


The greatest dimension to be the depth. The ends should be well bolted down to 



bolsters. These are long stout sticks of timber, from 10 to 15 ins square, (accord¬ 
ing to the span,) laid across the abuts at the bridge-seat, for the chords to rest on. 

Frequently two are used at each abut, even in small spans; and we 
have seen but one, under railroad spans of 150 feet. Large spans may require three 
or more. They are not necessarily placed in contact with each other; but may be 
some feet apart, if required. 

Or for spans of about 15 to 30 ft, we may use somewhat lighter beams, and truss 
each of them as in Fig 52, by an iron bar use; and a center post p. In this case the 
following dimensions will answer; the total deflection of the rod being ]/ H of the 
clear span. The screw ends of the bars are supposed to be upset; but the areas are 
given for the body of the rods. 

For each beam. 


Span. 

Ft. 

Beam. 

Ins. 


Section of Rod. 

Sq Ins. 

Section of Post. 

Sq Ins. 

15 

12 X 15 


3H 

25 

28 

13 X 17 


4 X 

33 

25 

14 X 18 


a 

42 

30 

15 X 20 


7 

50 


It is better to have two rods instead of one under each beam; each rod being of half 
the section here given ; and the two placed several ins apart. This affords a better 
footing for the post. The ends of the beam should be at right angles to the direction 
of the rod; and be provided with ample washers e e, of wood or iron, for distributing 
the pressure from the rod, over the whole area of the ends. The ends of these wash¬ 
ers may extend a few ins each way beyond the sides of the beam, as shown on a larger 
scale at g. This allows the rods to be outside of the beam ; instead of requiring holes 
to be bored in the latter, for passing the rods through them. They may be nearer 
together at the foot of the post. 

The head of the post may be tenoned into the bottom of the beam ; and be further united to it by 
iron straps. To prevent the foot from being worn by the rods, it should be shod with iron. A cast- 
iron shoe, as at s, may be bolted to it; having ribs for keeping the bars in place. Or a stout wronght- 
iron shoe may be well secured to it. In either case the rod at * should be so united to the shoe as to 
check any tendency in the foot, to slide toward r or r, under the vibration of passing loads. Perhaps 
this can he most conveniently done by making each rod e s e, in two separate lengths, r, r; and by 
uniting their lower ends to the shoe at s by hooks and eyes: or by eyes and bolts. &c. Various 
methods are iD use for the heads and feet of the posts of large spans ; but we cannot here treat upon 
details which pertain more to the professional bridge-builder. 

This mode of trussing is also well adapted to long floor beams; and has been used in long oblique 
web inembers; as well as in long stretches of chords from one point of support to another. 











































STRENGTH OF MATERIALS. 


515 


Continuous beams. When a single beam, as a b, Fig 40, is supported not 
only at its two ends, but at one or more intermediate points, it is said to be con¬ 
tinuous, It is stronger than if it were cut into two parts, a c,b c, each supported 

at both ends; because the tensile 
strength of the particles at o (lower 
Fig) assists in counteracting the bendg 
or breakg tendency of loads on the in¬ 
termediate parts oib, on, of the lower 
Fig. These particles at o must be torn 
asunder before the beam (if properly 
proportioned) can fail. Such a beam, 
mn, if very long and flexible, will, 
under its own wt, assume the shape of 
the reversed curve msosn; or if it be 
stiff, and heavily loaded, the same 
effect will follow. The points s s , at which the curves reverse, are called tlie 
points of contrary flexure; and the spans are virtually reduced from m-o 
and no, to ms and ns. When the beam is supported at only 3 points, as in the Fig, 
and uniformly loaded, the point of contrary flexure is dist from the central support 
of the span ; so that each span, om, on, becomes virtually reduced about y /± part; 



Fitf 40 


and the defs will be but about jo as great as if there were two separate beams* The 
sections of the beam at s and s will then experience no hor strain ; but merely the 
vert one arising from half the wt between m and s, and n and s. The position 
of tlie point of contrary flexure varies with the number of interme¬ 
diate supports, and with the manner of loading; and in bridges, &c, where the load 
moves along the beam, it changes its place during the transit, so as to bring the points 
s s considerably nearer to the central support o; thus reducing materially the ad¬ 
vantage commonly supposed to arise from connecting together the ends of adjacent 
bridge-trusses; if indeed there is any advantage in so doing, which is doubtful. The 
principle, however, becomes very useful in the case of long rafters or girders, stretch¬ 
ing over several points of support, especially when uniformly loaded. Each interval, 
except the two end ones, will have two points of contrary flexure; and will then have 
nearly twice as much strength, under an equally distributed load, as a single beam 
no longer than said interval. 















516 


STRENGTH OF MATERIALS. 


Art. 33. 


of hollow 



Fig. 12. 



Fig. 13. 


Strength 
Beams. 

During the preliminary investigations 
relative to the construction of the Menai 
tubular bridge, a. few experiments were 
made on the strength of hollow cast-iron 
beams of circular, oval, square, and rectan¬ 
gular cross-sections, supported at both 
ends, and loaded at the center. The clear 
span between the supports was in every 
case 6 ft, the thickness of metal in each 
beam, % inch; area of solid cross-section 

of each, 4.12 sq ins. The mean depth o o , Fig 12, of the circular tube, 3% ins; of 
the square one. Fig 13, 2%; of the oval, 4%; breadth, 2%; and of the rectangular 
one, mean depth, 3%; breadth, 1.833 ins. From these experiments Mr. Edwin Clark, 
assistant engineer in charge, deduced the following constants, and rules for center 
breakgloads: 

Const lor circ lubes, .95 ; oval, 1; square, 1.14; rectangle, .91. Then, first 
finding the area of the solid part of the cross-section in sq ins, 

Area of solid 

Center breaking = i n sq j Ils 

load in tons -Clear span in feet.- 

Ex. Circular beam, mean depth o o , 3% ins; area of solid ring, 4.12 sq ins; clear 
span, 6 ft. Here, 

Area. Mean depth. Const. 

4.12 X 3 5 X 95___ 2.28 tons, or 5107 lbs, breakg load. 


v Mean depth, oo, y Corresponding 
* in ins constant. 


6 (length.) 


The thickness of the cylinder or tube is about of the diam; and as a mean 

of 3 trials, it broke with a center load of 2.287 tons, or 5122 lbs; span 6 ft. Hence 
we derive for similar tubes, the constant 530, to be used in the rule, Art 12; that is, cen¬ 
ter breakg load in lbs, of circular cast-iron tubes with a thickness of one-tenth of the 
Cube of outer diam (in ins)X 530 

outer diam = Clear span in feet-> su PP osin g Mr. Clark’s iron to have been 

of average quality. 

The average breakg load of 3 square beams was 2.152 tons, or 4820 lbs; of the rec¬ 
tangular ones, 2.3 tons, or 5152 lbs; and of the 6 elliptic ones, 3.207 tons,or 7183 lbs. 
To all the foregoing extraneous loads must be added half the wt of the beam itself. 
See Art 9. 


Our rule of t humb, p 495, and rule, p 488, give breakg loads about one-third 
greater than Mr. Clark’s results, except for the oval beam, where they agree closely. 
The discrepancy is probably due to difference of quality of material. 

Hollow beams of thin wrought iron were experimented on at the 
same time; and for these Mr. Clark deduced the following constants, to be used with 
his foregoing rule for cast-iron ones : 

Constants for thin riveted tubes, circular, 1.74 ; oval, 1.85; rectangular, 1.96. 

“ “ welded tubes, “ 1.09; “ 1.27; “ 1.51. 

Art. 34. The following experiments on riveted sheet-iron cylindri¬ 
cal beams are by Fairbairn. 1st. Cylinder 18 ft long; 1 ft outer diam; clear 
span 17ft; thickness of iron .037, or i of an inch; wt of tube 107 lbs. 












STRENGTH OF MATERIALS. 


517 


Center load. 

Der. 

Center load. 


Def. 

Lbs. 

Ins. 

Lbs. 


Ins. 

1360 . 


2368 . 



1920 ... . 


2480 . 



2114 . 


•2592 . 

% 


2256 . 


2704 . 




After bearing 2704 lbs. for 1% minutes, failed by crushing at top. 

2d. Cyl 16 ft 10 ins long; 12.4 ins outer iliam; clear span 15 ft ins; thickness of 
iron .110, or full l inch; wt of tube 392 lbs. 

•7 


Center load. 

Lbs. 

Def. 

Ins. 

4 

Center load. 

Lbs. 

Def. 

Ins. 

2000 . 



8000 . 


4000 . 



10000 . 


6000 .. 



11440 .. 



With 11440 broke by the tearing of the bottom across the shackle-hole from which 
the load was suspended. 

3d. Cyl 25 ft long; 17.68 ins outer diam; clear span 23 ft 5 ins; thickness .0631, or 


o iuch ; 

weight of tube 346 lbs. 



Center load. 

Def. 

Center load. 

Der. 

Lbs. 

Ins. 

Lbs. 

Ins. 

1000 .... 

.12 

5000 . 

.48 

2000 .... 

.21 

5280 . 

.51 

3000 .... 


5840 . 

.60 

4000 ... 


6120. 

.71 


With 6400 broke at bottom; 25 ins from center, by tearing through the rivet-holes. 
4lli. Cyl 25 It long; 18.18 ins outer diam; clear span 23 ft 5 ins; thickness .119, or 
scant inch; wt of tube 777 lbs. 


Center load. 

Def. 

Center load. 

Def. 

Lbs. 

Ins. 

Lbs. 

Ins. 

2000 . 

.15 

10000 .. 


4000 . 

.30 

12000 . 

.95 

6000 . 


13000 . 


8000 . 


14240 . 



Broke through the rivet-holes 3 ft 3 ins from center, after sustaining the load for 
half a min. 



The tubes were composed of sheets about 2^ ft wide; and so 
long that a single sheet sufficed to form the entire circumf of the 
tube. They were united by double-riveted lap-joints. The loads 
were placed on a platform, supported by a rod r, Fig 11, which 
passed through a hole h in the bottom of the tube s. This rod was 
attached at its upper end to a block of wood w, rounded at its 
lower surface, so as to fit the tube. 

Circular blocks of wood were fitted into the ends of the tubes, to 
prevent them from crushing at those parts under their loads; and 
the ends rested upon blocks hollowed out to correspond with their 
cylindrical shape, to a depth equal to about % part of their diam. 











































518 STRENGTH OF MATERIALS. 


Art. 35. Hodgkinson’S beams have nearly times the strength of a 

beam of equal wt whose top and bottom flanges 
n are equal. Mr. llodgkinson, having found that on 

an average, cast iron reqd about 6^ times as much 
force to crush it as it did to pull it apart, contrived 
the beam (of which Fig 15 is a cross-section at the 
center) in which the upper or compressed rib or 
flange, u, has but %, of the area of the lower or 
extended one, b* The top flange he therefore as¬ 
sumed to be safe; inasmuch as its area is some¬ 
what greater than the proportion of 1 to 6%: and 
hence, breaking will take place from the yielding 
of the bottom flange by extension. As the result 
of his experiments, he gives the following rule, 
when the load is applied at top, or equally on both 
sides of the beam. See Rem. next page. 

Area of bot flange .. depth oo Constant 
^ in ins ^ 2.166 



Fig. 15. 


Tenter 
Breakg load 
in tons 


in sq ms 


Clear length in feet. . 

When the lower flange is as much as about 2)4 inches thick, experiments show 
that part of the breakg load thus obtained should be deducted; because thick casting* 
are proportionally weaker than thin ones. Half the wt of the beam itsell must be 
deducted, for the neat breakg load; this, however, is necessary only when the beams 
are very long; for such as are used for ordinary building purposes, it may be ne¬ 
glected. If the load is equally distributed, it wili be twice as great; but the entire 
wt of the beam must then be deducted. 


Ex. The upper rib u = 3 ins X 1 inch = 3 sq ins area; bottom rib b = ins 
X 12 ins = 18 sq ins area; total depth oo, 15 ins; clear span, 20 ft. Here, 


18 X 15 X 2.166 
20 


584.82 

20 


= 29.241 tons, the reqd load, including % the beam. 


Now to find tlie wt of lialf the beam, we may proceed thus: Mult 

the entire area of its cross-section in sq ins, by the clear span in ins. This gives us 
the cub ins of iron contained in the beam; and these div by 8600, give the wt of the 
beam in tons; because 8600 cub ins of cast iron weigh about 1 ton ; or near 4 cubic 
ins l lb. Thus, if the vert rib contains 12 sq ins, then since the two flanges con¬ 
tain 21, the entire section is 33 sq ins; and the span being 240 ins, we have 33 X 240 

7920 ' ’ 

= 7920 cub ins of iron. And —— = .92 of a ton, the wt of the beam. One-half 


of this, or .46 ton, taken from the breakg load 29.241 tons, leaves 28.78 tons as the neat 
breakg load ; showing that in such cases as this it is scarcely worth while in practice 
to make the deduction. These beams are not always made of the same section 
throughout, (see Fig 16,) but diminish toward the ends; this method is therefore not 
always strictly correct, but no great accuracy is needed in such cases. 

To find the size of a If odgkinsou beam, reqd to break under 
a given renter load, having the depth. Mult the given load in tons 
by the clear span in feet. Mult the constant 2.166 by the total depth, oo, in ins. 
Div the first prod by the last; the quot will be the area of the bottom rib in sq ins. 
This, div by 6, will be the area of the top rib. The bottom rib is usually made from 
6 to 8 times as wide as it is thick; and the top one from 3 to 6 times, The thickness 
of the stem is usually a little greater at bottom than at top; the average thickness 
being from y 1 ^ to yy of the depth of the beam. See Rem. next page. 

To save iron, the width of the bottom flange, and of the top one also if thought 

proper, may be reduced by curves 
to about x /z as great at each end 
of the beam as at its center; as 
shown by the middle sketch of 
Fig 16, of w hich the upper sketch 
is a side view. Or, leaving the 
dimensions of those flanges un¬ 
altered, the depth of the vertical 
rib may be reduced toward the 
ends, as shown by the lowest 
sketch. The theoretical curve is 
here an ellipse. When the width 
is reduced, the very ends may, for stability, be widened out, as at e, which is a top view. 



The vert rib is generally strengthened by casting brackets 


* In practice l / A is much better and safer than 





























STRENGTH OF MATERIALS. 


519 


on each side of it, as in the upper sketch. These should not extend entirely to tne 
upper rib, as they then expose the beam to crack as it cools. To prevent this tend¬ 
ency, they may be attached alternately to the top and bottom ribs. The upper ones 
however, are rarely needed. 

In designing these beams, as well as in all other castings, it is important to avoid 
sudden transitions from thin to thick parts; and to keep all parts as nearly as possi¬ 
ble of the same thickness. Otherwise the castings are apt to warp and crack in 
cooling. Also, bear in mind that the resistance or strength per sq inch is considera¬ 
bly less in thick castings than in thin ones. 

Item. The above rule for breakg loads is safe when the load is equally disposed 
on top, or on each side of the vert web; and when said web and the flanges are pro¬ 
portioned to each other about the same as those used in Mr. Hodgkinson’s experi¬ 
ments. But subsequent investigators have found his beams to break with but little 
more than half the loads given by the rule, when applied to only one side, as bo, or 
uo, Fig 15, of the top or bottom flange. W. II. Barlow, C. E., London, experimenting 
since Hodgkinsou, finds that when a cast-iron beam is liable to be loaded on only one 
side of the flange, the top flange should have an area equal to % that of the entire 
cross-section of the beam; and for beams so proportioned, he gives the following: 


Center 
breaking 
load, 
in tons 


/ Area bot flange^ 
V in sq ins 


\ , /Half area of \ v / De P th f 
+1 vert rib ) XI center to center of )X 
J V J Vtop and bot flanges/ 


Constant 
2.333 


Clear span in feet. 


Other experimenters recommend that even for loads pressing vertically through the 
upright rib, tho lower flange should have but about 3 instead of 6 times the area of 
the upper one. Cast beams should always be tested. 

The average ultimate resistance of steel to compression being about twice that to 
extension, a Hodgkinson beam of that metal should have its lower flange of twice 
the area of its upper one. Much uncertainty exists in the whole matter. 

Art. SB. For the purpose of ready reference, we give a few ex¬ 
perimental results with cast-iron beams of various shapes: being the actual center 
breakg loads in tons of sound beams. Some beams of Sterling’s toughened cast 
iron gave results full % higher than those of common iron. 

Actual center breakg loads In tons, of cast-iron beams. Clear 
spans in feat. Breadths and depths in inches. 


1.5 x.5 

i 


2>. 5 x.5 




Span 4% ft. 

Br load 2 tons. 



The above inverted. 

Br load 2.3 to 2.9 tons. 


4 x 


54 - 





11 x\ 


Span 4J4 ft. 

Br load .125 ton. 


6 x i-§- mi 



li-H, 

8x1^ 

i 


Span 11% ft f 
Br load 20 tons. 


3x3. ^ * 

A 



8 4-xl 


Span 18 ft. 

Br load 22 to 28 tons. 



The above inverted. 
Br. load .4 ton. 


.32 

47 


2. 27x. 5 


t- 

ifi 


Span 4% ft. 

f3.7 tons 
Br load^ 




to 

4.2. 



Span 27% ft.J 
Br load 29% tons. 


* As shown by dd. Fig 15. 

t “After bearing 17 tons, the beam was unloaded, and its elasticity appeared to be but little If at all 
injured.” Def under 4% tons, .15 inch ; tons, .3 inch ; 17 tons, 1.1 inches. 

J About two hundred of these beams were tested by center loada of 12 tons. Def % to % inoh. 






























520 


STRENGTH OF MATERIALS. 



Span 15 ft. 

Br load 12% tons. 



Span 4% ft. 

Br load o tons. 


2.33 x. 31 cp 

.27-1 

6.67_ 

x.66 


VTi 

eO 


Span 4% ft. 

Br load 10.5 to 
11.6 tons. 


3^x1 1 



0X1-J- FTg&sq y 


Span 19 ft. 

Br load 50 to 54 tons. 

By formula, p 4 C 8, 
it should have been 
but 40 tons. 


4 J xl£r f 9 A 

> NT 

> O* 

4± $ 

^ 1 2— 

15x2 J| I v' \.V^N\ .NXX^ ^ 

HL 


Span 30% ft. 

Br load 58 tons* 


In describing such beams, it is better to give the entire, depth of the beam; foi 
when the depth of the web is given, doubts arise whether it is meant to include the 
thicknesses of the two flanges, or not. Every writer, almost, that we have seen, 
leaves us in this doubt. 

Rf.m.. In beams either larg-er or smaller than these, but whose 

eross-sections are proportioned exactly as these are, and whose spans are the same 
that these have, the center breakg loads will be as the cubes of their cross-section 
lines. Thus, in a beam which is t , % , % , 2, 3, or 10 times as large every 

way, except in span, —r---* 

the breakg load will be TTH) (T ’ 2T ’ > 8,27, or 1000 times as great. 

If the spans also differ, first find the load as above, as if they were the 
same; then say, as the new span, is to the span given in our Figs, so is the breakg 
load thus found, to the actual breakg load for the new beam. Thus, suppose we wish 
to make a cast-iron beam, 4 times as large every way as the dimensions given in the 
first of these Figs; except its span, which is to be, say 10 ft, instead of 4% ft. Here 
the first breakg load is found to be 4 X 4 X I = 64 times as great; or 2 tons X 64 
««= 128 tons. Next, 

New Span. Span in Fig. First load. Actual load. 

10 : 4.5 ;; 128 ; 57.6 tons. 


In such cases we must, however, have regard to Rem 1, Art 11, p 494. The fore¬ 
going process applies equally to beams of any other shapes, such as the following 
ones; or whether solid or hollow, &c; and of any other materials; so that if we 
have all the dimensions, and the breakg load of any beam whatever, we may find 
that ior another one of the same material, and of the same proportions of cross-sec¬ 
tion. It may become advisable in important cases, to even make one or more model 
beams of some hitherto untried form ; and to break them in order to find the breakg 
weight of the actual beam of the same material. In doing this, the defs should also 
be measd, in order to see whether those of the actual beam may not be too great. See 
Art 26, &c. 

Art. 37. I beams, fig 18; Channels, fig 19; and Deck beams, fig 

20, are made by A. & P. Roberts & Co, Pencoyd lion Works, office, 261 S 4th St, 
Philadelphia; by New Jersey Steel & Iron Co, Trenton, N. J. (Agents, Morris, 
Wheeler & Co, 16th and Market Sts, Philadelphia); by Carnegie Bros & Co, Lini, 
Union Iron Mills, Pittsburgh Pa; by Phoenix Iron Co, Phcenixville Pa, office 410 
Walnut St, Philadelphia; by Passaic Rolling Mill Co, Paterson N. J., office As- 
tor House, New York City ; and others. Price, Pliila. 1886, 3 cts per lb, cut to 
specified lengths not exceeding 30 ft. For extra lengths, punching, framing etc 
address as above. Special discounts on large orders. 

Our tables, pp 522,523, give the principal sizes (includingthe largest and 
smallest) made in America. We give the maximum and minimum thick¬ 
nesses of web for each pattern. Any desired thickness between these extremes 
can be furnished. The width of flange increases equally with the thickness of 
web; but the thickness of flange, at root and at base, for any given pattern, are 
the same for all thicknesses of web. 


* This iron was “ Sterling’s toughened, ‘ having about 16 per cent of wrought scrap melted in it. 
Each of the 89 beams was tested by a center load of 20 tons, which produced defs of from % to % 
iuch. Entire length, 34% ft. 


















w 




■O X 


-Y 

Fig 18. 


S 


Q 


521 


Y 

Fig 19. 


Fig 20. 


In Belgium, I beams are rolled as large as 21.5 ins deep; 334 lbs per vd. 
Agents; J. H. Jackson & Co., 208 Franklin St, New York; Esherick & Co,263 S. 
4ih St, Philadelphia. 

in any bar, beam etc, of wrought iron, of uniform cross section, 

Weight per lineal _ area of cross section, v 
yur<l, in pounds in square inches ^ 

Area of cross section, _ weight of beam per lineal vard, in pounds 
in square inches--■——- : - 

Strength. Beams of good wrought iron do not break under a gradually ap¬ 
plied load until after they have bent so much as to be useless. The ultimate 
load is that one which so cripples the beam that it continues to yield indefi¬ 
nitely without increase of load. The web is supposed to be placed vert, as in 
our figs; and the beam supported at both ends, and stayed against yielding 
sideways; the dist apart of the side supports not exceeding 20 times the width 
of the flange. Then 

Tabular coefficient for cen ult 
Center ultimate load load* for the given size 

in lbs, including half wt of = - in ft - 

clear span of beam clear span in tt 


Extraneous cen ult load = cen ult load so found — 


half wt. of clear 
span of beam 


Distributed ultimate load = twice cen ult load 


Safe cen or distd load = 


ult cen or distd load 


required factor of safety 

The factor of safety should in no case be less than 3. When the load is 
subject to variation (see p 435) or vibration, or is liable to be suddenly applied, 
use from 4 to 6. The factor should be further increased when the length of beam 
between lateral supports exceeds 20 times the width of flange; until, when it 
reaches 70 times said width, the factor should be double that for a length = 20 
X width. 

Caution. With very short spans, the safe loads thus found, although safe 
as regards a ruptvre of the beam itself, may be so great as to endanger a crush¬ 
ing of the ends of the beam, or of the waif etc under them, unless the beam has 
at its ends a greater length of bearing than would otherwise be needed. See 
pp 436 to 438. 

To find what beam is required to sustain safely a given extraneous 
load in lbs, whether uniformly distributed or applied at the center. If uniformly 
distributed, divide it by 2. The quot is the equivalent center load. Then, in either 
case, add half the wt of the clear span of the beam itself. Mult together the sum, 
the required factor of safety, and the clear span of the beam in ft. Use any desired 
beam whose coefficient for ult cen load is not less than the prod. 

For the deflection in ins at the center of the length of the beam, under any 
load less than half the ultimate load : 

„ „ ^. Load in lbs X Cube of clear span in ft 

Deflection ___ ,, - 

in ins = Wt of beam 

in lbs per yd ^ 


Square of depth D v Constant, 
of beam, in ins given below. 


Constants for the above formula: 

For I beam, loaded at center.11200 

“ “ uniformly loaded.18000 

Tlie strength ofa beam or channel when used as a column, 

may be found by the formula, p 439, using the least radius of gyration, as given in 
the following tables. 


For channel beam, loaded at center, 10000 
“ “ uniformly loaded, 16000 


* The coeffs are tbe cen ult loads in lbs (including half the wt of clear span of beam) for beams of 
one ft clear span. 






























522 


STRENGTH OF MATERIALS 


Pencoyd I Beams.— See page 521. 



These three patterns are from the list of the New Jersey Steel «fc Iron 
Co. Trenton, N. J. The 20 inch are the largest, and the inch is the small¬ 
est, now made in America. 

W 

















































STRENGTH OF MATERIALS. 


Pencoyrt Channels.— See page 521. 


a 

s 

fc 


£ 

J= 

c 


30 

31 

32 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 


Q 

x 

<-» 

Cm 

o> 

O 


15 

15 

12 

12 

12 

12 

10 

10 

10 

10 

9 

9 

9 

9 

8 

8 

8 

8 

7 

7 

7 

7 

6 

6 

6 

6 

6 

6 

5 

5 

5 

5 

4 

4 


o = 

x ”7 

•- V 

3 

c 


4% 

4 

m 

2i# 

m 

m 


2U 

$ 


3 
3 

2% 

2 
2 

m n 


2/?t 
2s 5 
2 st 
2% 
2ft 
2M 
2 

2% 

019 

“bf 

2ft 

i-el 

2jV 

% 

2% 

2 

Iff 
1 % 
>34 
i SI 

153 
1 32 

\C 

m 

1-52 


1% 

1 5 5 2 

l5 3 2 


J3 

E-i 


% 52 


% 


n 




K 


Coefficient 

for ultcen load, 

in Sis. 

Least radius of 

gyration, in ins. 

Momet 

lner 

About 

X Y. 

its of 
tia. 

4 bout 
\V Z. 

1,040,400 

1.10 

557.44 

24.74 

84‘.,7u0 

1.13 

451.51 

19.05 

626,400 

.90 

268.51 

12.96 

426,300 

.92 

182.71 

7.42 

404,700 

.72 

173.51 

5.26 

274,200 

.74 

123.71 

3.22 

366,900 

.82 

131.04 

7.13 

259,500 

.84 

92.71 

4.29 

294,300 

.67 

105.16 

3.88 

206,700 

.69 

73.91 

2.33 

282,300 

.67 

90.66 

4.17 

200,100 

.68 

64.34 

2.47 

186,300 

.58 

59.85 

2.05 

135,600 

.59 

43.65 

1.31 

210,000 

.71 

60.00 

4.28 

139,800 

.71 

40.00 

2.17 

143,400 

.60 

41.03 

1.94 

98,700 

.60 

28.23 

1.06 

170,100 

.65 

42.57 

3.08 

117,900 

.65 

29.51 

1.71 

111,300 

.58 

27.86 

1.65 

73,800 

.58 

18.46 

.90 

126,300 ; 

.68 

25.12 

2.56 

85,500 

.67 

18.37 

1.46 

124,200 

.61 

24.35 

2.03 

72.900 

.60 

17.60 

1.15 

90,200 

.52 

16.73 

1.07 

54,300 

.51 

11.67 

.59 

87,400 

.58 

14.20 

1.55 

57,600 

.56 

10.29 

.86 

62,500 

.47 

12.54 

.73 

37,200 

.45 

6.67 

.37 

47,800 

.52 

6.49 

.85 

36,300 

.50 

5.16 

.54 

36,COO 

.49 

4.97 

.57 

28,800 

.48 

4.14 

.41 

21,600 

.47 

2.31 

.42 

18,900 

.46 

2.03 

.32 

10,000 

.43 

.80 

.21 

7,600 

.32 

.88 

.10 

6,700 

.31 

.48 

.08 

2,300 

.17 

.38 

.01 



523 



































524 


BRIDGES OF I BEAMS. 


Rolled Iron I beams for railroad bridges of short span 

(See table, p 522.) A single 7-incli beam, 88 lbs per yard, under each rail, will Shifin 
fur 3 or 4 ft span ; one of 15 inch, 200 lbs, for 8 or 10 ft; two of 12 inch, 194 lbs, sid< 
by side under each rail, as in Figs 50 A and 50 B, for 12 to 14 ft; and two of 15 inch 
145 lbs, under each rail, for 15 ft. By employing a greater number of beams, or b\ 
introducing a truss-rod, r r, Fig 52, p 514, the spans may be increased, or lighte' 
beams be used. Care must be taken to insure lateral stability, by means of hor tie* 
and struts. 




INCHES FEET 


Figs 50 A and 50 B, for which we are indebted to the courtesy of Mr. Jos. M. 
Wilson, C E, Engr of Bridges and Buildings, Penna R R, illustrate a standard 
form of rolled beam girder in use on that road. For spans of 14 ft, each beam is 12 
inch 180 lbs per yd; for 15 ft, 15 inch, 150 lbs. The two beams, B B. forming a 
girder, are held at the proper diet apart from each other by separators, S, each of 
which consists of a short piece of channel iron placed vertically, and having its 
flanges riveted to the webs of the beams. On some roads, cast-iron separators, and 
bolts which pass through them and through the webs of the beams, are used in¬ 
stead. The longitudinal dist, R R, Fig 50 A, between the separators, is about 5 times 
the depth of the beam. 

At the same points, R R, &c, of the span, are placed transverse tie-struts, T, Fig 
50 B, each composed of two 4-inch 21 lb channel bars placed back to back (so as to 
form an I) but % inch apart. Between the two channels are riveted angle-plates, 
A A, % inch thick, the flanges of which are fastened to the webs of the inner beams 
of the girders by the same rivets which hold the separators. At their centers, the 
two channel-bars are separated by a piece of bar-iron, % inch thick X 4 ins square, 
riveted between them, as shown. The cross-ties, notched to the girders as shown, 
give additional lateral support. 

At the ends of the span, the lower flanges of the beams are riveted to rectangular 
“ bolster-plates,” P, which rest upou slightly larger wall-plates, W. 
The rivets are counter-sunk under the plate, P. Both plates are of rolled-iron, % 
inch thick. They are held in place on the abuts by bolts passing through both plates 
as shown. At one end of the span, the bolt holes in the bolster plate are slightly 
elongated, to allow for contraction and expansion. 






























































ANGLE AND T IRON. 


525 


Angle and T Iron. 


1 The following tables give the sizes made by A. «& P. Roberts «*r <’o 
Pen coy <1 Iron Works. Phila. They also make “ Square-root" Angles* 
' Fig 1, from 1 in X 1 in X ^ in to 4 in X * in X % in. Many of the sizes in these 
tables, and others, are made by other mills. Prices, Flula, 1886, approximate: 
Angle iron, 2)^ cts per lb; T iron, 2% cts. 



The sizes A and B are, in all cases, measured from out to out, as 

indicated in the Figs. 

Alib le iron, of any given dimensions, A and B, in the tables, can be rolled 
to any desired thickness between the maximum and minimum thick¬ 
nesses given for that size. The dimensions A and B vary slightly with the thickness. 
Those given are the dimensions corresponding to the minimum thickness. In order¬ 
ing angle iron of thicknesses between the max and min, give either the thickness 
in ins, or the wt in lbs per yard, wanted ; but not both. 

The thickness of each size of T iron is fixed as given in the table, 
and cannot be varied except by changing the shape of the rolls, or making new 
j ones. This is always expensive. It is important, in designing,or in ordering rolled 
' iron of any kind, to bear this in mind, and not to introduce sizes that have to be 
specially made. 

The area in square ins of cross section of any bar of rolled 

. „ its weight in lbs per yd 

iron of uniform dimensions throughout, is =-—--• 

The ultimate or crippling: center load, for a beam consisting 
of a single bar of angle or T iron, supported at both ends, may be found by the 
formula, p 488, using the moment of inertia as given in the following tables. Or, 
approximately, and much more simply, 


Fit cen load 

in lbs 



2800 V -^ rea Cross- v Depth of 
section in sq ins * Beam in ins 

Clear span in ft. 


The ultimate distributed load is twice the ult center load. 

For the safe center or distributed load, divide the ultimate, center 
or distributed load (as the case may be) by the required factor of safety, which 
should in no case be less than 3, and should generally be from 4 to 6, according to 
circumstances. 

Under any load less than half the ultimate load; 


Reflection in ins 

at center 

with load at center 


1 


Load in lbs X Cube of clear span in ft 


68000 X 


Area in 
sq ins 


X 


Square of depth 
in ins. 


Reflection in ins l 

at center > 

with distributed load) 


Load in lbs X Cube of clear span in ft 


108000 X 


Area in v Square of depth 
sq ins in ins. 


The breaking’ load, for a column consisting of a single bar of angle 
or T iron, with flat ends, firmly fixed, may be found by the formula, p 439, using the 
least radius of gyration as given in the following tables. 










































526 


ANGLES AND T IRON 


Angles with equal legs. Fig. 2. 


Dimensions, A 
and B, in ins. 

Thickness, ins. 

tT 

t- n‘ 

a 

*j •<— 

Moment of In¬ 

ertia about XY 

Least Rad of 
Gyrat on in 
ins. 

Dist d from 
base to ueutr’l 
axis, ius. 

Dimensions, A 

and B, in ins. 

Thickness, ins. | 

tT 

>-» 

fee 

c. a 

Es 

Moment of In¬ 

ertia about XY 

Least Rad of 

Gyration in 

ins. 

Dist d from 

base to neutr'l 

axis, ius. 

1 X 1 

1/ 

2.3 

.02 

.20 

.30 

2MX2M 


22.5 

1.23 

.49 

.81 

tt 


4.4 

.04 

.20 

.35 


A 

13.1 

.95 

.55 

.78 

X VA 

i/ 

3.0 

.05 

.26 

.36 

u 

% 

25.0 

1.67 

.54 

.87 

bt 


5.6 

.08 

.26 

.40 

3 X3 

54 

14.4 

1.24 

.60 

.84 

i y 2 x i y 2 

A 

5.3 

.11 

.31 

.44 

44 


33.6 

2.62 

.59 

.98 


% 

9.8 

.19 

.31 

.51 

3^X3^ 

f* 

24.8 

2.87 

.70 

1.01 

X 1 % 

A 

6.2 

.18 

.36 

.51 

it 

/k 

39.8 

4.33 

.69 

1.10 

»( 


11.7 

.31 

.35 

.57 

4 X4 

% 

28.6 

4.36 

.81 

1.14 

2 X 2 

A 

7.1 

.27 

.40 

.57 

it 

% 

54.4 

7.67 

.80 

1.27 

U 

% 

13.6 

.50 

.39 

.64 

5 X5 

TB 

41.8 

10.02 

1.00 

1.41 

2^ X 2^ 

% 

10.6 

.50 

.45 

.65 

it 

i 

90.0 

19.64 

.98 

1.61 

tt 

XB 

17.8 

.79 

.44 

.72 

6 X6 

& 

50.6 

17,68 

1.19 

1 66 

2J* X 2^ 


11.9 

.70 

.50 

.72 

tt 

i 

110.0 

35.46 

1.17 

1.86 


Angles with unequal legs. Fig. 3. 


Dimensions, B 
and A, in ins. 

00 

a 

00 

m 

o 

a 

ja 

la 

H 

Wt per yd, in lbs. 

Moment of 
Inertia. 

Least Rad of Gy¬ 
ration, in ins. 

Dist in ius 
from base 
to neutral 
axis. 

Dimensions, B 
and A, in ins. 

flfi 

p 

a 

09 

09 

a 

o 

2 

Wt per yd, in lbs. 

Moment of 
Inertia. 

Least Rad of Gy¬ 

ration, in ins. 

Dist in ins 
from base 
to neutral 
axis. 


About 

W Z. 

a 

s 

. - 

About 

X Y. 

About 

W Z. 

a 

s 

CO 

X 

u> 

A 

11.9 

1.09 

.39 

.46 

.99 

.49 

5 

X3 

% 

54.4 

13.15 

3.51 

.69 

1.84 

.84 

tt 

s 

22.5 

1.92 

.67 

.46 

1.08 

.58 

5 

X3^ 

/H 

30.5 

7.78 

3.23 

.80 

1.61 

.86 

3 xP/ 2 

t 5 b 

16.2 

1.42 

.90 

.54 

.94 

.68 


“ 


58.1 

13.92 

5.55 

.79 

1.75 

1.00 


k 

25.0 

2.08 

1.30 

.54 

1.00 

.75 

5 

X4 

% 

32.3 

8.14 

4.66 

.87 

1.53 

1 03 

3 Ax*A 

A 

17.8 

2.19 

.94 

.56 

1.14 

.64 


tt 

l 

80.0(18.17 

10.17 

.86 

1.75 

1.25 

bt 

a 

27.5 

3.24 

1.36 

.56 

1.20 

.70 

&AX&A 

% 

32.3 10.12 

3.27 

.81 

1.82 

.82 

33^X3 

u 

21.2 

2.53 

1.72 

.64 

1.07 

.82 


bt 

% 

52.3 15.73 

4.96 

.80 

1.91 

.91 

bt 

% 

36.7 

4.11 

2.81 

.64 

1.17 

.92 

6 

X3^ 

/s 

39.6 14.76 

3.81 

.82 

2.06 

81 

4 X3 

Vs 

24.8 

3.96 

1.92 

.67 

1.28 

.78 


«t 

1 

85.0 ! 29.24 

7.21 

.81 

2.26 

1.01 

t. 

✓4 

39.8 

6.03 

2.87 

.65 

1 37 

.87 

6 

X4 

t 7 b 

41.8 15.46 

5.60 

.92 

1.96 

.96 

4 X&A 

% 

26.7 

4.17 

2.99 

.74 

1.20 

.95 


tt 

1 

90.0] 30.75 

10.75 

.91 

2.17 

1.17 

t. 

% 

43 0 

6.37 

4.52 

.73 

1.29 

1.04 

61^X4 

T^g 

44.0 19.29 

5.72 

.94 

2.18 

.93 

4^X3 

% 

26.7 

5.50 

1.98 

.69 

1.49 

.74 


tt 

l 

95.0'38.66 

11.00 

.93 

2.38 

1.13 

4b 

5/ 

78 

43.0 

8.44 2.98 

.68 

1.58 

.83 

7 

X3 14 

% 

61.7 30.25 

5.28 

.85 

2.57 

.82 

5 X3 

% 

28.6 

7.37 

2.04 

.70 

1.70 

.70 


tt 

l 

95.0|45.37 

7.53 

.84 

2.71 

.96 


T iron with equal legs. Fig 4. 


Dimen¬ 
sions, ins 

A and B. 

Thickues 

Stem, B. 

ssg 8, ins. 

Base, A. 

T3 

fc" • 

a> ae 

a£ 

oment of 
rtia about 
X Y. 

T3 

rt • 

* t 
so 

ce v. 

e £ 

O *■> Ji 

L “ ® 

£ Z o 

*“5 

-•-* z z 


At 

At 

At 

At 

is 

^ p 

J w 

— oj ce 
Q £ 


root. 

edge. 

root. 

edge. 


>—< 


JP 

1 X1 

A 


A 

3 5 2 

3. 

.03 

.26 

.30 

1 A x 1*4 

tt 

TB 

tb 

t 3 b 

4.5 

.07 

.27 

.37 


tt 

tt 

bt 

tt 

tb 

tt 

tt 

tt 

6. 

7.1 

.13 

.21 

.32 

.37 

.45 

.50 

2 X 2 

ftt 

A 

IS 

% 

10.5 

.38 

.43 

.60 

•M x 2*4 

k 

tt 

% 

tt 

11.75 

.52 

.50 

.61 


t r b 

tt 

5 

IS 

tt 

12. 

.54 

.47 

.65 

2^X2^ 

tt 

tt 

1 3 
32 

tt 

17.52 

.97 

.53 

.75 


tt 

tt 

tt 

tt 

19.5 

1.12 

.55 

.75 

3 X 3 

u 

TB 

t. 

t» 

26.0 

2.10 

.62 

.90 

3^X3^ 


bt 

A 

7 

TS 

31.0 

3.47 

.74 

1.00 

4 X 4 

tt 

tt 

x 9 e 

tb 

36.5 

5.26 

.84 

1.14 




























































































ANGLES AND T IRON, 


527 


T iron with unequal legs. (Using the lettering of Fig. 4.) 




Thicknesses, ins. 


♦a 



Dimensions 





*6 

*© o 

T? 

S £ . 

in ins. 

Stem B. 

Base A. 


“ 'S • 

« t* 

« K 

g 3 X 

£ ® c 







L~ 

oj c r’ 

-O 

u"- 


B. 

At 

At 

At 

At 

e -X 

O t- 

0 ) o 

C« 

*r * 

A. 

root. 

edge. 

root. 

edge. 

P* 

^ o 


Q y. 

X> 

2 

9 

IB 

t 5 b 

i 5 b 

k 

X 

X 

5.88 

.01 

.13 

.17 

It 

l 


tt 

tt 

7. 

.05 

.26 

.27 

tt 

VA 

5 

IB 

tt 

A 

% 

tt 

8.75 

.16 

.43 

.43 


IB 

i k 
m 

tt 

TB 

tt 

6.5 

.01 

.12 

.18 

2 M 

tt 

k 

t. 

tt 

9.1 

.10 

.33 

.32 

2% 

% 

% 

M 

TB 

18.75 

.56 

.55 

.66 

tt 

2 

tt 

tt 


tt 

21. 

.83 

.55 

.75 

3 

l l A 

\\ 

% 


X 

11.2 

.19 

.41 

.37 

tt 

% 

A 

ft 

% 

23.8 

1.38 

.63 

.82 

tt 

A 

ft 

IB 

28.25 

3.12 

.61 

1.10 

4 

2 

ft 

5 

TB 

TB 

t. 

20.4 

.68 

.58 

.54 

tt 

3 

5 

« 

tt 

25.25 

2.09 

.82 

.84 

tt 

tt 

% 

% 

IB 

% 

25.9 

1.94 

.86 

.77 

tt 

z'A 

IB 

k 

IB 

41.8 

4.65 

.88 

1.09 


tt 

Vh 

n 


• t 

44.5 

5.27 

.91 

1.16 

5 

2^ 

TB 

i 7 b 

k 

% 

30.7 

1.61 

.72 

.67 

tt 

tt 


tt 

IB 

k 

IB 

k 

33. 

1.63 

.70 

.64 

tt 

3 M 

% 

lB 

k 

48.44 

5.37 

1.04 

1.05 

tt 

4 

3§ 

tt 

tt 

44.1 

6.24 

1.09 

1.08 



/ 






























528 


BEAMS WITH THIN WEBS. 


RESISTANCE OF OPEN BEAMS. 

Beams, «trc, in which all of the longitudinal resistance if 
regarded as being exerted by the flange# 




Art. 1 . On page 486 we saw that the strength ot "closed'' beams, or beams ol 
solid cross-section, is proportional to the squares ot their depths: because the resist 
ing moment of each fibre is proportional to the square of its distance from the neu 
tral axis. 

Art. 2. But, as in an open beam, or truss, ixea Fig 6 (or in a closed beam with 

a thin web w, Fig 22, p 537) we may place at 
many of the fibres as possible in the chords, i j 
and e a Fig 6 (« and b Fig 22) or as far as possi¬ 
ble from the neutral axis n Fig 6, so that they 
may exert their maximum resistance. The 
chords are thus made to bear most of the hori¬ 
zontal or longitudinal strain. For conveni¬ 
ence’s sake we assume that they bear it all 
For the resistance of the web, see Art. 5, p 529. 

It is plain that the breaking moment of the 
load is the same as in the closed beam Fig 2, J 
479, — (Fig 6) the load X its leverage ( — ea oi 
ix) about the neutral axis n. But the resisting 
moment of the beam now’ consists of 



( longitudinal tensile its leverage in* 

resistance of the X about the neu 
upper chord i x tral axis n 


>\ /longitudinal compres- its leverage s 
- J + (sive resistance of the X ne* about the ] 


\ lower chord e a neutral axis n) j 


sum of the longitudinal re- 
sistances of the tw o chords 


X half depth* of beam 


But w'e may express this more simply by regarding the neutral axis or fulcrum at 
being at the inner end i or e of one of the chords ; and by writing 


Moment of resist¬ 
ance of beam 


longitudinal resistance of 
either one chord ix or ea 


X 


whole depth 
* of beam 


xe 


So long as the beam sustains its load, the longitudinal resistances of the two chord? 
are equal, although the longitudinal strengths of one or of both may be much greatei 
than this resistance. If the strength of either chord becomes less than 


Resistance = 


moment of rupture 


depth* of beam 


whole depth* 
i e of beam 


then that chord fails, and the beam gives way. 

Hence the 

maximum or ulti- ultimate longitudinal 
mate moment of = strength of the weaker X 
resistance of beam chord ix ore a 

In “closed beams,” as explained on p 485, the longitudinal tensile and com 
pressive resistances exerted by the fibres against rupture, are aided, to an iniportun 
extent, by the mutual adhesion of the fibres, which resists their sliding upon eacl 
other; for without such sliding, rupture cannot occur. But in “open beams,” such ai 
we are now treating of, the thickness of the flanges or chords is so slight, comparei 
with the depth of the beam or truss, that the sliding betw’een their fibres become! 
insignificant. The resistance due to the adhesion of the fibres to each other ii 
therefore neglected, and the flanges are regarded as sustaining only longitudina 
tensile or compressive strains. 

Art. 3. Let Fig 6 be a hor open beam 1.5 ft deep from i to e, and projecting 
ft from a wall into which it is firmly fixed by its flanges or chords i and e; an , 
let the concentrated load E at its outer end be 1 ton. This load tends to pull th 
beam into the dotted position by stretching or tearing apart the fibres of th 
upper chord at i. Now with how great a moment of rupture does it tend to d 
this, and how strong must the fibres of the chord be at i in order that their me 
ment of resistance may oppose it safely ? We shall here leave the wt of the bear 
itself out of consideration. When required to be included see Case 6, p482. 

Regard the lines i e and e a as the two arms of a bent lever resting on its fill 
crum e. This lever is plainly acted upon and balanced by two equal moment! 
one at each end a and i; namely at a Jne moment of rupture of the load, equal t 
(1 ton X 6 ft leverage ae) = G ft tons; and at i the resisting moment of the bean 
equal to the hor pull or st rain on the fibres at the chord i X 1-5 ft. leverage i . 
But we do not yet know what amount of hor pull by the fibres at i is required t 


nro monsiir.-rt lmm till! C.liters of aravitu of cross section of Mi 



























'• =?■ S’-* 


BEAMS WITH THIN WEBS. 


529 


is 

of 

St- 

en- 

ith 

ns 

it 

■si- 


balance the moment of the load. It is however very easily found by merely divid¬ 
ing the 6 ft tons moment of the load by the 1.5 ft leverage of the fibres, that is, by 
the depth of the beam. Thus we get (6-f-1.5) = 4 tons pull at i ; and we then Have 
the 6 X 1 — 6 ft tons moment of the load, balanced by the 1.5 X 4 = 6 ft tons mo¬ 
ment of the fibres. Therefore in order just to balance the moment of the load, 
the chord at i must be strong enough to bear a hor pull of 4 tons; or for a safety 
of 3, 4 or 6, Ac, strong enough to bear a pull of 12, 16, or 24, <fcc, tons. The web 
members of course carry this 4 ton hor pulling strain from the upper chord to the 
lower one upon which it acts as an equal hor compressing one. 

In shape of a formula the above stands thus. 


Hor strain at any Moment of load Load X its leverage 

point in a hor flange _ at that point at that point 

of an open cantilever Depth of beam — Depth of beam 


Hence if we know the size and of course the ultimate longitudinal tensile and 
compressive strength of the flange or chord, we have by transposition the ulti- 
““ mate or breaking load of the hor open beam, thus, 

Breaking load at any _ Ultimate strength of flange X Depth of beam. 
point of a hor open cantilever Leverage of load at that point. 


And for a safety of 3, 4 or 6, &c, we have 

Safe load — ^ l/ ± or 3^> &c, ttie ult strength of flange X Depth of beam. 

Leverage of load at that point. 


Art. 4. Also in a hor open beam or trnss supported at both 
is ends, after having found the moment of the load at any point (by “moments,” p 
479, &c) the strain on the beam as also its load in lbs or tons are found in the same 
way or by the same formulas. 

Rein. 1. The longitudinal strains on the flanges of hor closed beams 
with thin webs such as common rolled I beams, as well as their loads, are also fre- 
s quently computed in this same ready way, instead of the more troublesome one, p 488. 

- The webs are left entirely out of consideration as regards the hor strains. Although 
not strictly correct, it is sufficiently so for ordinary practice, and is safe. With these 
assumptions the dimensions or sectional areas of the top and bottom flanges are 
proportioned to the safe unit strains of the material. Thus Hodgkinson having 
found that the ultimate compressive strength of cast-iron averaged about 6 times 
as great as its tensile one, gave his upper flange only one-sixth the area of the 
lower one, in order that both should be equally strong. In wrought-iron the ten¬ 
sile strength is somewhat the greatest, which would lead to making the lower 
flange the smallest, but here this consideration is outweighed by the practical 
ones of greater ease of manufacture and of handling or placing which require 
equal flanges. 

Rem. 2. If the flanges are not horizontal, although the beam or 
truss itself may be so, the longitudinal strains on the flanges will be increased; 
and the transverse or shearing strains on the webs will also be changed as stated 
in Art 6. If the beams are inclined, modifications arise which we shall 
not treat of. Strangely, most of our standard authorities on bridge building do 
not even allude to them. 

Rem. 3. The principle of the bent lever in open beams explains why the 
strength of a truss is as its depth, (the length of its vert lever-arm) 
instead of as the square of its depth as in closed beams. The strength however is 
inversely as the length in both kinds. 

Art. 5. The web members of an open beam or common truss like Figs 
10 and 11, p 558, uniformly loaded, carry the vert or shearing forces of the load 
and beam from the center each way, up and down alternately from one chord to 
the other, until finally the end ones deposit it as load on the supports or abut¬ 
ments. For each member receives and carries its share of the shearing force in 
the shape of an end load, thus changing the shearing tendency into an alternately 
pulling and compressing one according as the alternate members are ties or struts. 
In doing this any web member that is oblique is (on account of its obliquity) 
strained to an extent that exceeds its load in the same proportion that the oblique 
length of the member exceeds the length it would have had if it had been vert, as 
explained in Art 11, p 557, &c. This excess of strain over the load on the obliques 

35 










530 


BEAMS WITH THIN WEBS, 


exhibits itself at their ends as hor pull along one chord, and hor compression along 
the other; and these hor strains on the chords are the same as those fouud by mo¬ 
ments. Thus it is seen in Figs p 558 that the hor strain at the center of each chord 
(as there found by tracing up the diff loads or shearing forces in their journey along 
the obliques) is set down at 16 tons. Now’ the whole distributed load on one truss 
of 64 ft span, and 16 ft depth, is 32 tons ; and by Case 10, p 483, we find that at the 
center the moment of this load is 16 X 16 = 256 tons; and this divided by the depth 
of our open truss or 16 ft gives 16 tons for the hor strain at center as before; and 
so at other points. The oblique web members are plainly the only ones that can 
convey their loads laterally, that is in directions tending tow ard or from the abut¬ 
ments. Vertical members merely convey their loads vert up or down from one 
chord to the other, at which last they transfer them to.oblique members which 
can convey them laterally. If both a pull and a push act at once in opposite 
directions on a web member, their diff is the actual strain. 


Rem. As a mat ter of economy in small spans it is often better not to 
proportion the sizes of the individual members to the strains they have to bear; 
out to give to the flanges throughout their entire length the same dimensions as 
are required at their most strained part, namely, at the center; and to make all 
the web members as strong as the most strained or end ones. This avoids the 
extra trouble and expense of getting out and fitting together many pieces of 
various sizes. 


Art. 6. Oblique or curved flanges. We have hitherto supposed 
the beams and their flanges to be horizontal; but a beam may be hor, and yet 
have one or both of its flanges oblique or curved as 
at A and B. In such cases the longitudinal strains 
along the flanges become greater; and the vert or 
shearing strains across the web in most cases less. 

See Rem at end of Art. It is plain that such flanges 
must as it were intercept to some extent (depending 
on their inclination) the vert force at any point, 
and convert it into an oblique one along the flanges, 
somewhat as the oblique web members of an open 
beam do. 

To find these new strains at any point o, 

Figs A, B, of either an upper or lower oblique or 
curved flange, first ascertain by “ Moments,” the 
hor strain at that point for a beam with the depth 
oe; and by “Shearing,” the shear also. 

Then from that point o draw a hor line h equal by 



scale to the hor strain; and from its end draw?; 


vert and ending either at the flange (produced if necessary) if straight as in A; 
or at a tangent l from o if the flange is curved as at B. Then will l in either fig 
give by the same scale the longitudinal strain along the flange at o; and h and v 
are the components of that strain. As a formula, the Rule reads thus, o being the 
angle formed by h and l at o. 


mom of rup 


Strain along . . , 

oblique Mange ~ hor strain - cosino of 0 = depth of beam 


cosine of o. 


Here v shows by the scale how’ much of the vert or shearing force has been con¬ 
verted into a longitudinal one; and if it be taken from the total shearing force 
before found, the remainder will show how much of said force still operates on the 
web at o. For exceptions, see Rem. The foregoing applies also to oblique flanges 
of open hor beams. 

In tli<» hor triangular flanged brain D with a concentrated load at 

its free end, draw a o vert and equal by scale to the load, 
and draw o c hor. Now here the whole load rests upon the 
upper end a of the oblique flange a n. which therefore sus¬ 
tains all of it as an end load, which it deposits as vert press¬ 
ure at n , and thus entirely prevents it from exerting any 
shearing force whatever upon any part of the beam. The 
shaded web is therefore of no use here. The line a c meas¬ 
ures the strain along the oblique flange; a o the vert pressure 
atra; and o c the hor pull of the load all along the upper 
flange a e. Also a o and o c are the components of a c. 

So also iu Figr E, with a concentrated load l suspended by a string from c. 
The string carries the load up to the two oblique flanges c a, c b, which convert 
its shearing tendency into two oblique pulls along themselves. At 

































































BEAMS WITH THIN WEBS. 


531 


the abutments or supports these pulls along cor and cb become converted into ver¬ 
tical pressures, together equal to the load l; and into hor pressures compressing a b. 

Here also the shaded web is unnecessary; as would 
likewise be the case if the load were transferred to e, 
and a single vert post (shown by the dark line) provided 
to carry it down to c, as the string before carried it up 
to c. If there is no such post the web acts, and the 
strain on either oblique flange is found as for A and B. 
But it is only in a few similar cases that the oblique 
flange entirely supplants the continuous web. 

Humber gives for timting the strain at 
any point of an oblique or curved web as follows. 
First find the shear as before as for a horizontal 



flange. Then 

If the compressed flange is inclined down to the nearest support, or 
Jf the st retched flange is inclined down from “ “ 

take the diff between the vert component v and the shearing force. But 
If the compressed flange is inclined down from the nearest support, or 
If the stretched flange is inclined down to “ “ “ 

take the sum of the vert component v and the shearing force. 


Rem. Hence in these last two cases (which do not include any of our above 
figs) the vertical force on the web is increased. 

As Humber remarks, in girders or beams with curved or oblique flanges the 
greatest strain in the web is not always where the greatest shearing strain is 
produced. 







532 


VERTICAL STRAINS IN BEAMS, ETC, 


VERTICAL STRAINS IN BEAMS, ETC. 


VERTICAL. OR SHEARING STRAINS IN BEAMS, ETC. 
Art. 1. When a loaded beam, a v. Fig 1, rests upon two supports, i and »% the' 


weight of the beam and load between the supports, and the upward reactions of the 

two supports, tend not only to bend the beam, as in *ig 
3, p 550, but also to cut or shear it across vertically, as 

* Fi 1 l 

1D In practice, beams rarely fail by shearing 1 , 

and then only when heavily loaded close to their points 
of support, as in Fig 1. 

But the xert forces which fcnrt to pro- 
<1 uce shearing exist in all beams ami 



trusses. In the latter, they cause all the strain on the verts and a part of that 
in the obliques (whether web members or flanges); and in beams with thin \\ebs 
and considerable area of flange, such as box and plate-girders, ps 537, &c, it is usual, 
and sufficiently (although uot strictly) correct, to assume, for convenience, that all 
of the vert strains are borne by the i veb, while the flanges are regarded as resisting 
only the longitudinal strains. The following instructions are therefore given for 
finding the amount of vert or shearing strain in the different parts ot a beam under 
different conditions of loading. For the sake of simplicity, wo neglect the wt of 
the beam itself, unless otherwise stated. | 

Art. 2. Imagine the beam, av,Fig 2 (supported at both ends, and , 
loaded at its center with 3 tons), to be divided into a number of slices,' 

u s t, Ac, by the vert planes whose edges are 
shown in the fig. If we take any two adjoin-:; 
iug slices, as m and n, to the right of the load, j 
it is plain that we may regard the Ze/Miand 
slice, tw, as being pressed dowrnoard by that ; 
portion (1^ tons) of the load which goes to, 
the right-hand support,r,while its neighbor-, 
ing slice, n, on the right, is upheld by the 
equal upward reaction of the right-hand sup¬ 
port. r. There is, therefore, a vert strain, 
equal to 1)4 tons, or to one of these forces, 
tending to shear the beam across on the vert 
plane separating the two slices; and this ten¬ 
dency must be resisted by the cohesive force 



of the beam in that vert plane.* Since the downward pressure exerted upon the 
right-hand support undergoes (in this case) no increase or diminution between the 
load and the support (there being no intermediate load or support), and since the, 
upward reaction of r is also exerted, unchanged, at every point between r and c, it 


follows that the shearing strain is equal at all those points. And since the load is 


at the center of the beam, the upward reaction of the left abutment, l, is equal to that 
of r, and there must, therefore, be an equal shearing strain of 1% tons at each point 
in the /e/f-hand half of the beam. In other words, a load, concentrated at 
the middle of a beam supported at both ends, exerts a uniform 
shearing strain, equal to half of said load, throughout the beam. 

Art. 3. But while, to the right of the load, each slice tended to slide downward 
past its neighbor on the rigid; the reverse is the case to the left of the load; each 
slice there tending to slide downward past its neighbor on the left. In other words, 
to the rigid of the load the vert downward strain on each section comes from the 
left , and vice versa. 

Art. 4. At the vert section immediately under or over a coneen- 
trated center load there is. strictly speaking, no shearing tendency between 
the two slices to the right and left of that section, because they evidently have no 
tendency to slide past each other. But there is, in the two slices, a combined crush¬ 
ing strain equal to the entire load; because each of them sustains half the load, and 
is pressed upward by the equal reaction of the abut. If we suppose a v to be a truss, 


* So long as the joint between m and n remains intact, n is of course also pressed downward by 
the half load, and m is also upheld by the support. But this does not affect the shearing strain in 
the joint, because. In order that n may receive the downward pres of the half load, said pres must be 
transmitted to it from m through the joint in question ; and so with the upward reaction transmitted 
from n to m. It is the transmission of the original action and reaction through uuy given joint that 
causes the shearing strain in it. 





























VERTICAL STRAINS IN BEAMS, ETC. 


533 


and the vert line between k and m to represent a vert post, then said post will have 
to bear the combined crushing strain, which, in the beam, comes upon slices k and 
m (= the entire load of 3 tons), and its pin at c in the lower chord will sustain a vert 
shearing strain of 3 tons in addition to the lior shearing strains from the chord. 

, Art. 5. If, in Fig 2, we make a l and vr eacli equal, by scale, to the upward 
• reaction of its abutment, or (in this case) equal to half the load; and draw Ir lior; 
tlj then vert lines (of uniform length in this case) drawn between a v and Ir will give 
tij the shearing strain at each point in the beam, except of course at the cen, c, as ex¬ 
it plained in Art 4. 

yj Art. 6. Figs 3 and 4 show the application of tlie foregoing- to 
common forms of trusses. In each fig, a load of 3 tons is supposed to be 
n|J sustained entirely by only one truss of the span. The membeis sustaining tension 
jinj are shown by light lines; and those under compression, by heavy lines. It is plain 


C 



that each vert member, in either fig, is strained tons.* Each oblique sustains a 
total strain greater than tons ; but the vert comp of each is only tons. Also, 
'o the downward strain on each web member.to the right of the load comes from the 
left, and vice versa. We see, also, that while the two central obliques have no ten- 

I dency to slide past each other, their combined vert strain (compression in Fig 3; ten* 
:i ! sion in Fig 4) is equal to the entire load; and the pin at c in either fig sustains a 
lil 1 vert shear equal to the entire load. And the cen vert rod in Fig 3 sustains a pull 

equal to the entire load, or 3 tons. Drawing a l and v reach equal to the upward 
s reaction of its abut; Ir hor; and the vert dotted lines; the latter give the vert 
strains (uniform in this case) at the several panel points, except of course at the cen, 

II as just explained. 

!1 | The arrows pointing downward represent the downward pressures caused by the 
H load; while those pointing upicard represent the upward reactions of the abuts. In 
“ Fig 3 the downward pres at each section is applied at its foot, and the upward pres 
' • at its head; and the vert members are therefore in tension. In Fig 4 the reverse 
W Of all this is the case. The directions of the arrows should be carefully noted, as 
jrj they have an important bearing upon what follows. The lengths of the arrows indi- 
il cate the amounts of the forces which they represent; and their positions show 
tl whether those forces are applied at the upper or lower chord, and whether the force 
tl comes to the section from the right or from the left. 

! Thus, in Fig 3, the arrow pointing downward, immediately under the load, and at 
(1 the bottom of the diagram, shows that the force represented by it is applied imme- 
I) diately at the joint (coming neither from the right nor from the left) and that said 
joint is in the lower chord. 

I This force is equal to the entire load, or 3 tons. The two equal arrows lmme- 

II diately to the right and left of the cen line, and at the top of the diagram, are made 
each half as long as the central arrow just referred to, in order to show that the 

11 force represented by each is half as great as that represented by the central arrow, 
it These two arrows represent the upward reactions of the two abuts, coming from 
the right and left, respectively, and meeting at c in the upper chord. 

11 Item. A single concentrated load produces its greatest 
: shearing strain when placed at one end of the span, immediately over the 

I ed<>-e of an abut; at which point the shear is then equal to the load; but there is 

II then no shearing strain in any other part of the beam. As the load moves along iho 
1 beam, the shear in front of it increases uniformly, and that behind it decreases uni¬ 
formly until the load reaches the other end of the span. At whatever poi nt 

the load may be, the shear at any instant is uniform 
throughout either one of the two segments into which the load 
divides the span. See Figs 5 and 6. 

* Except that the center vertical member, in Fig 3, belongs, as it were, to both 
halves of the truss, and therefore performs the same duty as two side verts, by 
supporting the entire load of 3 tons = twice the halt load, as explained in 

Art. 4. 

























534 


VERTICAL STRAINS IN BEAMS, ETC 


Art. 7. In Figs 5 and 6, the concentrated load Is not at the 
ceil of the beam. Therefore the upward reactions of the abuts are unequal, 
as are also the shearing strains in the two portions of the length of the beam. 

Hor dist from cen of 
1 lie reac- load grav of load to the 

lionot = 
either abut — 



other abut 



Fig. 7 


a 



b 


V 

"4 

_ 

I 

_j_ 

—i— 

— 

7 


Fig. 3 


a 



w 

1 




+ 

I 


•F 

i 

i- 

i 


-+ 


+■ -t + 
L 1_1 


Fig.O 


a 


W 


i 

i 




I 

-+ 

I 

-4 


I 

4 

i 

4 

i 


-i_j 


L1J 

Fig.lO 


upward reaction of abut at a 
between v and I* —upward reaction at v = 9 


Span. 

Art. 8. In Fig 7 we have two concentrated 
loads of 3 tons each, and each placed at a dist 
from an abut equal to one-third of the span. 
Here it might be supposed that the shear at 
any point might be found by simply adding 
together its shear by Fig 5 and that by Fig 6; 
but this is not the case, except for those points 
between an abut and the load nearest to it. At 
such points the vert forces in Fig 5 and those 
in Fig 6 act in the same directions, and thus 
assist each other when combined as in Fig 7; 
while at the sections between the tiro loads, 
they act in opposite directions, and consequent ly 
counteract each other. 

Thus, if we compare section c in Figs 5 and 
6, we will see that in Fig 5 slice m tends to slide 
downward past k ; while in Fig 6 the reverse 
is the case; so that in Fig 7, which may be 
regarded as a combination of Figs 5 and 6, these 
two equal and opposite shearing 
tendencies counteract each other, 
and there is consequently no shear at c. 

Art. 9. Similarly, in Fig 10, the shear in 
W D is equal to the difference (1 ton) between 
those (1 ton and 2 tons, respectively) in W D in 
Figs 8 and 9; while that in a W is equal to the 
sum (5 tons) of those (1 and 4) in a W in Figs 8 
and 9; and that in D v is equal to the sum (4) 
of those (2 and 2) in Dr in Figs 8 and 9. 

Art. 10. The following general rules 
are illustrated by the foregoing: 

Rule 1. The shearing or vert strain at 
any point of any beam, fixed or supported at 
one or at both ends, and loaded in any manner, 
is equal to the diff between the upward vert 
reaction of either abutment, and any load or 
part load on the beam between that abut and 
said point. To find the upward reaction of 
either abut, see Art 7, above.. 

Killed. Let all that part of the load to 
the rigid of the given section be called R; and 
that to the left of it, Ij. Then the shearing 
strain at that section will be equal to the diff 
between that portion of R that goes to the bft- 
haud support, and that portion of L> that goes 
to the right- hand support. 

Rem. 1. In applying either of these rules 
to a section immediately under or 
over a concentrated load, as at W, 
Fig 9, or concentrated portion of a load, as at 
W, Fig 10, we must, theoretically, consider the 
section as being the dividing line between the 
two portions of said concentrated load or part 
load which go to the two abuts respectively ; 
and must regard said portions as forming parts 
of the loads on the twm portions into which 
the given section divides the beam. 

For instance, in Fig 10, at any point, P, be¬ 
tween a and W, Rule 1 gives shearing strain = 
load between a and P — 5 — 0 = 5 tons; or, — load 
4 = 5 tons; and Rule 2 gives shear* 


7 


V 


w 


44 

i 



































































VERTICAL STRAINS IN BEAMS, ETC. 


535 




Wl 


.jNX 

W' => 


7 


1 ing strain — portion of R going to a — portion of L going to v — 5 — 0 = 5 tons. 
But, at W, the wt of 6 tons, resting there, divides; two-tinrds of it, or 4 tons, 
going to a , and one-third, or 2 tons, to v. Therefore “ L ” (or load to the left of 
W) = §W = 4 tons ; and “ R” (or load to the right of W) ='the remaining 2 tons 
of W, + D, 3 tons, = 5 tons. Here, Rule 1 gives shear at. W = upward reaction 
at a — load between a and W = 5 — 4 = 1 ton ; or, shear at W = load between v 
and W — upward reaction at v = 5 — 4 = 1 ton ; and Rule 2 gives shear at W = 
one-third of D, or one ton of R going to the left, abut at a. 

But in practice it is safer to neglect sucli refinement, and 
to consider the section W as having a shear equal to the greater of the two shears 
on each side of it, or, in this case, equal to the shear at any point between a and W, 
or 5 tons. 

The following are applications of Rules 1 and 2: 

Rem. 2. The shearing force at any cross section of a cantilever (a project¬ 
ing beam fixed at one end and free at the other), no matter how the load is disposed, 
is equal to the wt of that part of the beam and its load which is between said section 
and the free end. 

Art. 11. When a beam, av, Fig 11, supported at liotli ends, is 
uniformly loaded throughout, the shearing strain is greatest at the 
abuts: at each of which it is equal to half the load. From 
each support it diminishes uniformly to the center, where it 
is zero. Therefore, if we make a l and v r each equal, by scale, a 
to half the load, and from l and r draw straight lines, Ic and 
r c, to the center, c, of the span ; then a vert line, as o s, drawn 
from any point, o, in the beam, to either line, as Ic, gives the l 1 
shearing force at said point, n. 

Art. 12. When a beam, a v , Fig 12, supported at 
both ends, is uniformly loaded from one of its supports, as r, 
part way across, say to n, as when a train, as long as the span, or longer, 
comes part way upon a bridge, the greatest 
shear is at the abut, r, and is equal to the 
portion of the load borne by that abut. From 
that point it decreases uniformly to zero at a 
zero point, z, which is always within the load 
itself. To find the dist, n z, of the 
zero point, z, from n, when the load 
extends from v any given dist, as vn, toward 
a, 

Twice . Length, vn, of bridge . . Said covered , ng or nz= vn 
the span * occupied by load * " length, vn ' 2av' 

The zero point of shear is also the section of greatest moment of rupture. 

From the zero point, the shear again increases uniformly to the end.n, ot the 
load, and at the same rate of increase as from z to v. At n, and from n to the 
other abut, a, the shearing strain is equal to the portion of the load sustained 

b 'Vrt! ia” 1 ’ Fig 13 shows the shearing strains which take place 

e 


Fig. 11 


Nr 


W 


i 


7 - 


a 


n 


11 z 

3^1 


m 


V 




! I ! I j' r 

l 


Fig. 13 


Nj 


1 N 


-'1 


i \ 


III, 

X} ! I ii i_ 

A ! ! L*-r • • ! ! x ™ 

/! I y\ \ i i i i i 'iv 

/ ! ! z A'' i ! ! ! i i | | ! : ~~N— _<Z 

— ii r-7 ii 5 4 * NL NX 


Fig. 13 








_ _- 


mccesslvelv at a given point, fi, in a bridge, 0-8, while a train, as long 
is the bridge, conies upon it, passes across it, and leaves it, all in the same direc i<> . 



























536 


VERTICAL STRAINS IN BEAMS, ETC. 


It also shows the successive shears at each abut during the passage of 
the train. The train is supposed to move in the direction of the arrow, or from 
right to left, and, for convenience, is supposed to l>e of uniform wt per ft run through¬ 
out. The several shearing strains are found by Art 12. The vertical distances, 
S-e and 8 -g, are made each equal, by scale, to half the wt of the train, which is 
equal to the shear at 0 or at 8 when the head of the train is at 8; and when, con¬ 
sequently, the train just covers the span, as in Fig 11. (It must be borne in mind 
that we neglect the wt of the bridge itself.) 

The vert lines, 6 -d, 11-/, Ac, show, by the same scale, the amounts of shearing 
strain at 8 when the head of the train comes, respectively, to 6,11, Ac. Similar vert 
lines, drawn from points in the hor line, 16-0, to the lower curve, 16-gr-0, would show 
the corresponding vert strains at 0; and the heavy vert lines, 4-m, 6 -d, 8-h, 10-», 
Ac, from 16-0 to the heavy curve, 16-14-ci-0, show the successive shearing strains at 
the point 6, as the head of the train reaches 4, 6, 8, 10, &c. 

It will be noticed that at each abut the shear, just before the train touches 
the bridge at 0, is zero ; and that it increases until the train just covers the bridge, 
when it is equal to half the wt of the train, as in Fig 11. It then decreases, reaching 
zero again when the rear of the train leaves the bridge at 8.* But at any interme¬ 
diate point, as 6, the shear increases from zero until the head of the train comes to 
said point. During this time the shears at the point are the same as those at the 
abut 8 beyond it (see n. Fig 12), and consist solely of shearing strain passing through 
it to that abut. But now the point, 6, begins to pass strains to both abuts, and con¬ 
tinues to do so until the rear of the train has passed it. These opposing strains 
partly neutralize each other (see Art 8); and the resultant, or remaining, shear at 6 
diminishes until the head of the train reaches such a point, z , that 6 becomes the 
zero point. It then again increases until the head of the train reaches 14. The rear 
of the train is now at 6. From now on all the shear going to 0 (and no other) passes 
through 6. Consequently, the shear at 6 is, from now on, the same as that at 0, aud, 
like it, decreases, and becomes zero as the rear of the train leaves the bridge at 8. 

The greatest shear that can occur at any given point is when the 
longer segment of the span is loaded to that point. It is then greater at that point 
than when the whole span is loaded. 


* But, as indicated by the two curves, O-e-16 and 0-#-16, the shearing strains at 
the two abutments, 8 and 0, do not increase and decrease uniformly or equally. 
Thus: at the left abutment 8 (see upper curve O-e-16), the increase of shear, as the 
train comes upon the bridge, is at first slow, and afterward more rapid as the head 
of the train approaches 8. Similarly, the decrease at 8, as the train leaves the bridge, 
is at first slow, and afterward more rapid as the rear of the train approaches 8. At 
the right abutment 0, the exact reverse of this is the case. The shearing strains at ! 
the two abutments are not equal at any given moment except when the train cover* 
the entire bridge, and when there is no train on the bridge. 





RIVETED GIRDERS. 


537 


RIVETED GIRDERS. 

Art. 1. Plate girders. Figs 21,22, and 23: and box-girders. Figs 24 
and 25; with hor flanges,and supported at both ends. In these.it is usual,and suf- 



<t 




V 




Fig'. 23. 

ficiently correct, to assume for convenience that the flanges sustain all of the hor 
strains, and no other; and that the web sustains only the vert strains. 

On this assumption, 


The total hor compressive 
or tensile strain, in lbs, in 
either tlange, at any point in 
the span, is 


Moment of rupture, in ft-lbs, at that point, 
as found by pp 481, &c 

The vert dist in ft between the centers of 
gravity of cross-section of the two flanges 
at the same point. 

If the moment of rupture is in inch-tons, the vert dist must be in ins, and the 
flange strain will be in tons, &c, &c. 

Flange strain per square inch ( = Total strain in one flange, found as above 
of cross-section of flange j Area of cross-section of one flange in 

sq ins. 


Art. 2. The total strains on the two flanges at any given point in the span, are 
plainly equal to each other ; but that on the upper flange is of course compressive; 
and that on the lower one, tensile. 

Inasmuch as the vert dist between the cens of grav of cross-section of the two 
flanges is approximately the same throughout the span, the strain on either flange 
at any point may be taken as proportional to the moment of rupture of the girder 
at that point. Therefore, if we draw a diagram of these moments, as directed for 
various methods of loading, in pp 482, &c, and let any one of the ordinates equal 
by scale the flange strain at that point; then the other ords will give, by the sum* 
scale, the flange strains at their respective points. 

Art. 3. Areas of cross-section of flanges. 


Minimum allowable area ' 

in sq ins of effective cross- 
section of lower flange at 

any point in the span. 


Tensile strain in lbs on the flange at 
tlie given point, as found by Art 1 

Safe tensile strength of the metal in 
lbs per sq inch 












































538 


RIVETED GIRDERS. 


The* effective cross-section of the lower flange is = the cross 
section of the flange plate, plus that of both legs of both the angles that fasten it to the 
web. minus the rivet holes. When the rivets are staggered, as in Fig 25 A, it is 

usual to consider the effective section as equal to a 
net section taken on a staggered line, X A B C D Y, 
between the rivet holes, the oblique lines, A Band 
C D, being taken at only three-fourths of their actual 
lengths. But if this should give a greater area than 
would be obtained by deducting the area lost in all 
four of the holes, ABC and D, from that of the full 
cross-section on the straight line, X Y, then this last 
is taken as the effective area. Fig 25 A is a plan of 
part of the lower flange of a plate girder. F F are 
the projecting edges of the flange plate. H II are :■ 
the horizontal legs, and V V the vertical legs, of j 
the angles by which the hor flange plate, F F, is 1 
joined to the vert web, W, of the girder. 

For girders supporting quiescent loads, such as walls, floors, &c, the safe ten¬ 
sile strength is usually taken at about 10000 lbs per sq inch for wrought-iron 
such as is used in girders; but, for railroad bridges , from 6000 to 7000 lbs per sq inch. 

Having decided upon the shape of the upper flange, its area of cross-section 
may. be found by means of the rules for iron pillars, p 430 etc. 

If the flange is of the usual T shape: 

Square of least radius) _ (Approx) square of width of flange 
of gyration J in ins -=- 22.5. 

The effective cross-section of the upper flange is equal to its entire sec¬ 
tion, because the loss of metal occasioned by punching the rivet holes is compen¬ 
sated by the rivets themselves, which also resist compression. 

Art. 4. The width of the upper flange is governed by its length of 

the greatest longitudinal distance between those supports which prevent the girder ; 
from yielding sideways. Thus, in railroad girders, these supports consist of the j 
transverse bracing; and, in order that the flange may contribute sufficient lateral i 
stiffness to the girder, it is usual, in single-web girders, to make its width not less | 
than about one-twelfth of the greatest longitudinal distance between the points of 
attachment of that bracing. In buildings, where the load is quiescent, one-twentieth I 
is the usual minimum. In these, however, there is frequently no sideways support | 
between the abutments, so that the proper width of flange becomes one-twentieth 1 
of the span. 

Since the lower flange is in tension , its width is a matter of minor importance (pro- I 
vided sufficient area is given), and is generally fixed by considerations of practical 1 
convenience. 



Fig. 25 TV 


Art. 5. As the moments of rupture increase as we proceed 
from the ends of the span toward its center, the total flange strains, 
and the required area of cross-section of flange to withstand them, increase in the 
same proportion. The width of each flange is usually made uniform throughout 
the span, and the required increase of area is given by increasing 
their thickness. This is done by increasing (toward the center) the 
number of plates of which each flange is composed. The plates and 
angles composing a flange are riveted together throughout their 
length. When a flange-plate, or flange-angle, is too long to be 
made in one piece, the two lengths which compose it must be 
, joined, where they abut together, by special splices or “covers” 

—i-•— 4 (see butt-joints in “ Riveting,” p 469). In thus splicing the angles , 

, . „„ „ h h, Fig 25 B, rolled “ angle-covers,” cc, about two feet long. 

Fig. 25. B. are usod. 


















RIVETED GIRDERS. 


539 




Art. 6. Rivet*. The vn-t. rivets in the flanges, which join together the 
flange-plates and the hor legs of the flange-angles, are generally so spaced that their 
“pitch,” or dist apart from cen to cen in a line parallel with the length of the 
girder, is lrom 2*4 to <5 ins. Where the flange is made up of two or more plates, as 
in Art 5, the added plates are made so much longer than the length required by the 
moments of rupture, as to give room, outside of said required length, for a sufficient 
number of rivets to transmit to the plate its share of the longitudinal flange strains. 

The hor rivets joining the vert legs of the flange-angles" to the web-plate; and 
those used in splicing together the several lengths of the web-plate; may be spaced 
by the following rules : 


Greatest allowable strainl _ i^oon V Crippling area of 

on each rivet, in lbs ) ^ one rivet, in sq ins. 

The crippling area of a rivet, in sq ins, is = its diam, in ins X the thickness 
of the web-plate, in ins. 

In buildings, &e, where the load is stationary, 16000 may be used instead of 13000. 


Number of rivets in the 

depth of the girder between 
flange-rivet lines; or in a 
length of flange equal to said 
depth 


Total vert or shearing strain 
at the joint, in lbs, 

Greatest allowable strain 
on each rivet, in lbs 


The vert or shearing 1 strain may be found by pp 532, &c. 

If this, in any case, makes the pitch of the rivets less than about 2% ins, the thick 
ness of the web-plate should be increased. 


Art. 7. The web has to resist the vert forces acting npon the girder, and to 
transmit them to the abuts. In doing so, it is regarded as acting like the web mem¬ 
bers of the Pratt truss, Fig 31, p 595. In that truss the compressive strains are re¬ 
sisted by the vert posts; and the tensile strains by the main obliques, c c c. In the 
plate girder, vert stiffeners of angle or T iron, riveted to the web, take the place of 
the vert posts, and resist the crushing strains; while the web itself, in the panels 
between the stiffeners, takes the place of the obliques, and resists the diag tensile 
strains. For the vert strain at any point in the span, see pp 532, &c. 
The cliag tensile strain on the web will be = 

Said vert strain X length of diagonal drawn across a panel 
depth of girder. 

This last strain, however, need not be specially considered ; because, if the web is 
made strong enough to resist the crippling tendency of the rivets, it will also be 
strong enough to resist the diag tensile strain. 

Art. 8. The longitudinal hor dist between two stiffeners is usually 
made about equal to the depth of the girder; except that it is seldom less than about 
3 ft, or more than 5 ft, whatever the depth of the girder may be. The stiffeners 
are generally placed somewhat nearer together toward the ends of the span than at 
its center. 

In such railroad bridges as Nos 4 and 6, in our list, p 545, in which the road is car¬ 
ried by transverse floor girders; a heavy stiffener is placed at the end of each floor 
girder. The stiffeners are thus about 8 ft apart; and, as this exceeds considerably 
the usual limit, a lighter stiffener is placed between each two of the principal ones. 
See foot-note fl, p 545. 

Art. 9. Dimensions of stiffeners. Find by pp 532, &c, the vert strain 
on the girder at the point where the stiffener is to be placed. Then, having decided 
upon the shape of the stiffener, find its area of cross-section by means of the rules for 
iron pillars, pp 439 etc. 





540 


RIVETED GIRDERS. 


If the stiffener, ns usual, is of angle iron, or T iron, riveted to the opposite side: 


of the web plate; 

Square of least ) _f 4 „ n „ Air \ 
radius of gyration ) ~ (A l* prox) 


Square of width of stiffener in ins 
measured transversely of the girder 

22.5 “ 


In order to have as great a radius of gyration as possible, the narrower leg of ar 
angle stiffener (if the legs are unequal) is riveted to the web, leaving the wider one 
projecting. 

The area of hor cross section of the small portion of the web between the twe 
angles or T’s of the stiffener, is included in that of the stiffener; but that of any 
packing pieces (see foot-note ?, p 545) is not,because the latter have no firm bearing 
upon the flanges of the girder, and therefore give comparatively little support tc 
the stiffener. 

Art. 10. As already stated, if the web is proportioned as in Art 6, so as to be 
of sufficient thickness to be safe against crippling by the rivets, it will also be strong 
enough to resist safely the tensile strain. 

Art. 11. The web may also be regarded as resisting the shearing and 
buckling tendencies of the vert strains in the girder. The amount of the vert 
strain, at any point in the span, may be found by pp 532, Ac. 

The average ult shearing strength of rolled bridge plate is about 45000 lbs per sq 
inch ; and its safe strength say 9000 lbs; but, owing to the considerable depth of the 
web as compared with its thickness, it is more liable to fail through budding. In 
resisting the buckling tendency, it acts like a flat column; and its ult load may be 
found approx by the following formula, which is similar, in principle, to that used 
for columns: 


lilt buckling loadj 

in lbs per sq inch of vert )■ 
cross-section of web I 


_30000_ 

^ depth 2 , in ins 

1000 X thickness-', in ins 


in which 30000 is taken as the ult crushing load in lbs per sq inch, of wrought iron 
in short blocks. 

Ult buckling load 

Safe buckling load = Factor of safety, say from 3 to 6 according to 

circumstances 


Art. 12. In tlie box girder. Figs 24 and 25, if we could be certain that 
the strains were equally divided between the two webs, that on either one would 
of course be half that as found for the whole girder; but in practice one web is likely, 
through unavoidable inaccuracies in riveting, Ac, to receive more than its share of 
the strain ; and considerable margin should be allowed, according to circumstances, 
to cover this uncertainty. 

For this reason, plate girders are more economical than box girders. 
They have, also, the advantage of being more readily accessible for inspection, re¬ 
pairs, painting, Ac. On the other hand, box girders have greater lateral 
stability. 

Art. 13. Formulae for the ultimate crippling strength of well made 
plate and box girders, so proportioned as to be secure against buckling and against 
yielding sideways, and loaded with a quiescent weight; taking the breaking tensile 
strength of wrought-iron at 44800 lbs, and its elastic limit at half this, or 22400 lbs 
= 10 tons, per sq inch. 


Quiescent cen "I 
crippling load, 


Area of cross 
section of 


Depth in ins between 
the cens of grav of 


including 34 r == lower flange ^ crosR section of the A * 
the wt of clear J in sq ins flanges, at cen of span 

span of girder J 



Span in feet. 

The load so found is that which would cripple the girder by bending it beyond 
recovery. Calling it W ; then 

half the w't of the 

Quiescent extraneous crippling load in lbs = W — clear span of the 

girder in lbs. 

Quiescent distributed crippling load in lbs, including ( . r . w 

the wt of the clear span of the beam j twice vv. 

Quiescent extraneous distributed crippling load, in lbs = 

(twice W) - weight of entire clear span 
v ' ol girder in lbs. 


In railroad bridges, a factor of safety of 3 is used with the above loads. 









RIVETED GIRDERS. 


541 


span under a uniformly dis- - 
tributed quiescent load ) 


Art. 14. Formulae for deflections within the limit of elasticity: 

Load in tons of 2240 lbs X Span 3 , ft 


Deflection in ins at center of ^ 


Deflection in ins at center of 
span under a quiescent con 
centrated center load 


'•}- 


220 X Depth 2 , ins X A / ea °* Cr ° 88 . 8eCti . on 
N f ^ of one flange in sq ins 

Load in tons of 2240 lbs X Span 3 , ft 
1 Q7 v T)pnth2 ins V A,ea of cro8S 8ection 

167 X Depths ins X of one flatlge iu sq jll8 


It is very important that the rivet holes in the web should agree exactly in size 
and position with the corresponding holes in the vertical legs of the flange-angles; 
as otherwise considerable deflection may take place while the rivets are coming to 
their bearings, and undue strains be brought upon the web. 

Art. 15. The following are the results of an experiment with a box 
beam like Fig 25 made by Trenton (N J) Iron Co. Channels 6 ins X 2% ins X 
inch. Sides inch thick, 18 ins deep. Weight 207 lbs per yard. Length 20 ft 
3 ins; .dear span 19 ft 5 ins. The ends steadied sideways, but otherwise unconfined. 


Cen load. lbs. 

Def. Ins. 

Cen load. fbs. 

Def. Ins. 

0 

1 

K 

76862 

IK 

12990 

3 

Interval of 26 days. 

19920 

TH> 

81342 

lyi 

24230 

% 

85524 at once a 

-|15 

6 

28744 

X 

8 

crackling noise 

32284 

commenced. In 

2 t\ 

37841 

9 

10 min, 

42387 

M 

In 1 hour, 

2 tV 

3 7 

40923 

11 

M 

90302 

51460 

With a side defln of 

1/8 

55985 

1 3 

T6 

This increased 

60553 

1 5 

T6 

until the side plates gave way 

65089 

ii 

at their bottom edges, in an 

69954 

n! 

hour. 



Art. 16 . 
VTeight 


of ) 

entire girder of ( 
uniform cross f 
section, in lbs ) 


(: 


■) 


Allowance 
, for heads 
' alone of 
rivets 


+ 


Allowance 
for vert 
stiffeners, 
&c. 


Area of vert 
cross section of Length 
plates and an- X of girder X 1° 
gles alone,in sq iu yards 

ins 

The weight of the two heads of a rivet, after driving, may be 
roughly taken as averaging about two-thirds of that of the eutiie livet. More 
exactly: 


% inch, 
.4 lbs. 


If the diam of rivet is % U % 

the weight of its 2 heads is .167 .2 .25 

Eiveted girders, erected, cost, per pound, about twice as much as the plates. 
See p 402. 

Art 17. The plates are usually from x /i to % inch thick, and from 1 
ft wide up to 20 ft long, to 6 ft wide up to 15 ft long The angles (see PP 52o Ac) 
are from 211 X 2U X % to 6 X 6 X 1 inch. The rivets (see pp 469, Ac) are 
from % to PX ins diam, usually K inch; and are spaced from 2U to C ins apart 
from center to center. This dist from ceu to ceu of rivets is called their pitch. 

























542 


RIVETED GIRDERS. 


Art. IS. Figs 26, 27, aDd 28 illustrate methods of attaching the vert stiffeners 
to the webs of the girders. Formerly, they were sometimes bent, both at top and 
at bottom, as at * i. Fig 28, in order to pass the hor angle irons of the flanges. Now, 
however, their upper and lower ends generally abut sajuarely against tlie 
hor flanges of those angles, as in Figs 28 A, 28 B, 28 C, and 28 D; and should be 



Fig. 28. 

trimmed to fit them closely. Different methods are employed to enable the 
stifleners to pass the vert flanges of the angles. Sometimes, as in 
Figs 28 A and 28 B, “packing pieces” (flat bars of rolled-iron) are placed between 
each stiffener and the web; sometimes the upper and lower ends of the flange of the 
stiffener are cut away, as at a a, Fig 28 C; and sometimes the stiffeners are crimped 
or bent slightly, as in Fig 28 D. 



Fig. 27. 




Art. 10. In cases where a girder is placed directly under each rail, the <list 
apart of the two girders of a single track bridge is about 5 ft. Frequently, 
however, and especially in long spans in order to give the bridge sufficient lateral 
stability, the main girders are placed further apart; and the cross-ties, T, Fig 28 B, 
on which the rails rest, are then carried by longitudinal stringers, S, of iron or tim¬ 
ber, which rest upon transverse floor girders, F; and these, finally, rest upon the 
main girders, G. In such cases the latter are generally about 12 ft apart for single 
track, or where 3 girders are used for double track. Where only two girders are 
used for double track they are placed about 16 ft apart. 

The transverse floor girders are generally about 8 ft apart. They have a vert stiff¬ 
ener under each rail, and frequently others. 

Art. 20. Where transverse floor girders are used, they, with light hor diag ten¬ 
sion rods, give sufficient lateral bracing; but where these are wanting, as in 
Fig 28 A, special transverse strut-ties, T, &c, are used. They are generally made of 
angle or T iron. 












































































RIVETED GIRDERS. 


543 












































































544 


RIVETED GIRDERS. 


Fig 28 B shows one of the side girders, G, and parts of the transverse girder, F, &c, 
of bridge No 6. As in No 4, the lower tiauge of the track stringer, S, consists only 
of the hur legs of the two angle-bare, and has no flange-plate proper. It rests upon 
plates, P, % X %'A X ins, through which pass the rivets which fasten it to the 
flange of the transverse girder. 

The track stringers are stayed by bent plates, B, which are riveted to them and 
to the upper flange of each cross girder. A is a transverse brace ot 2% X 2*4 X l /i 
inch angle-iron. One of these is riveted, by means of a connecting-plate, to the 
upper flange of each track-stringer, at its joint with the next one. The plate, O, ot > 
% inch iron, is placed between, and riveted to, the two inner stiffener-angles, K» 
(only one shown), and two angles, L, 3 X 3 X %> and two others, M, X 3 X %• 11 



On p 545 arc given the principal dimensions, weights, *c, 

for different spans. The numbers (l, 2, 3, 4, 5, 6) are our own, and are used merely 
for convenience of reference. Where the girders are 5 ft apart (Nos 1, 2, 3, and 5) 
the bridge is for single track, the cross-ties rest directly upon, and are notched to, 
the upper chords, and one rail is placed directly over each girder. No 6 is for double 
track, and the roadway is arranged as shown in Fig 28 B. No 4 is also for double 
track, with roadway as in Fig 28 B, except that the track stringers, S, rest upon the 
lower chords of the'transverse girders. 

The lower flange of each main girder is riveted, at each end, to a rectangular 
rolled-iron “holster plate.” which rests upon a slightly larger “wall 
plate.” The two plates are held to the abut by two bolts which pass through 
both of them. At one end of the span, the bolt holes in the bolster plate are slightly 
elongated in the direction of the length of the bridge, so that the bolster plate may 
slide on the wall plate when the girder expands ami contracts under the 
influence of lieat and cold. 






























































RIVETED GIRDERS 


545 


Standard Plate Girders, Pennsylvania Railroad. 
For loads, see page 546. 


-— 

No. 1. 

No. 2. 

No. 3. 

No. 4. 

No. 5. 

No. 6. 

Girders. 







Length. 

33 ft. 

49 ft. 

59 ft 6 ins 

61 ft6% ius 

64 ft 8 ins 

70 ft % ins 

Span * * * § . 

25 to 30 ft 

40 to 45 ft 

50 to 55 ft 

56 ft. 

55 to 00 ft 


Dist apart, ceu tocen. 
Approx gross wt,t fl)s 

5 ft. 

5 ft. 

5 ft. 

12 ft 2 ius 

5 ft. 

12 ft 2 ins 

Due girder, alone.. 

6000 

12000 

17500 


21500 


Two girders, with 





transverse brac'g 
for single track .. 

14000 

27700 

39300 


49300 


Middle girder, alone 



29500 

38000 

One side girder. 





alone. 




20000 


25000 

Three girders, with 





transverse brac'g, 
for double track.. 




115000 


141500 

Upper flange,1 





Width. 

Thickness, 

10 ins.... 

12 ins.... 

12 ins.... 

14 ins.... 

16 ins.... 

14 ins 

At ceu of span_ 

1% ins... 

23-16 ins. 

2 9-16 ins. 

4 ins. 

2% ins... 

4% ins 

Atends “ 

15-16 in .. 

1 in. 

1 in. 

1 in. 

1 iu. 

1 in 

Lower flauge.J 






Width. 

Thickness, 

10 ins.... 

12 ius.... 

12 ins.... 

20 ins.... 

16 ins.... 

20 ins 

At cen of span... 

1 % ins... 

1 % ins.... 

2% ins... 

3% ins... 

2)4 ins... 

3 7-16 in* 

At euds “ 

15 16 in .. 

1 IU. 

1 in. 

1 in. 


1 in 

Web-plate. 






Depth. 

36 ins .... 

48 ins ... 

58 ins.... 

60 ins.... 

64 ins.... 

72 ins 

Thickness. 

% in. 

Vs in. 

Vs in. 

Vs in. 

Vs in. 

Vs in 

Ang-le stiff- 







eners. 







Size§ at ceu of span.. 

2MX3X 3 /s 

3X4X96-- 

3X4X%.. 

3X4X%.. 

If 

3X3*sX% 

If 

“ at ends 

3X3)4X!4 

3X4X%.. 

* 

3)4X'i)4X% 

* 

Dist apart 






At cen of span. 

3 ft 6 itis.. 

4 ft 3 ins.. 

4 ft 10 ins 

4 ft 2 ins. 

4 ft 11 ins 

4 ft 2 ins 

Near ends II of span 

2 ft 4 ins.. 

3 ft 6 ius.. 

3 ft 11 ins 

4 ft 2 ius. 

3 ft 11 ins 

4 ft 2 ins 

Transverse 







Birders. 







Dist apart cen to cen. 
Flanges. 






8 ft 3 % ins 

V'X%" 

19"X&" 




9"X3*"... 

19”X%".. 


Web !. 





Track 





stringers. 

Upper flange. 




8"X1116" 

»Vs"X)4”- 

l2WXVs" 


8"X%" 
8%"X7-16'* 
12 )4"XVs” 

Lower “ . 





Web. 











* Bv “span ” here is meant the clear distance between the abuts, taken just below the coping. 
The span to be used in ascertaining moments of rupture, &c, is measured between the centers of the 
wall-plates on which the girder rests ; and is generally front 1 to 2 ft less than the length of the girder. 

t These weights include the weights of the entire rivets (shanks and heads), and that of the plates 
and angles as ordered from the mill, and before any subsequent reduction by trimming, or by punch¬ 
ing for rivet holes, be. 

} Under “ flange ” we include not only the hor flange- plates, but also the hor legs of the two angle- 
bars by which said flange-plates are fastened to the vert web. The thickness of these angles is in¬ 
cluded in the flange thickness. Together they are generally narrower than the flange-plates. 
By “ width of flange” we mean its greatest width, or the width of the widest flange-plate. 

§ Each vert stiffener has, between it and the web, a “packing” consisting of a flat bar of 
rolled iron as wide as that leg of the angle stiffener which is fastened to the web of the girder, or 
wider, and as thick as the angles of the upper and lower flanges. As shown in both figs, the stiffeners 
extend between the hor flanges of the upper and lower angles; but the packing pieces only between 
the edges of their vert flanges. The packing pieces, by keeping the stiffeners away from the web, 
render it unnecessary to bend them, as at 11 , Fig 28, or as in Fig 28 D, or to cut away part of their 
flanges, as in Fig 28 C, in order to enable them to pass the flange-angles. 

II At cacli end of the span, over the abuts, two or more vert stiffeners are placed on each side 
of the beam and quite near each other, in order to withstand the severe strains at those points. One 
wide packing piece is placed under these on each side of the girder. Across each end of the girder a 
“cover-plate ” is riveted to the end stiffeners. It is about % inch thick, as wide as the flanges 
(tapering when these arc of unequal width), and as high as the extreme end depth of the girder. 

11 In Nos 4 and 6. ihe stiffeners, at the points where the transverse floor-girders are attached, 
consist each of two angle-bars placed together with a plate between them, so as to foim a T. The 
angles are about 3)4 X 5 X % inch, and the plates between them are about )4 X 8 inch. The inter¬ 
mediate stiffeners are single angle-bars with packing pieces, as in the smaller spans. See Art 8. 
































































































546 


RIVETED GIRDERS. 


The bridges in the table are required to carry safely either one of the three fol¬ 
lowing moving loads, A, B, or C. If the bridge is double track, it must carry such 
a load on each track at the same time, the two loads headed in the same direction. 


“Typical ” 
consolidation 
locomotive. 


Tender. 


“ Typical ” 
consolidation 
locomotive. 


Tender. 


Train. 


.00000 
w o o o o ® 

~ o o o o o 

~ -* T* -f 

(N W (M « 


8 § s ? 


O 3 
OOO 
co CO CO 


o OOOO 00 


o o 


00000 

00000 

rf -f -r -r 

»-• Cl (N (M 'N 

oOOOO 


O O 
o o 

8 g 


o o 
o o 

s g 


0000 



ft. 7.5 4.5 4.5 4.5 10.5 5.0 5.5 5.0 8.0 7.5 4.5 4.5 4.5 10.5 5.0 5.5 5.0 3.0 


B 


“Typical” 

passenger 

locomotive. 


Tender. 


“Typical ” 
passe nger 
locomotive. 


Tender 


Train. 


.0000 

2 S 8 8 S 

— 0000 


OOO 

OOO 

S o o 

o <c 


to 


0000 
o o o O 

8 g 8 8 


0000 

8 8 18 

to o to to 


O - 3 


o o 


00 


0000 


O O 


OO 


8 £ ? 
w 8 *- 


OOOO 


ft. 5.5 y.O 8.0 9.5 5.0 5.5 5.0 8.0 5.5 9.0 8.0 9.5 5.0 5.5 5.0 3.0 


Shitting 
locomotive 
“ class M '* 
Penna R R. 


C 


Tender. 


o 

o 

o 


o 

CO 


o 

o 

s 

01 


o 

o 

o 


o 

o 

o 


OOO O 


OOO 


ft. 6.0 4.7 13.3 4.8 3.7 4.8 5.0 


The above “ typical ” engines were de¬ 
signed (in skeleton) by Mr Wilson, for 
use in planning bridges for the Penna 
Train. R R. They were purposely made some¬ 
what heavier than the engines actually 
—»—„ in use on the road, in order to provide 
for the constantly increasing dimensions 
and weights of the latter, which, how¬ 
ever, are fast approaching these “typ¬ 
ical ” figures. Indeed, the shifting en¬ 
gine C’, here given (“Class M ”), which 
is in actual use, produces, with certain 


O *- S 
CO ® *-• 


lengths of span and panel, greater strains in a truss than either of the two typical 
engines. In calculating the strains on web members, the cross-girder load under the 
foremost pair of drivers is to be considered as the head of the train ; any load upon 
the preceding cross-girder being neglected. 

Equivalent uniform loads. Owing to the great diversity in the design 
of locomotives and in the distribution of the load upon their several pairs of wheels, 
the method of specifying the actual or assumed wheel loads, as above, necessitates 
much laborious calculation of strains by bridge-builders. To obvinte this Mr. Geo. 
II. Pegram, C. E., suggests* that the strains in plate girders and in the chords 
(i a and mx, Fig 13 f, p 564) of trusses, be calculated from an assumed total load of 


/\ , _ 60000 lbsf\ 

^3000 lbs + B pan in ft J X span in ft, uniformly distributed over the entire span as 


in Fig41, p483; and that the strains in the web members (ax, cr. etc, Fig 13f) of trusses 
be calculated from an assumed load consisting of a train weighing 3000 lbs per ft run 
and ofa single concentrated load of 30000 lbs. In order to find the strains on the several 
web members in turn, we suppose the train to extend from one end, as m Fig 13 f, of 
the span, first to o, then to p, etc, and thus to each panel point in succession ; the con¬ 
centrated load being supposed to be placed always at the panel point n, o, etc next 
behind the head o, etc of the train. The first half panel load is to be neglected in 
making the calculations. Thus, if the train be supposed to extend from m to q Fig 
13 f, the concentrated load would bo assumed to be at p, making the original panel 
load at n and at o each = 3000 ft) X length of one panel in feet; that at p = 3000 
lbs X panel length vw in ft + 30000 lbs; and the half panel load wq would be ne¬ 
glected. ” "- " ... 

of 


ected. Mr. Pegram finds that by using 2900 It s and 25000 It s respectively, instead 
the above 3000 tbs and 30000 ttm, we obtain strains practically equal to the'greatest 


of those caused by any of the above arrangements of wheel loads; and that by using 
3500 lbs and 35000 lbs respectively (as would seem to be advisable in view of the rapid 
increase in the weight of rolling stock), we should add but about 10 per cent to the 
weights of bridges designed for the above wheel loads. 


* Transactions, American Society of Civil Engineers, June 1886 
t With 00000 lbs uniformly distributed, the breaking moment at anv point is the 
same as would be caused by 30000 ibs concentrated at that point 




























TRUSSES. 


547 


TRUSSES. 


Art. 1. Wh en the span of a bridge, roof, Ac, becomes so great that single solid 
) beams, supported at their ends, cannot be employed, we resort to compound beams, 
called trusses*. composed of several pieces so arranged and united as to furnish 
the reqd strength. The designing, construction, and erection of trusses of great 
span, especially when of iron, involve such a multiplicity of important detail, that, 
like the building of locomotives, cars, Ac, they have become a specialty, or a dis¬ 
tinct branch of business, to which persons confine themselves to the exclusion more 
. or less of other departments; and thus attain a degree of skill beyond the reach of 
the general engineer.* The latter, however, should possess a knowledge of the sub¬ 
ject sufficient at least to enable him to form a well-grounded opinion of the general 
merits of a design; and to guard him against the adoption of one involving serious 
imperfections. In a volume like this we can aim at nothing more than an attempt 
to illustrate some few general principles. We shall confine ourselves to such trusses 
as are in common use; showing first the effects of uniform stationary loads, as in 
the case of roofs; and then those of moving loads, such as an engine and train on a 
' bridge. 

Art. 2. Most of the bridge trusses in common use have two long, straight, par- 
1 allel upper and lower members It, up; and l t, a p, Figs 10, 11, called the 

. chords; or in England, the booms*. Vertical pieces placed between, and con- 
| necting the upper and lower chords, are called posts, when they sustain compres- 
sion; and vertical ties, or suspension rods, Ac, when they sustain tension 
or pull. The oblique pieces seen in these figs arc called braces, strut-braces, 
main-braces, Ac, when resisting pres or thrust; or tie-braces, tension- 
braces. main oblique ties, oblique suspension-rods, Ac, when 
resisting pulls. Sometimes the same piece is adapted to bear both tension and cora- 
i pression alternately; and may then be called a tie-strut or a strut-tie. The 
a oblique members alluded to are sometimes called main-braces, whether they are 
: struts or ties; to distinguish them from counter-braces, or counters. These 
n last are not shown in Figs 10 and 11, but are seen in Figs 28 and 31, crossing the 
j main braces diagonally. These posts, braces, counters, ties, Ac, serve not only to 
keep the two chords asunder, and to prevent them from bending; but to transform 
the transverse strains produced by the wt of the truss anti its load, into other strains, 
acting longitudinally, or lengthwise, along the difif members; and to conduct said 
jj strains along the truss, to the firm supports of the abuts. A load placed at any one 
d of these members is, of course, partly supported by each abut; one part of it tra\els 
, up and down alternately between the chords, and along the successive members, 
; until it reaches one abut; and the other part, in like manner, goes to the other abut. 
These members, therefore, perform the duty of the vert web of the Ilodgkinson 
beam ; or of the I rolled beams, or of the tubular girder; and on this account are col¬ 
lectively called the web members, in contradistinction from the chords. Each 
portion of any load, while being transferred by the web members, from the spot at 
j which it is placed on the truss, to its final point of support on the abut, produces a 
j strain equal to itself upon every vert web member along which it travels between the 
„ parallel chords ; while upon each oblique member encountered on its way, it produces 
j a strain greater than itself, in the same proportion that the oblique member is longer 
than a vert one. 

i Whether the web members are strained by compression, or by tension; or, in 
other words, whether they act as struts, or as ties, the amount of strain will be the 
' same. In either case the straining agent is the same identical force, namely the wt, 
r or vert force of gravity of the truss itself, and of its load; and (Art 18, of Force in 
; Kigid Bodies) whether this force exhibits itself as a push, or as a,pull, neither its 
amount, nor its direction undergoes any change. So far, therefore, as regards the 
■ broad principle involved in the duty performed by the web members, they might be 
1 divided simply into verticals, and obliques. We shall frequently so desig¬ 
nate them. 

Whatever amount of strain the upper end of an oblique produces in one direction 
against the upper chord, that same amount will its lower end produce against the 
parallel lower chord ; but in the opposite direction. That is, if the top or head of 
1 any oblique, pushes the upper chord toward the right hand ; its foot will pull the 
lower chord to the same extent toward the left hand. This, however, is not pro- 

* The first writer to whom we are indebted for a knowledge of correct principles on this subject is 
8. Whipple, C. E-, the first edition of whose book (beyond all doubt the pioneer one) bears date, 
Utica, N. York, 1H47. He was followed by How. of England, and Haupt, of this country, both in 1851. 
The Murphy-Whipple bridge (of which Mr. John W. Murphy, C. H., has built several of the best.) 
owes its name tc these two gentlemen. 








548 


TRUSSES. 


cisely correct, inasmuch as when the oblique is a strut, the pres at its foot is some* 
what greater than at its head, because the foot supports also the wt of the strut 
itself; or if the oblique is a tie, with its head attached to the upper chord, then the 
strain is a little greater at the head than at the foot; because then the head upholds 
the wt of the oblique, and the foot sustains none of it. This remark applies, of 
course, to verts also. Another exception is. when the ends of two obliques meet 
each other; as those at the center of the trusses, in Figs 10 and 11. If, in such cases, 
the ends of the obliques abut against each other, instead of being separately attached 
to the chord, they will at that point exert their strains against each other, instead of 
against the chord. 

In any oblique, as c d, Fig 1, the vert dist a r, between its ends; and the hor dist 
a d between the same, are called its vert and hor spreads, or stretches, or 
readies. 

Art. 3. There is a great diff in principle between two classes of trusses in 
common use. In some of them, two chords are absolutely essential, as in the 
Howe truss, p 594; the Pratt, p 595; the Lattice, p 596; the Warren, p569; and 
their various modifications, known as the Murpliy-Whipple. the Linville, the Latrop, 
Ac. Ac, which differ only in certain unessential details. In the Ilowe and Pratt 
trusses there is no diff whatever of broad principle. The distinction between them 
consisting chiefly in the fiict that in Ilowe’s the verts are. ties, or suspension rods; 
and the obliques, struts; while in Pratt’s, the verts are posts; and the obliques, ties. 
In all these the strains on the verts and main obliques (not on the counters) are least 
at the center of the truss; and increase gradually toward the end of it; ■while those 
on the chords (as in an ordinary wooden beam) are greatest at the center, and least 
at the ends. Hence, also, such are called beam trusses. The strains on the 
counters are also greatest at the center. 

But there is another class, called suspension trusses, of which the Fink, 
Figs 46 and 47; and the Bollman, Figs 44, 45, are the principal representatives- 
In these but one chord is essential for a perfect truss. From this chord the web 
members are suspended; and to it alone do they all transfer their strains; and 
the strain on this hor chord is uniform from end to end. Figs 45 and 47 show 
perfect bridges, with but one chord each. In Figs 44 and 46, n n, n n, appear to 
be chords; but strictly speaking they are not; they are merely longitudinal pieces 
for upholding the cross-beams of the flooring, when the roadway is placed at the 
bottom of the truss. They have not to resist tension, as in beam trusses. 

In all the foremeutioned trusso* the roadway may be placed on either the top or the bottom chord; 
constituting in the lirst case a top road, or n deck bridge: and in the second, a bottom 
road, or a through bridge. 

Art. 4. That part of a truss, such as Figs 10, 28, 31, Ac, that is comprised be¬ 
tween two adjacent verts, is called a panel; thus, in Fig 10, e ij d, dj kc,& c; ant ! 
in Fig 31, of Pratt, tynu\ is a panel. The Triangular or Warren truss, Fig 11, pi 
558 has no verts, as essential parts of it; and its subdivisions are called simplj 
triangles; and a panel is a length of truss equal to the width of a triangle. Verb 
are sometimes added to it when the spaces a b, b c, c d, Fig 11, become too long foi 
safely supporting the roadway without them ; thus dividing the truss into half 
panels. It is not a matter of practical importance as regards strength, whethe) 
the number of panels in a truss be odd or even; but it is usually even, with a veri 
at the center of a truss. 

A panel-point, as a, b, d, c, o, or n, Fig 1, is one at which web-members 
meet a chord, or a rafter in a bridge or roof, as Fig 14, b, c, k, &c, 

The lengthy of a panel is its horizontal measurement. The 
best inclination of obliques, as regards economy of material in the 

web, is when their least angle (lp o, or s o n, Ac, Fig 10) with the chord is 45° i 
This applies also to the admirable Warren truss; in which the triangles are, however 

usually made equilateral. When the span it 
great, and the height of truss correspondingly so 
if the panels be made square, or nearly so, witl 
a view to secure this inclination of about 45° foi j 
the obliques, the verts (as to, sn, Ac, Fig 10 
will become so far apart, that the stretches oi 
dists p o, on, Ac, become too long to be safe fo 
upholding their loads of engines, cars, Ac, with 
out additional precautions. When, therefore 
the expense or inconvenience resulting from tlii 
would be too great, the verts may, as in Fig 1, b< 
placed so near together as to make half panel 














TRUSSES. 


549 


* 

it 

ie 

1» 

if 

!t 

K 

d 

i 


s 

ir 


much higher than they are long; and the obliques (both the main ones, and the 
dotted counters) then run across one vert, as in the Fig; or across two, if neces¬ 
sary. In the Warren girder, the expedient is to introduce verts; or else a second set 
of triangles, as in Fig 1, omitting the verts. From 8 to 12 or 15 ft apart are ordinary 
dists for verts in bridges of moderately large spans. Frequently panels are made 
considerably higher than long; disregarding the economical angle of 45°. In large 
bridges, the main obliques, instead of being each in one piece, are usually made of 
two or more parallel pieces, disposed in such a manner as to let the counters pass 
between them diagonally, without mutual interference. Each lower chord (and fre¬ 
quently the upper one also) in large spans, is usually made up of several parallel 
beams of wood, or bars of iron, side by side. 


In European iron bridges, the chords are generally attached to the web tnem- 
» bers by riveting; but in America this is done by means of cylindrical iron or 
* steel pins as shown for a lower chord in Fig 61, p 612. 

d The tension members in the web, like the lower chord, generally consist of 
>i round, flat or square iron bars, with eyes (sx Fig 61) at their ends for the pas- 
t sage of the pins. 

i The upper chord, and the struts or compression members in the web, were at 
: first generally made of cast iron ; but they are now almost universally of rolled 
i. iron or steel. To give them sufficient lateral stability as pillars without an un- 
t due expenditure of material, they are made hollow. Many different shapes are 
e used. That shown in Figs 13 and 14 of “Trestles”, p 757, is a common one; fre- 
t quently with channel bars (p 521) instead of the side plates and angles of Fig 
8 14. The Phoenix segment column (p 449) is also very largely used for this pur- 
i pose. The ends of the struts, like those of the ties, are furnished with eyes for 
, the passage of the pins. 

When the web of an iron or steel bridge truss consists of inclined and verti- 
1 cal members they are so arranged that, the vertical , or shorter , members shall bear 
1 the compressive strains, and the inclined, or longer, members the tensile, strains, 
i as in Fig 31, p 595 ;* because (see lines 11 to 18, p 457) a short pillar is st ronger 
j than a longer one of the same material and cross section. In great spans, 
s where the truss is necessarily very deep, the obliques are often made to cross 
i two panels, as in Fig 1, thus intersecting each vertical post at its center. In 
such cases the oblique is sometimes fastened to the post by a pin at the point 
, of intersection, in order to further strengthen the post and to prevent the long 
! oblique from sagging. 

In wooden bridges, the verts are generally ties, and the obliques, posts, as in Fig 
10, p 558. The Howe, Fig 28, p 594, has oblique wooden struts, and vert iron ties. 

In long spans provision must be made for the expansion ami eontrac- 
f tion of the truss under the effects of lieat and cold. See p 614. 

If in Fig 10 we imagine lines crossing the panels diag, as the main obliques shown 
in the Fig do, but in the opposite direction, as shown in Figs 28 and 31, they will rep- 
< resent counter-braces, or counters. These, like the main obliques, are in 
some cases struts, and in others ties. Although important members, they are less so 
than the main obliques. They are unnecessary when the load is uniform and station¬ 
ary, as is usually assumed to be the case in roofs; and are required only when the 
load is unequal, or a moving one, as in a train crossing a bridge. In this last case they 
act chiefly while the span is but partially loaded. If the train at any moment covers 
the entire span, and is of uniform wt, their action ceases for that time. Their office 
is solely to counteract the deranging tendency of the unequal loading of diff parts 
of the truss, as shown in Figs 9]/ 2 . In Fig 9 b, an excess of load along a o 

would tend to derange the main braces ho and ta; and this would be counteracted 
by counters across co and ts. The same thing may be effected by arranging the main 
braces ho ta, so as to bear tension as well as compression. The bad effects of une¬ 
qual loads'must plainly become greater in proport ion as the load is heavier than the 
truss itself; and when the bridge becomes very heavy, so that the load must extend 
over several panels before its effects become serious, but little counterbracing is 
needed- and that at and near the center only; whereas, in a very light bridge, the 
counters should extend from the center, where they are most strained; to near the 
ends where the strain upon them is least. Inasmuch as we shall first speak of uni¬ 
formly loaded trusses, we shall not here say more respecting counters. See Kemark, 
Art 10. 

It would at first sight appear that the several parts of a bridge truss must be most 
strained when covered from end to end with its maximum load ; but this is true only 
of the chords; and of the main obliques and verts, as la, tp. Fig 10, at the rads of 
the truss The other web members are more strained by a part of the load as it passes 
along the truss ; so that if they be correctly proportioned for a full load, they will 

* Except that, when the roadway is on the lower chord, the two web members 
vesting upon the abutments are generally inclined struts. 










550 


TRUSSES. 


be too weak for a partial one. If all be made as strong; as the end ones, they will, 11 j 
is true, be safe for a passing load; but this would require an expense of material thaf! 
would be justified only in the case of moderate spajis, especially of wood; in which L 
the additional trouble and expense of getting out and fitting together pieces of , 
many diflf sizes, may more than counterbalance the saving in material. 

Art. 5. Trusses with moving loa<ls require calculation diff fron: 
that for uniform loads. We shall first treat of the latter only ; and in so doing shall; s 
not employ the shortest methods, but such as will render the general principles clear 
to any one acquainted with the simple elements of “Composition and Resolution of 
Forces.” The strains on trusses may be found with all the accuracy needed for prac-j 
tical purposes, by means of diagrams drawn to a scale. The same division of the scale | 
that answers for a foot of length, may also represent a ton, 1000 lbs, or any other , 
convenient wt, load, or strain, and may thus be used for measuring the lines which j 
represent such. 

The chords, verts, and obliques heretofore mentioned, constitute all the essential! 
elements of a complete truss: but other pieces are necessary lor a complete 
bridge; such as roof and floor beams; transverse bracing for connecting two par¬ 
allel trusses with one another, so as better to resist lateral or sidewise motion from 
winds or lurchings of trains; bars for tying the truss to the piers and abutments in 
some cases, &c. The same may be said of the extension frequently made at the ends 
of either an upper or a lower chord of a bridge as shown at v v in the bottom chord 
of Fig 31. Here the trusses are perfect without the extensions; but the bridge 

requires them, to allow the load to reach and to leave it. They may be needed for 
the same purpose in an upper chord of a top-road bridge; or for extending a roof 
over an entire span, &c. The eud vert postspu,of the same Fig are not parts of 
the truss, but supports for upholding it; also, the posts p and cf, Fig 28, are not es 
sential to the truss. 


Rem. Besides the forms of truss already mentioned, there are many others, in some of which 
arches are introduced either as principal members, or merely as auxiliaries; as Town’s Lattice. 
Fig 33 ; the Bow and String, Fig 35; aud the Burr, Fig 36, all of much merit. The Lattice 
and the Burr have both fallen into undeserved disrepute, from the fact that being the first trusses 
that were extensively introduced upoD the railroads in this country, they were built too weak for the 
heavy engines and trains of the present day, aud consequently failed. 

Art. 6. (’hurds. When a beam a b, Fig 3, supported at both ends, breaks 

either under its own wt, or under the 



action of a load placed on top of it, or 
suspended from it below, it does so be¬ 
cause the lower fibres, near its center 
I, are pulled asunder ; and its upper 
ones at u, crushed together to such an 
extent as to offer no effective resist¬ 
ance. The fig shows this in a some¬ 
what exaggerated manner. The ex¬ 
treme upper particles at «,and the ex¬ 
treme lower ones at l, being the most 


strained, give way first; and the strength of the beam being thereby diminished, the 
adjacent ones give way in rapid succession. The compressed particles of the beam 
are all above a certain point n ; while the extended ones are below it. If we imagine 
an infinitely fine needle to be held perp to this page, and in that position to be stuck 
through the point n, passing entireli/ through the beam, or page, then the infinitely 
fine hole thus made will pass along what is called the neutral axis of the beam. 
It is so named because the fibres situated in that line, and which were cut in two by 
the needle, are neither compressed nor extended, until the strain becomes so great 
that on its removal the beam will not entirely recover itself; or. in other words, 
until the strain exceeds the elastic limit of the beam. Within the limits of elasticity, 
the neutral axis may be assumed to pass through the cen of grav of the cross-section 

of the beam. Thus, if the cross-section be of 
any of the forms shown in Fig4, then so long 
as the beam is safe, or the load within the 
elastic limits, the line na will pass along its 
cen of grav; which is at the same time its 
neutral axis. But the chords of a truss 
differ essentially in condition from the 


n.*■ 


Fig. 4. 



. a 


fibres of the beam, as will be seen by comparing pp 485 and 48G with Art. 2 p 528 
where it is shown that while the resistance of a closed beam is in proportion to the 
square of its depth, that of a truss, or open beam, is proportional simply to its depth 
Ihe same quantity of material that composes the beam ab Fig 3, will present i 

morn rPKl’tftant'A tn lmmlinrr l>rn ol/ino. 5*';+ l.-v X... 1. u • 1 1 


far 


more resistance to bending or breaking if it be cut in two lengthwise along the hori 














TRUSSES. 


551 



Fig. 4%. 


zontal plane a b, and converted into top and bottom chords of a truss; because the 
leverage with which the resistance acts is thus greatly increased. Besides, the depths 
of the chords are so small compared with their distance from the neutral axis, that 
their fibres may be assumed to act unitedly and equally. Hence, practically, all the 
fibres in the upper chord must be crushed, or all those in the lower chord pulled 
apart, at the same instant, before the truss can give way ; whereas, in the solid beam, 
ab Fig 3, the extreme upper or lower fibres yield first; then those next to them,and 
so on, one after the other. 

Art. 7. In tiie designing of trusses, especially such as may have to 
bear unequal loads at different parts, as in a bridge, the 
point chiefly to be aimed at is to dispose its various 
parts so as to form a series of properly connected tri¬ 
angles, because in that shape they present more 
resistance to derangement of form, than in figs of a 
greater number of sides. Thus, in the three beams at 
a, Figs 4with a bolt at each junction or joint, the 
triangular form evidently cannot be changed by any but a force sufficient to either 
bend or break either the beams or the bolts. But in the 4-sided fig It, the form may 
readily be changed to that at c, by a force at n entirely too small to injure either 
the beams or the bolts. In a the bolts assist to prevent change of form; but in b 
they are merely pivots, around which great changes may easily take place. 

Before the strains can be calculated, and the truss propor¬ 
tioned to those strains, its weight must be known; for this tends to break it, as well as the extraue- 
ous load. But, on the other hand, we cannot learn its wt until we know the size of its diff members. 
In this dilemma we must assume for it au approximate wt, based upou our knowledge of somewhat 
similar trusses already built. This becomes the more necessary as the truss increases in size, so that 
its own wt becomes greater in proportion to that of the load. The table, p 605, gives safe assumed 
wts for bridge trusses; and p 580 will aid in the case of roofs. In very small spans, especially of 
bridges, the load is generally so much greater than the wt of the truss, that the latter might almost 
be neglected entirely. 

Bern. For finding the strains on a paneled truss by means of a drawing, it 
is best to represent each member by a single line, as in Figs 1,10,14,23, &c. Such 
is called a skeleton drawing or diagram of the truss. Each of the parts 
into which the panel-points divide either chord, or a rafter, is to be regarded as 
a separate member. 

As will be shown farther on, a load consisting of some portion of the wt of the 
truss and its load, is assumed to be supported at each panel-point. AU the forces 
which meet at any panel-point (namely, the aforesaid partial load, and the forces 
acting lengthwise of the members which meet there) hold each other in equili¬ 
brium. 

The forces act ins; upon a truss (omitting wind) are the downward 
one of the wt of itself and load; and the upward one of the reaction of the abut¬ 
ments ; and these two forces are equal. They produce all the strains along the 
members. 


I 



Fig 1 5 is the most simple form of a roof truss. 

equal rafters o a, ol; 
and a hor tie-beam a b. 

Here, as in roofs gen¬ 
erally, the entire weight 
of the truss, and of its 
load of roof - covering, 
snow, wind, &c, may be 
assumed to be uniformly 
distributed across the 
whole span. A roof con¬ 
sists of several trusses, 
placed usually from 8 to 

12 ft apart; but some- ^_ 

times much less, and at 
others much more. The 
trusses rest on longitudi¬ 
nal timbers, p, p, called 
wall-plates, stretching 
along the top of the wall; 
and serving to distribu te 
the wt of the truss and 
its load over a greater 

area. On the rafters, and at intervals of a few ft, 


It consists of two 


arc fixed pieces of timber called purlins, of 


























552 


TRUSSES. 


smalt scantling, running across from truss to truss : to which the laths or boards are nailed which 
support the shingles, tin, or slate, &c, which forms the roof-coveriog. 

A truss plainly supports all the purlins, roof-covering, snow, Ac, Ac, which occupy the space half¬ 
way on each side of it to the next trass. Thus, suppose a span of 30 ft, and each rafter to be lfi.8 ft 
long ; and that the trusses are say 12 ft apart from center to center, and assume (as it is generally 
well to do,) that tire wt of the truss, covering, snow, &c, may amount to 40 lbs per sq ft of area of 
roof. Then each truss has to sustain 33.fi X 12 X 40 — 16128 B>s, including its own weight. Strictly i * 1 
the wt of the tie-beam should be omitted; because in Fig 5 no part of it is upheld by the rafters. 
It is very trilling however in comparison with the load. 

To find the strains upon the different parts of a truss. 

Fi}*5. Fi ret calculate in the manner just shown, the entire wt in lbs of a truss I 
and its load. Through the cemer U of either rafter draw a vert line H r. From odvaw a hor iine i 
o H. Joiu H a. Now oil the vert line H r, lay off H I by any convenient scale to represent the entire 
uniformly distributed wt of one rafter and its load ; aud draw the hor line I E. Then will I K give 
by the same scale the hor force at the bead of the rafter; and H i£ the amount and direction of the 
oblique force which presses the foot of the rafter. The hor force at the foot of the rafter will be equal 
to that at its head ;* and equal also to the hor pull along the whole length of the tie-bean).t 

Or, consider the force of gravity, G It, Fig ( = H I) as resolved into two corn- , 
ponents; one, L R, in the direction of the length of the rafter ; and the other, G L, 



rtf right angles to it. The latter is the force which tends to break the rafter trans¬ 
versely, or like a beam. Since the rafter is uniformly loaded, the cen of grav G is at 
the center of its length. Hence, by the principle of the lever. Art. 54, p 359, op, — % 

G L, is exerted against the top of the other rafter; and the other half n q at the abut 
a. At the top, op causes, or is resolved into, two forces; first, the horizontal pres¬ 
sure ob ( = I E) against the either rafter; ami, second, a longitudinal thrust oz along 
oa.% This thrust [o z) is uniform from o to a. But at each point between o and a it 
is added to by a portion of the other longitudinal thrust L R which arises directly from i 
the pres of the load. Since the load is uniformly distributed, this last thrust in¬ 
creases uniformly from nothing at the top o, to its full amount L R at the foot, a. At 
the top therefore, the tobil longitudinal thrust is o z. At G it is o z 4- half L R. At a 
it ak = oz -|- LR; and combines with the transverse pres a q there, to form the 
resultant pressure a v of the foot of the rafter, which is of course the same as H E, the 
resultant formed by the load III and the horizontal pres I E of the other rafter. It 
will be noticed that the horizonhtl comjHinents, n q and s k, of a q and a k, are in oppo¬ 
site directions. Their difference (= tv) is the pull on the tie beam, and is = I E = o b. 
But their vert components, an and a are both donmward; and their sum ( = a t) is 
= III = the load on oa = the upward reaction of the abut a. 

The sizes in Fig 5 may 1 m* found in the following- manner. 

Take, for example, a truss of white pine, of 30 ft span, and 7J4 ft rise. The wt of the entire roof, 
snow, Ac, &e, 40 lbs per sq ft of roof area. Trusses 12 ft apart from center to center; so that each 
truss will have to sustain a total loa<r(including its own wt,) of 33.6 x 40 X 12 = 16128 lbs, which we 
may call 16000, or each rafter 8000 lbs. We will calculate each part with a safety of 3; which we 


* The foot of each rafter tends to slide or push outward horizontally in the direction of the arrows 
t and tr; each with a force equal to I E. But the tie-beam prevents them from so doing, and thus con¬ 
verts their pushes into pulls against each other; and thereby into a pulling strain along the whole 
tie-beam itself, to an amount equal to one of the forces; as two men pulling against each other at the 
two ends of a rope, each with a force of 10 lbs, only strain the rope 10 lbs. In other words, it requires 
two equal opposing forces of 10 lbs each, to produce one strain of 10 H>s. 

f And there is a hor strain to the same degree generated at every point along the length of each 
rafter. 

I It is immaterial whether we thus resolve o p directly into ob aud o z, (as though the head of the 
rafter rested agaiust a vert wall at ©>; or whether we hrst resolve it between the two rafters,into o e 
aud or. For iu the latter case we must add to o c a thrust (= or = c z) produced in o a by the trans¬ 
verse pres (similar to op) of the head of the oliter rafter; aud the sum of these two (o c and o r) is 














TRUSSES 


553 


ch 

If. 

ft 

»7 

"J 

it 

i. 

8 

se 

re 

re 

tt 

si 

I- 

f, 




t| 

' 




think is abundantly sufficient, with the assumption of 40 lbs per sq ft. First prepare a diagram 
of the truss, on a scale of say % inch to a ft. This diagram will consist of but three lines. We will 
use the same scale of ^ inch to represent 1000 lbs of either wt or strain. Make HI by scale equal to 
1 inch ; that is to the 8000 lbs uniformly distributed wtof one rafter and its load. Also draw I E, and 
measure it. It will be equal in this case (accidentally) to H I, or 8000 lbs; and this is the amount of 
pull along the tie-beam.*Now we see by table, page 403, that average while pine breaks under a pull 

10000 

of 10000 Bis per sq inch; so that for a safety of 3, we must not subject it to more than—-— = 3333 

O 

lbs pull per sq inch. The weakest part of the tie-beam is where it is cut into, near the ends, for foot¬ 
ing the rafters; and even what is there left by the cut, is usually still farther reduced by the holes 
of the bolts or spikes driven into it through the feet of the rafters. Therefore, allowing for these 
things, we must give to the tie-beam at that point a transverse section, of solid wood, equal at least to 
8000 

= 2.4 sq ins. This would no doubt be sufficient to resist the pull; but there are other considera¬ 
tions, such as danger of sagging or breaking down if persons should get on it; or if a moderate load 
should chance to lie laid upon it, &c, which cause the tie-beam (even when unloaded even by the wt of a 
plastered ceiling below,t as is here supposed to be the case,) to be made about as large as a rafter. 

If, instead of a beam, we had used an iron rod to resist the 8000 Bn pull, we should have reqd one 
with a breaking strength of 8000 X 3 = 24000 lbs; and by the table of bolts, page 409, we see that a 
diam of full i-|- inch would suffice if upset; or of full 1.04 inch if not upset. See Rem p 408. 


Now, as to the rafters, each of them is an inclined beam, supported at both 
ends, and uniformly loaded with 8000 B>s ; which is equal to a center load of 4000 lbs. But for a safety 
of 3 against 4000, we will find its dimensions for a breaking center load of 12000. Its 

length must here be taken as if measd horizontally between ils end supports, or 15 ft. 

We will assume for it some probable approximate depth ; say 9 ins. Then by Art 20, p 497, we 
find that for a breaking center load of 12000 lbs, its breadth will be 4.94, say 5 ins. Therefore to be 
safe with 4000 lbs center load, each rafter must at its head be 5 ins broad, by 9 ins deep.f 

We may take G L (= 7200 lbs) instead of H I (= 8000 lb; as the load ; but then oa (= 16.77 ft) 
must be taken as the span instead of w a (= 15 ft). The result is the same in both cases. 

As to the strength of the rafter against its lougitudiual compressive strain, we find that G L Fig 
5)4 measures 7200 lbs ; therefore op = 3600 lbs. Drawing the parallelogram o bp z at the top of the 
rafter Fig 534 we find oz — 7200 lbs. The total longitudinal strain in the rafter is = oz at the top ; 
and = ak = oz + LR = 7200 + 3600 = 10800 lbs, at the foot. Thus, so far as the longitudinal 
strain is concerned, the rafter might be of less cross-section at top than at foot; but in practice the 
expense of cutting the timber to that shape would generally more than compensate for the slight dif¬ 
ference of material, even assuming that it could be saved. It is uncertain how the transverse and 
longitudinal strains are distributed through the cross section; for the least sagging of the rafter 
would throw most of the longitudinal compressive strain on those fibres (on the upper side of the 
rafter) which are already under compression due to the transverse strain. For safety we will add to 
the cross-section required by the rafter as a beam, a sufficient area to he safe in itself against the 
greatest longitudinal compressive strain, or that (a k) at the foot, which we have just found to be 
= 10800 lbs; and will make the area uniform throughout. Now we find by table p 436 that aver¬ 
age white pine or spruce crushes under a pressure of say 6000 lbs per sq inch. Therefore it will have 
a safety of 3 under 2000 lbs per sq inch ; so that we must provide for each rafter f = say 5H sq 
ins of area of cross-section in addition to the 5 X 9 ins already found. These 5^ sq ins may be added 
either in the breadth of the rafter, thus making it say 5.6 X 9 ; or to its depth, making it say 
5 X 10.2. 

In the next three trusses we shall not enter Into this detail 
of calculation; as we conceive that this example sulliccs to 
elucidate its principle. 


Art. 8. Next to Fig 5, in point of simplicity, is Fig 6; which represents a truss 

for either a bridge or a roof of mode- 



* The pull I E along the tie beam will thus be equal to the uniformly distributed load on the rafter 
only when the rise o w is one fourth of the span. If ow exceeds one fourth of the span, I E will be 
less than H I, and vice versa. ... «... 

t The weight of an ordinary lathed and plastered ceiling- is 

about 10 lbs per sq ft; and that of an ordinary floor of 1 % inch boards, to¬ 

gether with the usual 3 bv 12 inch joists, 15 ins apart from center to center, is from 10 to 12 lbs per sq 

ft In preliminary calculations it is well to take the two together at 25 lbs per sq ft. 

t This is not a bad proportion of breadth to depth. If we had assumed say lo ins for the depth, we 
should have got a rafter so thin as to be laterally weak. Frequently, two or three assumptions and 
calculations may have to be made before wc hit upon a satisfactory proportion. 
























554 


TRUSSES 


traveled up the king-rod to o, and from there down the rafters to a and b ; or, indirectly , by a cir¬ 
cuitous route. That the king-rod sustains all between y and y, will be evident when we reflect that 
a beam a b, when firmly suspended at its center n, may be regarded as two separate beams nb, n a. 
One-half of the beam n b and its load would, in that case, manifestly be borne by the wall x, and 
the other half by «; and so with n o. Therefore, » upholds one-half of the beam a b and its load ; 
or, in other words, all between y and y. The king-rod trausfers the wt of and on y<y, to the heads 
of the rafters at o. This wt may, therefore, be considered precisely in the light of oue resting upon i 
o; and we may proceed to find the strains which it produces upon the rafters and tie-beam, by Art 
33. of Force in Rigid Bodies. Namely, on o n make o t, by scale, equal to said total wt of 

yy and its load, and the wtof the king-rod itself. Complete the parallelogram of forces omtd ; 
and draw its hor diag m d. Theu will o in, o d measure the straius produced by said total weight 
only, along their respective rafters; and cm, c d the pulling forces produced by the same wt only 
along the tie-beam ab; causing strain all along it equal to oue of them. 

The strain om or o d must be added to the other longitudinal strains oz and LK (Fig 5M) in the 
rafter, found as already' explained. 

For light railroad bridges, a truss like Fig 6, of 30 ft span and 10 ft 

high, may have a chord of 15" by 18"; rafters 10" by 10"; one rod of 2%" diarn; or 
two rods of 1%" tliani, aud several inches apart transversely of the bridge; which 
is far better than oue. 


The pull on the tie-beam will be I E added to cm or cd. Find the 

safe area by dividing their sum by 3333, which is the number of lbs per sq inch, giving a safety of 3. 
Theu regarding half the length of the tie-beam supported at both euds. aud loaded at its center with 
only one-fourth of the wt of and on the entire tie-beam, find its safe dimensions by the rules, p497, 
or by table, page 499. The resulting area, added to the safe area for the pull just- found, will be the 
eutire section of the tie-beam, unless some addition be made to the depth, to allow for what is cut 
away for the feet of the rafters.• See Rem, p 500, also Rem, p 355. 

As to the vertical king-rod, ri o, it must be strong enough to bear safely 

a pull equal to its own weight, added to the weight of and upon y y. If the rod is of good bar irou, 
it should have one square inch for a safety of 3, of cross-section for each 20000 lbs of said weight; 

If of wood, it must, for a safety of 3, have at least one sq inch for about each 
3333 lbs of said weight. A safety of 3 will be enough if the bar is not liable to vibration. 

When the kiug-rod is of wood, it is improperly termed a king post. Since a post is intended to sus¬ 
tain a load on its top, the term might lead to the inference that the upper ends of the rafters rested 
upon, or were upheld by the king-post; whereas, as we have seen, they actually upheld it. 

We add the calculated approximate dimensions for a truss. 

Fig 1 6, of 30 ft span; and 7V£ ft rise. Trusses 12 ft apart cen to cen. Wt of 

rafters and load on top of them, 40 lbs per sq ft of area of roof. Wt of and on the tie-beam, includ¬ 
ing floor, ceiling, load, and momentum, 100 lbs per sq ft. Timber white pine. Safety of each piece 3. 
Rafters 8>£ ins broad, by 11 deep. Tie-beam 8}4 broad, by II deep, 

without any allowance for cutting at feet of rafters. King-rod ljg inch diam if upset; or full if 
not upset, t 


With no floor or loading on the tie-beam, except its own wt, say 

500 lbs, we have, approximately enough, rafters 6% ins broad by 9 ius deep. 

Tie-beam, say same as rafter, or X 9- King rod, y A inch diam ; but 

it would be expedieut to make it rather more. Trusses 12 ft apart center to center. 

Art. 9. In Fig 7 we have a truss consisting of two rafters, a b, a d; a tie-beam, 
b d\ a king-rod, o c; and two struts or braces, e c, h c. Either the rafters or the tie-beam, or both, 
may be supposed to be uniformly loaded. 


- --O- y y 

of the tie-beam, and of any load of 
floor, ceiling, people, &c, that maybe 
placed upon that portion ; together 
with its own weight. But it also sus¬ 
tains. in addition to these, the weight 
of the two struts e c. he; part of the 
weight of the portions z r, aud x u, 
of the rafters ; and part of the weight 
of the roof-covering, snow, &c, that 
may rest on said portions. That it up¬ 
holds itself, y y, and the struts, is al¬ 
most self-evident; but that it upolds 
part of z r, aud a: u, and their loads, is 
not at first sight so apparent. Such 

struts are introdneed 
into trusses when the rafters 

become so long as to be in danger of 
bendiug too much, or of breaking un- 
... . . der their loads; or else requiring the 

use of inconveniently large timbers to make them of. They act like posts in affording partial sunnort 
to the rafters. They carry a part of the strain upon the rafters down to the foot c, of the king-rod- 
and the king-rod carries it from there up to the tops, a. of the rafters. From a it passes down through 
the entire length of the rafters to their feet. Thus, it is seen that the action of the struts consists in 
relieving the ratters from a transverse , or cross-strain which would endauger their safety; and in 

*Iu cases where no appearance of sagging would be admissible, it is not alwavs enough that the 
rafters and tie-beam be safe; for they may be perfectly safe, and yet sag too much for some purpose* 
When such is the case, refer to table, page 512. 1 ' "* 

t We have known country road bridges, Fig 6, of 30 ft span, and 7>* ft rise, of two trusses 18 ft 
apart, in which neither the timbers, nor the probable loads, were larger than in this example. 





















TRUSSES 


555 


converting it into a longitudinal strain in the direction of their length, in which they can resist it 
with less danger. As we proceed with the subject of trusses for bridges as well as roofs, it will be 
seen that this is the grand duty of such struts and obliques generally. In roofs they thus assist the 

I rafters ; and in bridges the chords. 

Where each rafter is a solid unbroken piece, as in our figures, it is uncertain what portion of the 
load zr or xu is actually borne by the strut ec or ft c. The introduction of the struts thus renders 
it impossible to calculate the strains with certainty by the above described method. For this reasou, 
aud for safety, we adopt auother method, in which we begin with an assumption which is not strictly 
I correct, but which simplifies the problem and enables us to calculate with exactness for each mem¬ 
ber a strain which is sufficiently near to the probabie true oue for most practical purposes. This as¬ 
sumption is that the rafter is in two parts, aU and U ft ; that these parts are connected by a per¬ 
fectly flexible joint at U, so that all of the load of aud on z r rests upon the strut e c ; aud that the 
load of and on xj and j z rests directly upon j. Draw e o aud ft n vertically ; and make each of them, 
by any convenient scale, equal to the weight in lbs of either zr or xu, and its load. From o and n 
draw the dotted lines, ot, nw, parallel to the struts; and ok, nv, parallel to the rafters; thus com¬ 
pleting the parallelograms of forces, ekoi, and hw nv. Draw the horizontal diagonals i k, and v w.* 
Then by Composition and Resolution of Forces, either e k or ft v, measured by the same scale as be¬ 
fore, will give the longitudiual strain in lbs upon each one of the struts. This strain presses the 
struts lengthwise from head to foot. They are also strained longitudinally and transversely by their 
own weight, as the rafters in Fig 5 were strained by their own weight and that of the roof; but in 
practice these strains in the struts, due to their own weight, are so trifling compared with that from 
the roof portions which they sustain, that they may be neglected. 

Therefore, each strut may be regarded as if a vert pillar, 
bearing' a load equal to ek or hv. Now, the strain ek, along the strut 

ec, is compounded or composed of the vert strain es, (which is equal to half ot e o, or oue-half of the 
wt of and on z r ;) aud of the hor strain sk. And the strain ft v along the strut ft c, is compounded 
of the vert strain ht, (which is equal to halfot hn, or one-half of the wt of and on in) ;* and of the 
hor strain tv. These two hor strains sk and tv neutralize or counteract each other, by pressing 
against each other at the feet of the struts ; and therefore only the vert ones es and ft t pull upon the 
king-rod ; and they pull it to an extent equal to half the weights of and on zr aud xu.* 

The Ring-rod, therefore, upholds in all, 1st, the weight of the two 

struts; 2d, the wt of and on yy. 'Ad, half the wt of andonzrand »iz;*and 4th, its own wt. It 
must, therefore, have sufficient sectional area to safely sustain a pull equal to the sum of these four. 
This area may be found by means of the table of bolts on p 409. 

Make a g by scale equal to the sum of these four wts, plus the weight of xj and jz, which rests di¬ 
rectly upouy - . Draw gm, gl parallel to the rafters; and Itn hor. 

For the dimensions of the rafters, ab, ad, commencing with what 

they require as beams, supported at the ends, bear in mind that the introduction of the struts ec, h c 
converts each rafter, as ab, into two shorter ones, ac, eft; each of which sustains, in the present 
case, only oue-half the load of aud on the whole rafter; or only 14 of it as a center load. Find the 
safe dimensions for the short beam, with its smaller center load, by rules, p 497, or by table, p 499. 

The compressive strain on a U is a m. That on U 6 (or the greatest comp strain) is = a m + er. 
Divide this sunt by 2000 (or by whatever other number of lbs may be considered the safe crushing 
strength of the timber). The quot is the safe area in sq ins reqd for that purpose. Add it to the area 
previously found for the rafter as a beam. The sum is the eutire area required. 

The tie-beam. The pull on the tie-beam is fm + si. Divide it by 3333, the 

safe pull in lbs per sq inch. The quot will be the safe area reqd for that strain. Then consider one- 
half ot the tie-beam to be a uniformly loaded beam supported at each end ; and find the safe dimen¬ 
sions by rules, p 497, or by table, p 499. To these dimensions add the area just found for the hor 
pull; the sum is the entire area reqd for the tie-beam, unless some addition be made to compensate 
for the cutting away at the feet of the rafters.t 

Below are the calculated dimensions for tw r o trusses. Fig 7, 
of 40 ft span ; 10 ft rise : and 12 ft apart from center to center. In the first of 
these the tie-beam with its floor, ceiling, and other load, are assumed at the rate of 100 lbs per sq ft 
of floor; while, in the second, no specific load is assumed for that member, for reasons before given. 
In both, the wt of the rafters, with their roof-coverings and load of snow, and wind, is taken at 40 lbs 
per sq ft of roof surface between the centers of two trusses. The safety of each separate part is taken 
at 3 ; except that the unloaded tie-beam is fixed.by rule of thumb. Timbers white pine. The great 
est dimension in each case is the depth. Dimensions in inches. 1st. Rafters 8 X 10. 
Tie-beam 8 X 15. Each strut 4)4 X 4)4. King-rod 1% diarn if upset; or 2 ins if not upset. In prac¬ 
tice it is better to make the struts as broad as the rafters. 2<1. Rafters 6 X 8. The 
tie-beam requires, theoretically, only 16 sq ins area; we will make it 6X8, like the rafters. Each 
strut 434 X 4*4; (the same as in the other.) King-rod % diam if upset; or scant 1 inch if not upset. 
If a tie-rod were used instead of a tie-beam, its diam would be \]4 inch if upset; or 1.6 if not. 

Art. 10. Fig 9 is a truss with a tie-beam a b ; two rafters to a, zb ; two queen- 
rods,! or queens, wt, zt, and a hor straining beam d. Il may represent a roof uniformly loaded 
along the rafters and straining beam ; and having a uniform load along the tie-beam. Or only one 
of these loads may be supposed to exist, as in a bridge with a load along aft ; or a roof with its load 
along aw zb. The queen wt supports, besides its own weight, all the weight ot and on the part sy 


* Each strut will thus bear halfot the wt of and on zr, or zv, only when, as in Fig 7, the incli¬ 
nation of the strut is the same as that of the rafter. If the strut is steeper than the rafter, it will 
bear more than half; but if it is less steep than the rafter, it will bear less than half; the remainder 
being in every case borne by the rafter. The parallelogram of forces will of course show all this. 
When the inclinations of a rafter and strut are not equal, we cannot draw hor diags ik, v w; but 
from the points i,k,v,w, we must draw hor lines to the vert diags eo, and ft n. 

t When a tik-bbam is so i.ono that it must be spliced, allowance must be made for the weaken¬ 
ing effect of the splice. For Splices, see p 611; and for other joiuts, p 613. 
j The queens are frequently made of wood. 










556 


TRUSSES. 


of the tie-beam ; and the other one z t, that of and on ny; sand it each being halfway between a 

queen and au abut. These are the 
only strains on the queens; so that 
their proper diarns can be found by 
table of bolts, p 409. The parts of 
the tie-beam from s and it to the 
abuts, or walls, as well as whatever 
loads those parts may bear, are sus¬ 
tained directly by the abuts. 

The queens transfer, as it were, 
the weights of themselves and of s y 
and uy, with their loads, directly to 
w and z. To find the strains on the 
various parts of the truss, first from 
the center U of a rafter aw, draw a 
vert line U H ; aud from w draw a 
hor line tcH to meet it. Join Ha. 
Make H I by scale equal to the wt of 
only one rafter ami its uniformly 
distributed load. Also draw og vert, and equal, by the same scale, to the wt upheld by the queen-rod 
wt, added to one-half the wt of the strainiug-beam d. and its load; for it also presses vert at o. 
Draw g m hor, or parallel to the straining-beam; aud g c parallel to the rafter; thus completing the 
parallelogram o eg m of forces. 



The strains on the straining-beam d. The hor line IE and oc to¬ 
gether, give all the hor pres against the end w of the straining-beam d; audit is plain that a similar 
process on the other side of the truss, would give an equal pres against the cud z. These two equal 
pressures reacting against each other, produce a strain, equal to one of them, throughout the entire 
length of the straining-beam ; and therefore, the beam must be regarded as a pillar with a load equal 
to this strain, on its top ; and the dimensions and area of section, for safely supporting it, may be 
found by the rule, p 458 ; or table, p 459.* 

But beside this, the straining-beam, if loaded, must be regarded also as a beam supported at both 
ends; and the area necessary for this, as found by tables, page 499 or 512, must be added to that al¬ 
ready found. 






The strains oil the rafters. First, consider a rafter w a as an inclined 
loaded beam supported at both ends; and find the proper dimensions and area, by the rules on page 
496; or by the tables, p 499, or 512. 

Second, add together om and the other longitudinal strains oz and r, R (found as in Fig 5X) in the 
rafter, aud divide their sum by 2000 (or by whatever other number of lbs may be considered the safe 
crushing strength of the timber). The quot is the safe area in sq ins reqd for that purpose. Add it 
to that already found for the rafter as a beam. This last sum is the total area reqd. 

The tie-beam. The hor strain, or pull on the tie-beam, will be equal to the 
push on the straining-beam; and is represented by I E and o c together. Find the safe area by table, 
page 463 ; or by dividing the hor strain by 3333, which is the pull in lbs per sq inch that ordinary 
building timber will bear with a safety of 3. 

Then, since in this truss the queens divide the tie-beam into three lengths, each of these must be con¬ 
sidered as a separate beam, (loaded or unloaded, as the case mav be,) supported at each end. Its safe 
dimensions being found, add the area just found for resisting the pull. Add, if reqd, an allowance 
for the cutting away at the feet of the rafters. 

Below are the calculated approximate dimensions for two 
trusses. Fig 1 9, of sixty ft span ; 15 ft rise; and 12 ft apart from center to 

center. All the conditions the same as for the preceding example of Fig 7. 1st. Rafters 12 ins broad, 
by 14 ins deep. Straining-beam 12 broad, by 12 deep. Tie-beam 12 broad, by 12 deep. Each queen 
rod IX ins diam if upset; 1 X if not.t 

2d. Rafters 10 X 11%. Straining-beam 10 X 11. Tie-beam, say 10 X 12. Each 

queen-rod X inch diam. Unloaded tie-rod, 1 X- 

The proper size for each piece, so that they shall all be stiitable 

for the. truss, cannot be determined at once. We must find any dimensions that 
will answer for each piece by itself: and afterwards adjust them by recalculation, perhaps 3 or 4 
times. Great accuracy is not necessary in doing this. See Note, p 573. 


* A strut or tie cannot be strained along the direction of its length 

by a force acting at one end, unless there is at the other end an equal force acting in the same straight 

line but in the opposite direction, and which may be either one 
single force, or the resultant of two or more forces neither of 
which acts in that direction. Hence if in 

Fig 9 we place a load at Z only, a parallelogram v egn of forces 
will not give the hor strain v e along the beam Z W, because 
there is then no equal reacting hor force at the other end in the 
direction from H towards W. In that case a load at Z only, 
(represented by z c in Fig X) produces at z the two strains z n, 
ee\ which last pressing towards a tends to make z 5 revolve 
around b as a center, thus forcing z downwards, and the joint w 
upwards, thereby causing the distortions seen in Figs 9)4, 9^. 
The force z e therefore evidently tends to break the joint w\ 
and with a moment equal to the force z e (in tons or lbs, &c) 
If the moment of resistance of the joint can withstand this the 



mult by its leverage w o perp to z a _ _ 

truss will remain unchanged; but a simple strut from z to a would remove all danger, bj ..astaiuing 
the whole of the force z e effectively, and thus relieving the joint w entirely, 
t See Rem, p 408. 




















TRUSSES. 


557 



Fig. 9%, 


^SaTc 

Fig. 9'A. 


Rkm. The truss in Figs 9 and 9*4 affords a good opportunity for alluding, in a general way to 
the principle of couutcrhracing, and to the necessity (as stated in Art 7) of ad- 
tiering to a triangular arrangement of the parts of a truss. So loug as this truss is uniformly loaded 
throughout its length, it is well arranged for sustaining the resulting strains; because the strains on 
each side of the center are equal, and balance each other. But if a heavy load be placed along a o 
only (i ig 9*4) its teudency to depress ac will produce the derangement shown in Fig 9% ; because 
the horizontal pressure from a toward t, Fig 9*4, will then become greater than that from t toward *. 
Jne two triangular portions will still retain their original fig; but owing to the ease with which the 
4-sided portion, nmce, has been deranged, and changed to s tee, their position becomes altered to 
the dangerous one in Fig 9The diag c m 
has been lengthened to ct; while the diag <5 

en has been shortened to es. Now, if there 
had been a strong bar of iron reaching from 
c to m, with its ends firmly attached to those 
points, it would have divided the whole truss 
into triangles ; and then the diag cm could 
not have become lengthened to ct by any 
strain less than one sufficient to break this 
iron bar by pulling it apart; therefore the truss would have remained safe, and unchanged in figure ; 
for the bar, while preventing cm from lengthening to ct, would, as a consequence, prevent en from 
shortening to es. Or, omitting the iron bar at cm, suppose a stiff, unbending inclined post to 
be inserted between e and n. This also will divide the whole truss into triangles; and it is then 
plain that en could not be shortened to es by any strain less than one sufficient to break the post by 
crushing it. Therefore, in this case also, the truss would have remained safe, and unchanged in 
figure ; for the post, while preventing en from shortening to e s, would, as a consequence, prevent cm 
from lengthening to ct. Either the bar or the post would be a counterbrace against the effect of un¬ 
equal loading. With a uniform load it is not needed. Neither are additional counterbracing pieces 
reeded in bridge trusses of the forms Figs 10,11, 12, 13, provided each web member is so constructed 
as to bear alternately compression and extension. 

The next Fig 9 6, shows the bad effect produced in a __ A _ 

truss longer than Fig 9%, when the web members are 
not so constructed. In the Burr bridge, Fig 36, ana in 
some others, although the truss is divided into trian¬ 
gles. yet the inclined braces, i c, &c, are often impro¬ 
perly adapted to bear compression only; their ends not 
being firmly attached to the chords. Consequently, 
with a heavy load at a, the derangement shown in Fig 
9 6 (analogous to that in Fig 9*4) takes place. To pre¬ 
vent it, counterbracing must be resorted to, either by 
inserting struts or ties along the dotted diagonals; or 
by making the braces capable of resisting tension as 
well as compression. The last method shows that counterbracing can be performed without the ad¬ 
dition of pieces specially called counterbraces, an l denoted by the dotted diagonals. All that is re¬ 
quired in the principle of counterbracing, is to so arrange and connect the several web members, 
that the strain produced by unequal loading at any point, as a, between the abuts; or along any 
portion of that distance, shall be properly transferred by them to both abutments. 


i/ 

X 

X 

X 

X 

\ w 

Fig. 9« ^ a 0 8 

2 n pF 


Fig. % 



Art. 11. The strains in such trusses as Fig's 10 and 11, 

may be found by three very simple processes when the truss and its load are uniform 
from the center each way.* When this is the case it is usual and safe to assume that 
the half load e p, Fig 1U or 11, on the right hand of the center e, rests on the right 
hand support p ; and that the half load e a on the left hand of the center e, rests on 
the left hand support «.f It is often assumed also, for simplifying the calculations, 
that the entire weight of the truss and its load is distributed along one chord only. 
This is plainly incorrect; but inasmuch as the extraneous load (such as the covering 
of slate, snow, etc., on a roof, and the travelling load on a bridge) in many cases ac¬ 
tually does rest on one chord only, and is great in comparison with the weight of the 
truss alone, the error arising from the assumption in such cases is not of practical 
importance. 

But in bridges of great span the weight of the truss may bear a large proportion 
to that of its load ; or there may be an upper and a lower roadway, one resting on 
each chord; and a roof truss may have to bear not only the covering, snow, &c., on 
its upper chord or rafters: but a floor with a plastered ceiling beneath it, and all 
the load incident to any ordinary room, on its lower chord. In such cases the entire 
weight of the truss and load must be properly distributed along both chords before 
we can correctly find the strains. But this will in no way affect the principle of the 
three processes which we are about to explain, and as we proceed we shall give di¬ 
rections for both cases. 


* It is not necessary that the entire load should in itself be uniform ; but merely 
uniform each way from, the center. Thus at e may be say 1 ton ; at d and m each say 
5 tons; at c and n each 2 tons, &c. 

f This assumption is untrue, and opposed to the unvarying law that 
every individual portion ot the entire weight rests partly on each suppoit. thus, 
one portion of the load at o rests partly on p and partly on a; and so with every 
other portion; and on this fact depends the difference in the methods of calculating 
the strains from uniform, and ununiform or moving loads. When, however, the 
weight of the truss and load is uniform each way from the center we obtain correct 
results, and more readily by adopting the erroneous assumption. 




















558 


TRU8SE8. 

















































TRUSSES. 


559 


Beginning' then with uniformly <listribute<l weights of 
j truss and load, and assuming all of said weights to rest on the long chord a p, 
j prepare a correct skeleton diagram of the truss (or at least of one-half of it), such as 
j Figs 10 and 11, in which the height or depth e i, Fig 10, is the vert distance between 
J the centers of the depths of the two horizontal chords. A scale of from to % of 
an inch to a foot will generally be large enough. 

Then the first process is the very easy one of ascertaining how much of the 
i total uniform weight is to be considered as sustained at each point of support along 
| either the top or the bottom chord, as the case may be; remembering that one half 
of each end panel is sustained directly by the abut nearest to it, as in the preceding 
! cases. 






In order more fully to illustrate the following Articles, we shall assume each of 
the trusses, pp 558, 570, 571, 572, 574, 585 and 586, to be 64 ft long, and 16 ft high, 
and to be divided into 8 equal panels. Total uniform wt of one truss and its load, 
32 tons; or 4 tons to a panel. Consequently there will be 9 points of support to each 
truss. Thus, in Figs 14, 15, and 16, in which the load is supposed to rest on top of 
the truss, and in Figs 10 and 11, in which it rests upon the bottom, the points of 
support are at a, b, c, d, e, in, n, o, p. Some of these are not shown in the first three 
Figs. If both chords are loaded, there will be points of support in the short one 
also. Thus, in Fig 10 there will be 7, and in Fig 11 there will be 8 of them. Now, 
in Figs 10 and 11, w, x, y, etc., being midway between the points of support, it is 
plain that (assuming all the weight to be on the lower chord) the point o must sus¬ 
tain that portion of it comprised between w and x; nail between y and x; while 
the abutp sustains directly the portion from w top. The same principle applies to 
all the other trusses; and equally so whether the panels be of the same width or 
not; each point of support is assumed to sustain all the uniform wt of truss and load 
between itself and the two points midway to the adjacent points of support, how¬ 
ever unequal the two distances may be. In our Figs 10 to 16, the strong dotted lines 
of the web members represent ties; the full lines, struts. The dots intimate that 
chains may serve as ties. When the panels are of equal length, p o,o n, etc., the dis¬ 
tance from p to w will be but half a panel; so that each abut will directly sustain 
but half as much wt as each other point. Therefore, to find the amount of wt sus¬ 
tained at each of the nine points of support, we have only to div the total wt (32 


tons) by a number less by 1 than the number of points. 


32 

The quot — = 4 tons, will 


j be a full panel-load, to be at each point, except the two end ones, a and p, at the 
abuts; at each of which it will be but half of one of the full panel-loads, or two 
tons.* The amounts of these panel and half-panel loads should at once be figured on 
the sketch at their proper points, as is done in our Figs ; a 2 being placed at each end 
of the truss; and a 4 at the other points. Each of these panel-loads of course causes 
a vert strain equal to itself where it rests. As the strains on one half of the truss 
are the same as those on the other half, the numbers need only be written on one of 
them; indeed, the sketch, as a general rule, need show but one-half of the truss. 

If there is a load on the other chord also, it must be in the same 
way divided among the points of support of that chord, and be figured as before. 

The second process. All the panel-loads are of course eventually trans¬ 
mitted through the truss to the abuts; as is manifest from the fact that each abut 
sustains half the total load. But each panel-load, while travelling, as it were, up 
and down alternate web members from its original point of support, to the nearest 
abut, places, so to speak, an additional load, or more correctly produces an addi¬ 
tional vert strain equal to itself, at every intervening point of support in each 
chord.f Our second process consists ^n finding the amount of this additional vert 
strain at each point of support. 


* This of course is only when the end panels are of the same length as the others. 
When not so, the loads at the points of support and on the abutments will plainly 
vary from the above. 

f In trusses like Figs 10 and 11, with two horizontal chords, the panel loads are 
transferred directly from their points of support via the web members to the abuts. 
In such trusses the strains in the web members are least at the ccn of the span, and 
greatest at its ends ; while those in the chords are greatest at the cen and least at the 
ends. This is indicated in tin; Figs by the dill thicknesses of the lines representing 
these members. But in Figs 14, 15 and 16, the panel loads divide at their respective 
panel points, as explained in Rem p 573 ; a portion of each panel load going directly 
to the nearest abut via the sloping rafter; the remainder going first to the cen e via 
the web members, whence it finally reaches the abuts via the rafters Figs 14 and 16, 








560 


TRUSSES. 


In Figs 10 and 11, with parallel horizontal upper and leaver chords, the vert strains 
are very easily found, thus : Remembering that only half of the center panel-load 
straiu at e goes to each abut, begin with the 4 tons at e. 

In Fig. 10 these 4 tons first go up the tie ei to i, where they produce a vert strain 
of 4 tons, which figure as in the diagram, lint at i these 4 tons separate; 2 of them 
going to the abut p, and the other 2 to the abut a. The last 2 first pass down i d to 
d, where also they produce a vert strain of 2 tons, which also figure, as must be done 
with all that follow. At d these 2 tons unite with, or as it were take up and carry 
along with them the 4 tons already there ; and the entire 6 tons go up the tie dj to j, 
where they produce a vert straiu of 6 tons. From j these 6 tons go down the strut 
jc to c, where they also produce a vert strain of 6 tons. At c these 6 tons take up 
the 4 tons already there, and the entire 10 tons go up ck to k; and thus the process 
continues until 14 tons find their way to the abut «, where they meet the 2 tons of 
load already there; thus making 10 tons, or one-half the wt of the truss and its load; 
which is a proof that our work is correct so far. 

In Fig 11, the 4 tons at the center e separate there; 2 of them going up e t to i , and 
thence to the abut a as before. 

In either Fig if there is a uniform load on each chord there is no 

difference in the second process; for after having by the “first process” divided each 
load among the points of support of its own chord, the portion at each point must 
be taken up as it occurs, and carried on with the others to the abutment as before. 

The third and last process cousists in completing our sketch, or diagram, 
in such a manner as to enable us to measure by scale the strains produced along 
every member of the truss, by these vert strains thus accumulated at the diff poiuts 
of support, a, b, c, l, k, etc. 

To do this in Figf 10 (that is, whenever the web members are alternately 
vertical and oblique) from each point of support of one chord only, beginning at the 
center apex t', draw a vert line as iv,jv, etc., to represent by any convenient scale, 
the vert strain figured at said point; except at the center one t, where the vert line 
must represent only half the vert strain, inasmuch as that is all that goes to each 
abut. Draw also the lior lines v u, v u , etc. Then will each oblique line i u,j u, etc., 
give by the same scale the strain (2.2, 6.7,11.2,15.7) along its own oblique web mem- 
uer, as figd. The lior lines give the hor strains exerted on the chords by each oblique 
at both its ends. We have figured all these strains (7,5, 3,1) at the head and foot of 
each oblique. Each of these hor strains extends from the ends of the oblique, to 
the center of the chord; therefore the end stretches of the chords bear 7 tons hor 
strain; the next ones 7 +5 = 12 tons; the next ones 7 +5+3 = 15 tons ; and the 
center one 7 + 5 + 3 + 1= 16 tons; all of which are figured along the chords.* 

We have said that the vert, hor, and oblique sides of the triangles give the strains, 
but it would be more correct to say that each of them gives a force, which being 
balanced by the other two, thereby causes a strain equal to itself, instead of mo¬ 
tion. 

The hor strains at the centers of the two chords will be 
equal in both Figs 10 and 11, whether one or both chords be uniformly loaded; 
or if the truss be inverted; with only the exception in the foot note.* 


or via the two inclined ties in Fig 15, of which i a is one. In such cases the strains 
on the web members are greatest at the cen of the span, and least at its ends ; while 
those on the tie beam and rafters, Figs 14 and 16, and on the tie rods and hor chord 
Fig 15, are greatest at the ends of the span and least at its cen. This also is indicated 
in Figs 14, 15 and 16 by diffs in the thicknesses of the lines, and is the exact reverse 
of the case of Figs 10 and 11. 

* When at the central apex i, Fig 10, the two ends of the ojjliques id, ini, which 
meet there, are so arranged as to butt tight against each other, then the ceuter hor 
strain of 1 ton at that point is not borne by the chords, but by the obliques them¬ 
selves; so that there will then be that much less strain at the center point of that 
chord than along the center stretch d tn of the other chord. But if instead of this, 
they abut against, the. chord , at some little distance from each other, then the chord 
also receives the strain; so that the hor strains at the centers of the two chords 
become equal, as we assume to be the case in all our Figs of uniform trusses uni¬ 
formly loaded each way from the center. The same remark applies to the hor strain 
of .5 of a ton at the ceuter apex e of Fig 11. 





TRUSSES. 


561 


The strain along the center vert tie e i of Fig 10, will be equal to the 4 tons at e; 
and when the entire wt in assumed to be on the long chord , the vert lines at the other 
points of support will give the vert pulling strains on the other verts, as 6, 10, 14. 

But with loads upon both chords this last will not be the case; but 
the strain oil each vert tie will then be equal to the vert strain at its foot.* 

If the loaded truss is inverted, the verts become struts or posts, and 
the obliques ties; also the strain on each vert is then the one figured at its top ; but 
the amount of strain on each part of the entire truss will remain as before. 

In Fig 10 all the uniform wt is on the long chord, and the resulting strains are 
j all figured on the diagram. We add the strains that would occur in case there were 
[ an additional uniform load of 6 tons on the upper chord from l to t. This would give 
1 ton at each point of support along that chord, except the two end ones l and t, at 
i each of which it would be but .5 of a ton. All these must be figured on the short 
chord of the diagram, as were 4, 4, Ac, on the long one. The student may then 
work out the case for himself. We repeat that uniform trusses and loads require no 
counter-bracing. 


For Fig 10, but for a load on each chord. 


et' = 

4. 

tons. 

i d = 

2.8 

tons. 

a b or 

Ik = 

8.5 

tons 

dj = 

6.5 

44 

jc = 

8.39 

(4 

be or 

kj = 

14.75 

44 

c k — 

11.5 

u 

k b — 

13.98 

44 

c d or 

j i — 

18.50 

44 

b l = 

16.5 

a 

l a = 

19.01 

44 

d e or 

at i = 

19.75 

44 


For tile “ third process ” in Fig 11 (or when all the web members are 
oblique, whether equally so or not) after having found and figured the loads and vert 
strains at each point of support precisely as directed for Fig 10, then from every 
such point in both chords draw a vert line as ev, i v, d v, &c ; and on it lay off sepa¬ 
rately by scale both the vert strain that comes to that point through a web member 
from towards the center of the truss ; and the one that goes from it through another 
web member towards the abut; except that at the very starting point e. Fig 11, there 
i is but one vert strain (the one of 2 tons going from it); and at the very end a also 
I there is but one, namely that of 14 tons coming to it along the oblique l a ; for the 
*2 tons at a are not to be included, because they do not reach a by means of a web 
member. Therefore both at e and at a only a single vert strain is to be laid off.f 


* It is so in botli cases, for in any of these trusses under stationary loads 
the strain along a web tie whether vert or oblique may be considered to commence 
at its lower end, that being the end at which the panel loads first act on their route 
' to the abut,and up which they as it were work their way. Hut under moving loads 
| the same member may have to act both as a tie and a strut; hence the remark 
will not apply to such. Referring to what is said above, when the entire wt is 
j on only one chord, the vert strains at the two ends of any tie in Fig 10 are equal. 
Hence a line drawn to represent the upper one, may be assumed to repre¬ 
sent also the lower one. But when both chords are loaded, the vert strains 
figured at the two ends of any tie are unequal, and we must then have regard 
i to the true principle. If the verls should he struts or posts (as 
if Fig 10 should be inverted) then any strain along them must be received from 
their tops, or the reverse of the case with ties. It will aid the student very 
much in what follows to familiarize himself with the idea that strains pass 
only down the struts, and up the ties. 

f When all the wt of truss and load is assumed to be on the long chord as in our 
Fig 11, then the vert strain that comes to any point in the short chord by one web 
! member is plainly the same in amount as that which goes from it by the other web 
member; and hence onl} r one vert measurement need be laid off for it, as is seen at 
i v,j v . k v, l v, in the Fig. But at the long chord (or at both chords when there is 
a load on both) the vert strain that comes to any point is less than the one that goes 
from it towards the abut, and is evidently the one last figured at that point, as tho 
2, 6, 10, Ac, tons at d , c, b, Ac, in Fig 11 ; while the one that goes from that point 
towards the abut is as evidently equal to the sum of the two strains figured at that 
point, as the 6,10, 14, Ac, tons at d, c, b , Ac. When both chords are loaded there will 
be two vert strains to be figured at each point of support in both chords, except at 
the very starting point, and at each abut, where will be but one in any case. 

37 









562 


TRUSSES. 


Draw also the lior lines, thus forming a series of triangles (as ivu, ivu,jvu,jv u , 
&c, of the upper chord; and avu, bvu, b zu, &c, of the lower chord), each with 
one vert, one hor, and one oblique side. Then the combined lior strain exerted 
upon either chord, by the obliques at any point of support and by the part load (if 
any) supported there ; will be measured by the sum of the two hor lines directly oppo¬ 
site to said point, except at the center e, and at the end a. at each of which it will be 
measured by the single hor line v u, or v u, opposite each.* These hor strains (7, ft, 
3, 1 tons on the upper chord; and 3.5, 6, 4, 2, .5 on the lower chord) are figured close 
to the points of support at which they occur; and the total hor strains on the sev¬ 
eral stretches of the chords are figured midway of said stretches.f 


Tlie strain along 1 any oblique as j c, will be measured by the oblique 

side./ u, or c u, of either one of the two triangles on either side of it; and this will 
be the case whether one or both chords be loaded; or if the truss and load be in¬ 
verted. All the strains in Fig 11 (loaded on the long chord only) being figured on 
the diagram, we give below the strains that would occur in case there were an ad¬ 
ditional uniform load of 7 tons on the short chord from l to t; which would give 1 
ton at each point of support along that chord, except the two end ones l and t , at 
each of which it would be but .5 of a ton. All these must first be figured on the 
short chord of the diagram, as were 4, 4, <fcc, on the long one. The student may 
then work out the case for himself. 


For Fig: 11, hut for a load on each chord. 


e i — 2.06 tons, 
t d = 3.09 “ 
a j — 7.22 “ 
j c = 8.25 “ 


c k = 12.36 tons 
kb = 13.40 “ 
b l = 17.53 “ 
l a = 18.04 “ 


a b = 4.38 tons 
b c = 11.88 “ 
cd = 16.88 “ 

d e — 19.38 “ 
at e = 19.88 “ 


Ik— 8.63 tons. 
Tcj = 14.88 “ 

j i = 18.63 “ 

i to center = 19.88 “ 


Art 12. The strains in Figs 10 and 11 may readily he calcu¬ 
lated (after having by the “first and second processes” found the vert strains at 
all the points of support) whether one or both chords be loaded, or if the truss and 
load be inverted. Thus, divide the hor stretch of an oblique by its vert stretch ; the 
quotient will be the natural tangent (.5 for Fig 10, and .25 for Fig 11) of the 
angle (26° 34' in Fig 10, and 14° 2' in Fig 11) which the oblique forms with a vert 
line. Divide the actual length of an oblique by its vert stretch; the quotient will 
be the nat secant (1.12 for Fig 10, and 1.03 for Fig 11) of the same angle. Then 
the strain along any oblique in Fig 10 or 11, is found by multiplying the 
vert strain that travels towards the abutment along said oblique, by the nat secant. 

The hor strain on cither chord, caused by either end of any oblique, 
is in Fig 10 equal to the vert strain that travels along said oblique towards the abut¬ 
ment, mult by nat tangent. And in Fig 11, it is equal to the vert strain that comes 
to, added to that which goes from any end of an oblique, mult by nat tang; except 
at the center and end, where the single vert strain must be mult by nat tang. 

All this is simply because that if we assume the vert side of any one of the tri¬ 
angles to be radius or 1, then the hor side becomes by that same scale the nat tang 
of the angle which it subtends; while the oblique side becomes the nat secant of 
the same angle. 

The principle of Ihe mode of finding: the strains in Figs 10 and 

11 is this. We know that if three forces are in equilibrium with each other at any 
point, the lines which represent them will form a triangle. Fiow at every point of 


* Each of the upper hor lines uu, u u, <fcc, in Fig 11 is to be considered as com¬ 
posed of two separate ones v u, v u, &c ; the right hand one of which measures the 
hor strain caused by the vert strain that conies to i,j, Ac; while the left hand oue 
measures the hor strain caused by the vert oue that goes from i,j, &c, towards the 
abutment. Such lines as u u , u u, &c, occur only when all the wt is on one chord ; 
for when both chords are loaded, the vert strain that comes to, and that which goes 
from any point of support differ, therefore requiring two unequal vert measurements, 
and two unequal hor lines at each point of support of both chords, except at the 
center, and at the ends; at each of which will be but one. 


fThe common rule 


Total uniform wt X span 


= hor strain at center 


8 times the lieiglit 

of either hor chord, is not strictly correct except when both chords extend the full 
length of the span, and are both loaded throughout their entire length; or in the 
impossible case of the entire wt being on the long chord. Still in ordinary cases it 
is a sufficiently close approximation. On this subject see Art 19. 










TRUSSES. 


563 


support in Fig 10 we have one set of three such forces; and in Fig 11, two sets. In 
Fig 10 it was not necessary to show those at the long chord. Now each set, or each 
triangle represents a vert force, a hor one, and an oblique one, keeping each other 
in equilibrium at the point of support. It is true that there are other forces acting 
at the same point, but as they hold each other in equilibrium, they do not interfere 
with the first ones. Thus, both the 7 and the 12 tons hor forces along the chord at 
k are balanced or held in equilibrium by the equal ones from t and s, on the other 
half of the truss; without disturbing the forces represented by the sides of the tri¬ 
angles. Hence by measuring those sides we obtain the forces and strains themselves. 



The same principle either by diagram or by calculation 
applies to Figs 12 to 1.?^, when 
uniform and uniformly loaded. In those 


Figs all the weight is here assumed to be 
on the long chord; (but after what we 
have said, no difficulty can arise when 
placing loads on the other chord also.) 
All said wt is first to be properly dis¬ 
tributed among the points of support on 
said long chord, and there figured, as 
shown by the upper 4s and 2s along that 
chord in the Figs. This being done, we 
have figured all the other vert strains, 
thus providing the data for drawing the 
vert sides of the triangles; and these in 
turn give us the hor and oblique sides 
which measure the corresponding strains, 
and all of which are drawn on the Figs. 


A / 

A h 

/ ; « ;-i 

/ 1 / " 

/ i 2 15 

/ j / 

/ i p i‘_ 

/ 1 / 


e / V 

/ ! ! 

m 2 A. 4 

CL 4r 

112 8 4 

: 


All these trusses being uniform and 
uniformly loaded from the center each 
way require no couuter bracing. Bear in 
mind that the vert strains that accumu¬ 
late at an abut must equal half the wt of 
the truss and its load. 


Ill Fig 12, with no oblique at the 
center, the 4 tons at a having no oblique^ 
in contact on either side of them, go to b ; 
and on their way to b strain ab 4 tons. 
From b all 4 go along the w r eb members 
to the nearest abut e as figured. 


In Figs 12% and 13 the web mem¬ 
bers of each are to be regarded in some 
degree as belonging to two separate 
trusses, namely abode and mn op e in 
Fig 12%; and abode and mn o d e in 
Fig 13; and the vert strains at their ends 
are to be found on that assumption, as 
figured. In Fig 13, o d is a vertical tie. 


In Fig 13% there being at none of 
the 4 ton loads on the long chord an ob¬ 
lique in contact with them on either side, 
they (like that at e Fig 10, or at a Fig 12) 
pass each by itself vertically to the upper 
chord, where figure them. Of those at 6, 

2 go to the abut e by way of the oblique 
be; but the other two 4s all go to e, each 
by its own oblique. Each of the three 
hor lines gives the strain exerted upon 
the upper chord at the respective panel 
point; but the hor strain along the lower 
chord is uniform from end to end, because all the forces that produce it act at its 
ends only. It is equal to the sum of the three hor lines. 























564 


TRUSSES. 


In Fiff 13%. at the 4 ton loads at c and e, there is no oblique in contact with 
them on either side; therefore they pass at once vertically to t and y, where figure 
them. Then begin with 2 of the 4 tons at a. 

All loads that have no oblique in contact oil either side, whether sustained 
by vert ties or by vert posts, are to be thus transferred at once to the opposite 
chord in order to meet an oblique along which to travel to the nearest abut; and 
said vert ties or posts will be strained to the amounts of said loads. 



Art. 12%. Moving loads and eoiinterhrneing. The foregoing 

investigations refer to cases where the load is always in the same position, and of 
tiie same amount; and where, consequently, the strain on any given member is 
constant. But in many cases, as in railroad bridges, the amount, and mannerof 
distribution, of the load, change greatly from time to time, so that the strain on a 
given member may vary greatly in amount, and may even change from tension 
to compression or vice versa. In the present article we investigate these changes 
in the strains, caused by changes in the position of the load. 

To simplify the subject, we regard the weight of the locomotive as being uni¬ 
formly distributed over its length, as shown in Fig 13/; and this suffices to illus¬ 
trate the principle. But specifications frequently call for calculations based 
upon the actual load on each, pair of wheels of an actual or assumed engine and 
tender, as shown on p 546. 

For simplicity, also, we here seek only the maximum strain on each member, 
in order to determine where counterbraces are required. 1 n practice it is necessary to 
note not only the maximum but also the minimum strain on each member; for 
when the strain on a member is subject to great and fnquent variations, espe¬ 
cially where it changes from tension to compression or vice versa, the ultimate 
strength of t he member is reduced to an important extent. See p 435. 

We will calculate the maximum strains in a bridge Fig 13/of 120 ft span mx, 
divided into 6 panels, each 20 ft long and 30 ft high. For convenience of calcu¬ 
lation we will suppose the fig to represent the two trusses of such a bridge com¬ 
bined into one. \Ve first calculate the max strain in each member of this sup¬ 
posed double truss, and then divide each strain by 2 for the actual strain in the 
corresponding member of each actual truss. 

The weight of this double truss is 48 tons; or 8 tons per panel ; or .4 ton per 
ft run. The floor, and its several timbers, such as cross girders, hor bracing, 
Ac, which, although not portions of the truss proper , are essential to the bridge; 
and to be considered as so much permanent load, equally distributed along the 
truss, are assumed to weigh an additional 24 tons; or 4 tons per panel; or .2 ton 
per tt run of the double t rtiss. Therefore, the truss and its accompaniments to¬ 
gether, or in other words, the bridge superstructure, weighs 72 tons; or 12 tons 
per panel; or .6 of a ton per ft run. Since each panel is 20 ft long, and 30 ft 
high, each oblique is q/ 2 ^ + 302 = 33 f t long. and U)e secant 0 f t t,e ang i e which 
it. forms with a vert, (or of the brace angle,) is = length of oblique -f- its vert 
spread = 36 -s- 30 = 1.2: and the nat tang of the same angle is = hor spread 
-T- vert spread = 20 30 = .6667. 




















TRUSSES. 


565 


The moving load is assumed to consist of an engine, yz, weighing 36 tons: all ot 
which is supposed to he so concentrated on its drivers, as to stand upon the length 
of one panel. The engine is supposed to he fol¬ 

lowed by a tender and cars, weighing 21 tons per panel; or 1.05 ton per ft run. 

The greater danger from engines, arises from the fact that in many 
of them, especially in heavy freight engines, all the wt is concentrated upon their 
driving wheels; which are so close together that a 30 or 40 ton engine on 8 drivers, 
will often have its entire wt sustained upon only 12 to 15 feet length of the truss; 
thus bringing a great strain upon each individual web-member, as it crosses the 
bridge. This point is one of the greatest importance in arranging the preliminary 
data on which to form the basis for calculating the strains on a bridge truss; and we 
should allow liberally for the greatest load that can possibly be brought upon one 
panel at a time. When the panels are quite short, each one will have of course 
to sustain only a smaller portion of the wt of engine; it may, indeed, have to be di¬ 
vided among 2, 3, or 4 of them. 

The fact that the engine produces upon each panel in succession, a greater strain 
than an equal length of loaded cars can do, causes a modification of the calcula¬ 
tions ; as will be seen as we proceed. 

Such data must be prepared before beginning the calculations Our assumptions 
in the case before us have been made entirely w ith reference to ease of calculating 
our example with but few figures. Our truss, together w ith a load of cars extending 
from end to end, will weigh 198 tons; or 1.85 tons per ft run; or 33 tons per panel. 

Having prepared a diagram, begin by finding the strains caused in only the verts 
and obliques, oy the bridge itself, half the wt of the trussjaione being considered 
to be on the top chord, while all of the weight (24 tons) of floor rests of course 
directly on the lower chord. 

Bv our “ first process,” p 559, there will then be 6 tons each at h, c, and c; 3 tons 
each at i and a ; 8 tons each at «, o, p, q , and r; and 4 tons each at rn and x. By our 
second and third processes, the strains on the web members will be as follows: 


On ep . 8 tons. 

“ ho and cq, each.15 “ 

“ i n and ar “ .29 “ 

“ e o and e q “ . 8.4 “ 

“ hn and c r “ .25.2 “ 

“ i m and ax “ .38.4 “ 


Write these strains at the feet of the respective members. 

We now have the strains on the web members, resulting from the truss itself,and 
from its uniformly distributed permanent load of floor, &c; and are prepared to 
begin with the additional strains resulting from the moving load. We find those op 
the counters, on one-half of the diagram ; and those on the main obliques and ver¬ 
ticals, on the other half. First suppose the engine to be in the position y z, on the 
half diagram p x ; with the train reaching to the farthest end m of the truss. In 
this position of the moving load, the vertical a r, and the end oblique ax, will be 
more strained by it than in any other; and their strains will be produced by that 
portion of the load that may be regarded as being first concentrated at r ; and as 
passing thence up r a to a ; and from a, down a x to the abutment x. To find this 
portion, the truss may be considered as a simple beam or lever, marked off into 
six equal divisions at r, q, p, o, n. In that case we know (from the principle of the 
lever) that if any load, as for instance our engine of 36 tons, be placed uniformly 
along y z, its whole weight may be assumed to be concentrated at the middle point 
r •; and that five-sixths of it will be borne by the nearest abutment a?; and only 
one-sixth by the farthest abutment m. So also with the panel length y w of cars; 
its weight being supposed concentrated at q, four-sixths ot it must be borne by x; 
and two-sixths of it by to. Of the cars w v, three-sixths are borne by x\ and three- 
sixths by to. Of those along v u, two-sixths are borne by x ; and four-sixths by to; 
of those along u s, one-sixth byx; and five-sixths by to. The half panel of cars, 
s m, rests directly upon &, and produces no strains in the truss. 

By this process, then, we find that five-sixths of the engine,or 30 tons; four-sixths 
of the cars y w, or 14 tons; three-sixths of the cars w v, or 10.5 tons; two-sixths 
of v u, or 7 tons ; and one-sixth of u s, or 3.5 tons; making 65 tons in all, are borne 
by the abutment x, when the moviug load is in the position shown in our fig. This 









TRUSSES. 


rm 


loud of 65 tons passes from r to a; .anil from a to x.* The calculation is made as 
shown under our fig below the heading “ With engine at r.” 

We next suppose the engine to back into the position y w ; with the train reaching 
to m; and with no load between y and x. In this position the strains on qc , and cr , 
are greater than in any other; and by a process precisely as before, and as shown 
under our fig, below the heading “ Engine at 7 ,” we find that 45 tons of the moving 
load go to the abutx; passing from 7 to c; and from c to r, on their way to it. Then 
placing the engine at iv v, the cars reaching to m, and no load between w and x, the 
strains on p e and e q, are the maximum that they can be subjected to by an engino 
and train ; and are found like the others, as shown under “ Engine at p." We have 
now reached the center of the truss, and have obtained data for the greatest strains 
that can occur on the verts and main obliques on one-half of it. So far as regards 
the corresponding members on the other half, we stop here ; for it is manifest that 
an engine and train crossing the bridge in the opposite direction, must produce the 
same maximum strains upon them as we have already found for the others. That is, 
i m will be strained the same as ax; ni the same as r a; eo the same as e q, and so on, 
by a returning train. Write these strains near the tops of the verticals. 

But as yet, we have not sufficient data for determining where counters will bo 
required. To obtain it, we draw the two lines h p, and i 0 , parallel to the obliques 
on the opposite side of the truss; and taking it for granted for the present, that these 
two lines are counters, we back the engine to v u, and then to ms; performing each 
time, calculations precisely similar to the former ones: and as shown under the head¬ 
ings “ Engine at o,” &c. We have stated in a former Art, that a counter is most 
strained when the moving load extends from it to the nearest abut. Therefore, a 
counter h p would be most strained with the engine on v u, and the train reaching 
to m ; and i o, by the engine on u s, with a train to m. 

* With the engine at r, as in the Fig, the counters i o and h p are not required. 
Consider them removed, and it will be seen that the several sixths of the panel loads 
reach their respective abutments by the following routes: 

Of us, one-sixth to x, via nhoeqcrax: five sixths to m, via n i m. Of dm, 
two-sixths to x, via oeqcrax; four-sixths to m, via ohnirn. Of w r, three- 
sixths to x, via p e q c r a x\ three-sixths to m, via p eo h n i m. Of y w, four-sixths 
to x, via q cr a x; two-sixths to m, via q e o It n i m. Of the engine z y, five-sixths 
to x, via rax; one-sixth to m, via rcqeohni m. 

The one sixth of engine going to in, in passing from e to m, and the five-sixths go¬ 
ing to x, plainly bring upon each member the same, kind of strain as that brought 
upon it by the weight of truss and floor; i. e., tension upon the ties and compression 
upon the struts; and thus increase their strains; but in passing from r to 1 ?,the one- 
sixth going to m pulls the struts c r and e q, and presses tiie tie cq, thus diminishing 
the strains upon those members. 

And, similarly, in any position of the engine and train, each panel length of mov¬ 
ing load, except w v, divides at its point ot support; one portion passing up the ver¬ 
tical tie (and increasing the tension upon it) on its way to one abutment: and the 
other portion passing up the inclined strut (and diminishing the thrust upon it) on 
its way to the other abutment. 

4'iitition. When a web member is subjected to its maximum strain, no such 
relieving strain can thus come to it from the moving load. Thus c q has its maxi¬ 
mum strain (tension) when the engine is at q , with train reaching to m ; and there 
is then no load at r to relieve c 7 by sending a compressive strain through it. 

Th e final or resultant strain (which may or may not be the maximum strain) upon 
any member, for any gi ven position of the train, is the difference between the tension 
and compression upon it at that moment; and is of course a pull if the tension is 
greater than the compression, and vice versa. 

Thus, with engine at r, as in the figure, the final strain on the tie a r is = 

Tension. Tension. Compression. Tension. 

29 tons from bridge and , 65 tons of train go- 

floor, as per p 565 + ing to x, as above 

and this is also the maximum strain on a r, 
gine still at r) is — 

Tension. 

Five-sixths of en¬ 
gine which do not 
pass through c 7 

T C T 

35 — 6 = 44 tons 


Tension. 

15 tons from 
bridge and floor 


+ 


( 


65 tons 
going to x 


— 0 = 94 tons, 

But on c q, the final strain (with en- 




Compression. 
x / { ' of engine going to m, 
which presses c q, and 
thus relieves its tension. 


T 

= 15 


+ 


But this is not the maximum strain on C 7 ; for this takes place, as already remarked, 
with engine at 7 , and amounts to 45 tons. 





















































ERRATA. 


For TrautAvine’s Civil Engineer’s Pocket Book. 

9th to 15th Editions (1885 to 1891) inclusive. 

To face lower part of page 567. 

P. 565. Line 5 from foot: For upon x read upon m. 

Pp. 567, 568. Under Verticals: Strain on Vertical e p 
with engine at p. 

If the counter li p be made to act as a tie, the strain on the 
middle vertical e p is equal to the 28.5 tons upward reaction of 
the abutment x, as stated ; but otherwise it is equal to the 36 
tons weight of engine at p. 

In either case we have 36 tons engine at p, minus 28.5 tons 
upward reaction of ^,=7.5* tons, tending to shorten the panel 
e h 0 p along e o and to lengthen it along h p. 

Now if the ends of h p be secured, it may be sufficiently 
tightened to act as a tie, relieving the main brace e o of all com¬ 
pressive strain. We may then suppose e o removed. In this 
case the 28.5 tons upward reaction of x w r ould go via x a r c q 
ep to p, causing a live-load strain of 28.5 tons in e p, the same 
as in a r and c q. Half of the engine load (or 18 tons) would 
then go via // p to m. 

But the proper office of h p is to act as a counter strut when 
the engine is at o, and it should not be depended upon to act as 
a tie. Hence the true maximum live-load strain in the middle 
vertical, e p . is the 36 tons engine at p ; and the maximum total 
strain in e p is : 

live-load (entire weight of engine at p), - 36 tons, 

dead-load (see page 565) - - - 8 “ 

Total, 44. tons. 

instead of 36.5 tons as stated on page 568. 

*=18 tons (half weight of engine) going to 7 n, minus 10.5 tons (two 
sixths of o one-sixth of n) going to x. 




TRUSSES. 


567 



Willi Engine at p, 

there go to x. 

Tons. 

IS = 3-6 engine. 

7 = 2-6 t u. 

8.5 = 1-6 u s. 

2S.50 go to x. 

1.2 nat sec. 


34.2 on e q 

from load only. 


MAIX BRACES. 

With Engine at q, 

there go to x. 

Tons. 

24 = 4-6 engine. 

10.5 - 3-6 w v. 

7 - 2-6 v u. 

3.5 = 1-6 u s. 


45. go to x. 
1.2 nat sec. 


54. oner 

from load only. 


With Engine at r, 

there go to x. 

Tons. 

30 = 5-6 engine. 

14 = 4-6 v \t. 

10.5 = 3-6 w v. 

7 = 2-6 t ix. 

3.5 = 1-6 u s. 


65. go to x. 
1.2 nat sec. 


78. on a x 

from load only. 


Vertical e p. 

28.5 go to x. 

Total on e p from load only. 


VERTICALS. 

Vertical c q. 

45 go to x. 

Total on c q from load only. 


Vertical a r. 

65 go to x. 

Total on a r from load only. 


Counter i o. 


COUNTERS. 


Counter k p. 


Engine at it. 


Engine at o. 


6 =1-6 engine go to x. 

1.2 nat sec. 

7.2 on i o 

from load only. 


12 = 2-6 engine. 

3.5 = 1-6 u s. 

15.5 go to x. 

1.2 not sec. 

18.6 on h p 
from load only. 


Supposing the calculations to have been made as above, we have thereby found 
the several loads, say 65; 45; 28.5; 15.5; and 6; which go from train to x, from the 






















568 


TRUSSES. 


several points r, q, p, 0 , v, when the engine is at each of those points in succession; 
with a train reaching in each case to ru. Each of these loads, in travelling, as it were, 
from its particular point, to the abut x, first ascends to the top of its own vert: and 
then descends along the adjacent oblique; producing in said oblique a strain as much 
greater than the load itself, as the length of the oblique is greater than its vert 
spread. Now, each of our obliques, as before stated, is 156 it long, or 1.2 times as long 
as its vert spread 30 ft, (or the height of truss;) which 1.2 is therefore the nat sec of 
the angle which any of our obliques forms with a vert line. Therefore, to find the 
strain which each of our moving loads produces on the oblique next to it, on its way 
to the abut x, mult each said load by 1.2, as in the above calculations. We thus ob- ' 
tain for these strains, 78 tons on ax; 54 on c r; and 34.2 on eq ; all which write near 
the tops of the obliques. 

For the verticals and main obliques; Total max strain = truss strain 
-f moving-load strain. Thus: 


Verticals 

ep 8 + 28.5 = 36.5 tons 

ho, cq, each 15 + 45 = 60 “ 

in, ar, “ 29 + 65 = 94 “ 


Main obliques 

eo, eq, each 8.4 + 34.2 = 42.6 tons 
hn,cr, “ 25.2 + 54 = 79.2 “ 
im,ax, “ 38.4 + 78 *=116.4 “ 


For the counters, we go to the other side,/* m, of the truss. Beginning 
with the panel eh op, we examine its twodiags, hp, eo. and see that with the engine 
at v u, and the cars reaching tom, there is produced in the counter hp its maximum 
strain of 18.6 tons; which tends to cause in the truss the kind of derangementshovvn 
in Figs 9%, 9%, and 9 6 . Now, to resist this derangement, there is nothing but the 
8.4 tons produced by the truss, floor. Ac, upon the opposite diag eoof the same panel. 
Since, therefore, the deranging effect of the load is greater than the preventive effect 
of the weight of the truss, there must be a counter at hp, able to bear a pres equal 
at least to the difference between the two, or to 18.6 — 8.4 = 10.2 tons; or else the 
strut e o must be made so that, it can also act as a tie, capable of sustaining safely a 
pull of 10.2 tons. This last method relieves h o and e p from 10.2 tons each, but this 
relief comes when they have not their maximum load. 

We now go to the next panel, h i n o. But here we find that the deranging effect 
of the load on the counter i o, with the engine at u s, is but 7.2 tons; while the pre¬ 
ventive effect of the wt of the truss, exerted through the opposite diag h n, is 25.2 
tons. Hence, the moving load can produce no derangement of the truss; and con 
sequently the counter i o may be omitted. On this same principle each panel on on« 
side of the truss must be examined, when there are many of them ; and the insertion 
or omission of counterbraces be determined upon. When we thus arrive at a panel 
at which no counter is reqd, none will be needed between it and the nearest end of 
the bridge. Similar counters will, of course, be needed on the other side of the truss. 
In practice it is better to retain the first apparently unnecessary counterbrace; 
counting from the center of the truss. Thus, although calculation shows t o to be 
unnecessary, it is well to retain it. The lighter a bridge is, in proportion to its 
moving load, the greater will be the number of panels requiring counters. 

The strains on the chords are greatest when the truss is loaded 

from end to end; ami for Fig 13/, as well as for Fig 13 < 7 , may readily be calcu 
lated by Art 12; or found by a diagram with a max load. Or sufficiently close for 
most purposes, the hor strain along either chord at the center of the truss where 
the straiu is greatest, will be equal to 


Total weight of truss and load X span. 


8 times the depth or height of truss. 


which here is = 


198 X 120 
8 X 30 


23760 

-SO" = 99 t0n8 - 


Finally, each of all the strains in our double truss must be div by 2 ; for propor¬ 
tioning them among the two actual trusses, which we have all along supposed (for 
convenience) to be combined into one. 


Art 12%. The Warren or triangular truss. Fig. 13 g. 

Here the dotted web-members which supplant the ties in Fig 13/ are not 
vert; but inclined to the same extent as the struts or braces. Hence the hor stretch 
of each oblique will be but 6 a//the length of a panel, or 10 ft; or only half as great as in 

Fig 13/. Consequently, the length of each oblique will be V 10 * +/s 0 ' i = 31.6 ft; 

31 6 * 

and the nat sec of the brace angle will be —— = 1.05; and the nat tang of the same 
10 30 

angle will be — = .333. All the other data being the same as in the foregoing ex- 











truss only being supposed to be on the top chord, making, by our first process, p 559, 
4.8 tons each at e, d, c, and b ; 2.4 tons each at i and a ; 8 tons each at n, o, p, q , and r ; 
' *\d 4 tons each at m and x. By our second and third processes, the strains on the 
ib members will be as follows : 




‘ On p d and p e, each 
“do and c q, “ 

“ o e and q 6, “ 


4.2 tons 
, 9.24 “ 
17.64 “ 


On e n and b r, each.22.68 tons. 

“ « i and r o, “ .31.08 “ 

“ i m and a x, “ .33.6 “ 


| Vrite these strains at the feet of the respective members, as before. 

Having now the correct strains arising from the weight of the truss and floor: 
i next fiud, precisely as in the preceding example, how much of the moving load will 
go to x when the engine is at r , as in the fig, with the train reaching to m ; and 
i afterward with the engine at q , p, o, and n. in succession. These loads will of course 
be the same as in the former example, namely: 

Engine at n. Engine at o. Engine at p. Engine at q. Engine at r. 

6 go to x. 15.5 go to x. 28.5 go to x. 45 go to x. 65 go to x. 

Tltiplying each of these by the nat sec 1.05, we get the compressing strains which 
%y produce on the end oblique strut a x; and on the other obliques that urepar- 
let to it; namely : 


On e o. On d p. On c q. On b r. On a x. 

6.3 16.275 29.925 47.25 68.25 

hich write near the tops of said obliques. But they also produce precisely the 
same amount of strains, in the shape of tensions or pulls, on the respective dotted 
ies which carry them to the struts between x and p ; and on the struts which carry 
them to the dotted ties between m and]). Write them all near the heads of said 
obliques also.* 

ffow, if on the half truss x p, we add together the strains written at the head and 
'oot of each oblique separately, the sums will be the total or maximum strain (com¬ 
pressive on the struts, and tensile on the ties) which said obliques will be subjected 
to on the passage of the train. They will of course be the same on the other half of 
the truss when the train crosses in the opposite direction. 

Finally, as to eounterbracinK> we go to the other half, m p, of the 
truss. Beginning with the oblique d p , we see that with the engine at o, and 


* The remarks in foot-note p 566 apply also, in principle, to Fig 13g. Thus, with 
the engine atr, as in the fig, the five-sixths of engine, going to a;, increase the strains 
on r a and ax. The one-sixth, going to m, diminishes the strains in r b, bq, q c. and 
| cp (but these members have not now their maximum strains), and increases those 
in p d, d o, o e, e n , n i , and i m. 

The final strains, also, are found in the same way as in Fig 13f. Thus, with engine 
at q, train reaching to m, we have final strain on q b (maximum) 

Tension. Tension. Compression. Tension. 

= 17.64 from truss and floor + 45 going to* X 1-05 = 47.25 — 0 = 64.89; 

and on q c (not maximum) 

Compression. Compression. Tension. 

■= 9.24 from truss and floor + [(45 to x — % engine) X 1-05] — (§ engine X 1-05) 

c c T c 

= 9.24 + 22.05 — 12.60 = 18.69 tons. 



















570 


TRUSSES. 


the train reaching to tn, the deranging compressive strain, 16.3 tons, of the mov¬ 
ing load, is greater than the preservative tensile strain, 4.2 tons, of the truss 
and floor, acting on it at the same time. Therefore, d p, although a tie, the same as 
c p, is liable at times to be compressed rather than pulled. Therefore, it must be so 
arranged as to act also as a strut; at least so far as to bear a pressure equal to the 
difference between the 16.3 tons of pressure from the load and the 4.2 tons of tension 
from the weight of the truss and floor; or to 12.1 tons. The same end would be ac- ; 
complished by inserting a counter tie reaching from o to c. 

On the next oblique o d, which is a strut, the same as c q , the moving load on v m 
produces a pull of 16.3; while the truss and floor produce on it a pres of only 9.2 
at the same time. Therefore, although it is a strut, it is liable at times to be. pulled 
rather than compressed; and consequently it must be made able to bear a pull also, 
equal at least to 16.3 — 9.2 = 7.1 tons. The introduction of a counter strut reaching 
from e top would answer the same purpose. On the tie e o, the deranging compres¬ 
sive load strain 6.3 is less than the preservative tensile strain 17.6 of the truss and 
flnor acting upon it at the same time. Therefore, it may remain as a tie only ; or in 
other words, it requires no counterbracing. When this is thecase.no other oblique 
between it and the nearest abutment rn needs counterbracing. It is almost needless 
to remark, that the half xp of the truss requires the same as the half mp, when the 
engine crosses in the opposite direction. 

TIi« strains on the chords are found as directed on p 568. 



Art. 13. In Fig 14, as in Figs 10 and 11, the truss is of 64 ft span and 16 ft rise 
with an extraneous load of 32 tons; of which our first process gives as before' 

2 tons each at a and at the other end of the span, and 4 tons at each other panel point 
b, c, d, e , / etc. The wt of the truss itself is neglected, as before. 

Second ami third processes.* Each panel load divides at its point of : 
support, as explained more fully in Rem, p 573, one portion, rs'" etc. going directlv 
down the rafter to the nearest abut; the other, bs'" etc. by way of the web membe/s 
to the peak e of the truss, and thence via the rafters to the abuts. 

Thus, beginning at the panel load 6, nearest the abut, lay off by scale the vert 
hr = that panel load, 4 tons. Draw rh" f parallel to the rafter, and h f,/ s! n horizonta 1 . 
or parallel to the tie beam. Measure s"'r, which represents the portion (2 tons) of o 
going directly to a via the rafter, and write its amount <2) at a. Also measure 5.s'" I 
the portion (2 tons) ot b going to e via bkcj die, and write its amount (2) at c over ! 
the original panel-load (4 i of c ; thus making the final load at e = 6 tons Make I 
cr = this load, draw r h" parallel to the rafter, and h" s" hor. Then s"r ( = 2 tons') 
goes from c to a via the rafter; and c «" ( = 4 tons) goes to e via ejd i e. Write these 
amounts at a and d respectively. The final load at d thus becomes 8 tons Therefore 
make dr = 8 tons ; draw r h' parallel to the rafter, and h’ s' hor. Then s’r (2 tons) 
goes from d to a via the rafter; and d s' (6 tons) to e via d i e. Write these amounts 


* When wishing to know' merely the amounts of the several fined panel loads in a 
truss like Fig 14, uniformly loaded and divided into panels of equal length; without car¬ 
ing to trace the process of their accumulation from the original panel loads- our 

second process may be shortened, thus: 


original panel load at abut a = -th e entire w t of truss a nd l oad _ 

entire number of points of support abedefe tc minus 1 
Tlien, calling this original panel load at a “a”, the final panel loads are as follows: 
a | ^ ~ a > c== 3a; at d = 4a; and so on with any number of points of surmort 

along a rafter, except at the apex e, where the final load is twice the final load at 
the nearest panel point (d in our Fig) or = % the wt of the entire truss and its load. 
The final load at a = half the weight of entire truss and load. 


























TRUSSES. 


571 


v at a and e respectively. Now it is plain that the same process, on the other half of 
the truss, brings another 6 tons to e. Write this at e also. Thus we get for the final 
a load at e, 4 + 6 + 6 = 16 tons, or ^ the wt of the entire truss and its load. 

The 16 tons thus concentrated at e divide there, half going down each rafter to an 
abut. Draw er vert, and = these 16 tons. Draw rh parallel to ae, and As hor. 
Then esand sr are the 2 halves (8 tons each) of ei, passing from e to the abuts via 
c-jthe rafters. Therefore write 8 at a. 

It is by mere accident that the two vertical lines b l and er, representing the loads 
m Jat b and e, happen just to extend to the tie a i in our Fig. 

2' We thus find, for Fig 14, 


Strains along 1 the verticals. 


2 41ong b l = nothing, except weight of tie-ba 
from y to y. 


bar 


id i 


c k, = b s = 2 tons. 
dj. —e # = 4. 


Strains along the obliques. 

ut 

Along b k — b h — 4.47 tons. 

“ c j — c h = 5.66. 

“ di=dh = 7.21. 


ei — 2 tf s — 2 X 6 — 12; for while each other vertical tie bears only the vert strain brought upon it by 
the oblique strut next nearer the abut; the center tie e i, of course bears those from 2 obliques; 
one ou each side of it. 

Strains along the horizontal tie-bar a i. 

% wt of truss and load X % span 


At i = s A — 16 tons; also = 


height of truss. 


From,/ to i = s A + s h = 16 + 4 = 20. 


U 


(< 


k to j = sh+sh + s A = 16+ 4 + 4 = 24. 

r r r r r r r r r r r r 

aiok^=sh-\-sh-\-s h + s A = 16+ 4 + 4 + 4 = 28. 


Strains along the rafter e a. 

From e to d = h r = 17.9 tons. 

r 

“ d to c = h r + h r = 17.9 + 4.47 = 22.4. 

“ cto£> = Ar+Ar + Ar = 17.9 + 4.47 + 4.47 = 26.8. 

“ b to a = A r + A r + A r = A r = 17.9 + 4.47 + 4.47 + 4.47 = 31.3. 

It will be observed that the hor components A s, except the center one, have equal 
lengths; also those marked A r, parallel to the rafter; while the oblique ones have 
not. 


For a span of 100 feet, rise 20 ft, or i of the span; trusses 10 feet apart 
from center to center; loaded on top only; the following dimensions will be amply 
sufficient for a covering of slate. Kafters and tie-beam, each 10" X 12" deep. The 
rafters may, if preferred, be reduced to 9 X 12 at top. The verts of round bar-iron 
of good quality, % inch, % inch, 1 inch, and 1% inches diameter. The obliques or 
braces, 5 X 10, 6 X 10, 8 X 10; thus making the truss-thickness uniformly 10". See 
Table 2, p. 579. For shorter spans, see NOTE, p 573. 



Art. 14. In Fig 15, the process is the same as in Fig 14, except that the vert line? 
representing the strains at the points of support a, b, c. d, r, are to be drawn upward 


from 


i; 

8 

2 

2 

2 


and the strains l s'", k s", j s', are to be carried forward to the next 


Span 64 ft. 
Rise 16 ft. 


Fi«f. 15 

O 
















572 


TRUSSES. 


panel-load. Fig 15 is simply Fig 14 inverted, and those members which resisted f 
pressure, in Fig 14, resist pull in Fig 15, and vice versa. In other words, the struts 
become ties, and the ties, struts. All the strains are equal to those in Fig 14, except 
those on the verts , each of which is 4 tons greater than the coiresponding.one in Fig 
14, because the original panel loads of 4 tons each, instead of being applied directly 
to the ends of the obliques, as in Fig 14, have first to pass through the vert struts 
b I, c Ic, d j, e i; the total loads on which will be, respectively, 4, 6, 8, and 16 tons. 


6^6 


f 


4 e 



Art. 15. In Fig 16, the process is the same as in Fig 14, except that the lines h s,. 
Ac, must be drawn and measured parallel to the inclined tie a i ; instead ofbeinghor. 
As in Fig 14 b s'" is carried forward to c; c s" to d; d s' to e. In this way, we find, 
as before, the vert strain of 6 + 4 + 6 = 16 tons at e. But we must now add tc At 
these 16 tons, another strain generated by the obliquity of the tie-rod a i. This strain 
is found by mult the one at e, (16 tons, or half the wt of the entire truss and its 
load,) by the vert dist n i, (6 ft,) which the center i of the tie-rod is raised above 
the horizontal u u , and div the prod by the dist i e, (10 it.) 

That is, —— = 9.6 tons; which also write down as in the Fig; making the total If 

vert strain at e 25.6 tons, instead of the 16 tons of Fig 14.* 

Now, make e. r by scale, = 25.6 tons ; draw r h parallel to the rafter e a, and meet¬ 
ing the other rafter; also draw h s, parallel to the raised tie-bar » a. Then the strains) 1 
along the members of the truss will be as follows, taken from a Fig on a larger scale. 


St rains a Ion;; the verticals. 

Along the one at b — nothing, except weight of 
tie-bar for the width of 
one panel (8 ft.) 

Along co = 6 s"' = 2 tons. 

“ d z = c s" = i “ 

“ ei = 2d»‘ + 9.6 = 21.6. 


Strains along- the obliques 

Along bo — bh’" ~ 6.43. 

“ cz — ck" — 6.96. 


c z 
di 


dh =8.04. 


Strains along the rained tic-bar a i. 


At i ~ hs = 26 tons. 

From z to i = h s A' s' = 26 -f- 6.5 = 32.5. 

“ otoz = hs 4- h's' h" s" = 26 6.5 -f 6.5 = 39. 

“ a too = hs -f h' »' h"s" -(- h"‘ ~ 26 -j- 6.5 -f 6.5 -f- 6.5 = 46.5. 

Strains along the rafter ea. 

From d to e. = hr = 28.5 tons. 

“ e to d = Ar -4- h' r = 28.5 -f- 7.13 = 35.63. 

“ btoc = hr 4- h'r 4 - h'r = 28.5 -f 7.13 4- 7.13 = 42.76. 

“ o to b — hr + h'r h"r-\-h'"r — 28.5 -(- 7.13 -|- 7.13 -j- 7.13 = 49.9. 



* It is probable that the tie rod is sometimes raised in this manner by persons ignorant of the fact 
that they thereby greatly increase the straius on the rafters, &c. 

All the strains in I- igs 1-4, 15. anil 1G mav also be found by pre¬ 
cisely the same process as that for bowstring and cresceut trusses iu Art 19 

The tension in the tie-rod which brings the 9.6 tons additional load to e, causes at the same time 
an equal upward pull at a. Hence the final pres of the truss upon each abut remains 8 tons 
(= half the weight of truss and load) as in Fig 14. 










TRUSSES. 


573 


Rem. The reason for measuring only parts of the vert lines which, in Figs 14, 15, 
16, represent the whole panel-loads, is that the rafter a«, Figs 14 and 16; or the tie 
a i of Fig 15, being inclined, also bear a part of each panel-load ; and since that part 
does not go forward to the next point of support, but goes backward, along said in¬ 
clined member, to the abut at 
d li «, it must be omitted in the 

second process. Thus, in Fig 
17, if ba be an inclined rafter 
resting on an abut a: bg a 
strut; and hr a vert line repre¬ 
senting the load sustained at 
b by ba and bg: if we com¬ 
plete the parallelogram bmrn 
of forces, then will bm give 
by scale the strain along the 
strut; and bn that along the 
rafter. The strain along the 
c> strut is made up or composed 
of the portion b s of vert force; 
and the hor force sm. The 
vert portion bs alone goes to the next point of support; while s w.strains the tie ag 
hor. So also the strain bn is made up of the other portion (ho or sr) of the vert 
force/»?•; and of the hor force on; which, when the strain bn reaches a, become 
again resolved into two; one of which, bo, presses vert upon the abut; or, in other 
words, transfers to the abut the portion bo of the load resting on 6; while the por¬ 
tion on, which is equal to sm, strains the tie ag hor. 

But in Fig 18, where b r also represents a load resting on b, and supported by a 
strut b g, and by a hor chord b a, if we complete the parallelogram 6 m r n, we have 
the strain b m along the strut, composed of all the vert force b r, and the hor force 
r in. The whole of b r is transferred to the next point of support; while r m and b n 
produce only hor strains along b a and g y. 

Art. 16. The roof truss, Figs 19, 20, and 21, consists of two complete Fink 
trusses, an e and e k l, Fig 21. It is supposed to be of the same span (64 ft) and lit 



Fio' 17 


Fioi8 


NOTE. 

The following 1 may at times save trouble in designing roof 
trusses. After the dimensions of all the members of a roof truss of any span 
have been calculated, then those of any smaller span similarly arranged, and having the same rise 
in proportion to its span, and the same extraneous load per sq ft; but with the trusses at the same 
dist apart as in the large span ; may be found safely; and often near enough for practice, thus : 

Find the area of cross section of each member of the large truss, in sq ins. Then make the area 
of cross section of each member of the small truss less than that of the corresponding member of the 
large truss in proportion as its span is smaller. The small truss thus obtained will be stronger than 
the large one under the same extraneous load per sq ft. For instance, suppose a truss of 175 ft span 
has been designed to carry safely a total load (t e including its own wt) of 40 lbs per sq ft of ground 
covered Such a truss, by table, p 580, will weigh 8.05 tbs per sq ft of ground covered, leaving 
40 — 805 = 31.95 lbs per sq ft as its safe extraneous load of purlins, slate, snow, <fec. Now a truss 
of one-tifth this span, or 35 ft, proportioned by the above rule, would sustain safely one-fifth the 
same total load ; or (with trusses at same dist apart in both cases) would sustain the same total load 
(40 lbs) per sq ft. Rut the weight of the small truss itself, per sq ft of ground covered, is only one- 
fifth of that of the larger one, or 1.61 lbs, leaving 40 — 1.61 = 38.39 lbs per sq ft as its safe extraneous 
load, while that of the larger one, as shown above, is only 31.95 lbs. Reductions will, however, 
rarely be made to as small as one-fifth ; and where the short span is not less than half the long one, 
the method will answer very well in practice. For examples of reducing, see p 581. 

With the same total load per sq ft (including the wt of the 
truss itself) and with trusses at the same dist apart in all cases, the strains on 
the several members of trusses proportioned by the above rule, will be in the same proportion as the 
spans; as will also the areas of cross section and weights per ft run, of each member, the wt of the 
truss alone, per ft of span and per sq ft of ground covered, and the total load on a truss, including 
its own wt; but the total wts of the trusses alone will be as the squares of the spans. 












574 


TRUSSES. 


(16 ft) as Figs 10, 11,14, 15, 16; and to have the same number '9) of points of sup 
port for the weight (32 tons,) supposed to lie uniformly distributed along its top 
Consequently from our first process there will, as before, be 2 tons of panel 
load at each of the end supports; and 4 tons at each of the others. Write thes 




































TRUSSES. 


575 


I down as in Fig 20.* The part truss ex a may be regarded as being composed of 
t three separate trusses e x a, e g c, c m a; as will appear more plainly from ena,tic, 
and coa, in lig 21. These may be called first and second secondary trusses. In Fig 
19, the halt e y p exhibits a truss on the same principle, but having a greater num¬ 
ber of points of support for the uniform wt. That half truss consists of first, second, 
and third secondaries, as shown by eyp , egi, and esw. However far this subdi¬ 
vision may be carried, if the struts occur at equal dists, each of the smallest divisions, 
as coa, Fig 21, is to be regarded as a complete truss in itself, loaded at its center 
only. One-half, therefore, of its wt must rest upon each of its supports a and c. 

Thus, with our second process at one of the shortest struts, as 6 to, Fig 20, 
2 of its 4 tons go to c at the next longer strut, cx ; and 2 of them to the end a of 
the rafter, as written on the Fig. Then, at the other of the shortest struts, d g, 2 
tons go to c; and 2 to the end e of the rafter. 

YY e will suppose, for the present, that the end e, and the corresponding end of the 
other half truss e. k l. Fig 21, rest upon an abutment at e, as a rests upon its abut. 

YY e thus have 4 tons at b; 4 at d ; and 8 at c. 

In like manner we now regard the first secondary truss axe (see ane, Fig 21) as 
loaded at its center, c, only, with 8 tons as just explained. Of these 8 tons, 4 are of 
course supported by the abut at a, and 4 by the similar abut supposed, lor the pres- 
j ent, to be at e; both of these 4 tons are therefore set down as at a and e. When 
1 there are more points of support, as along the rafter ep. Fig 19, the process is pre¬ 
cisely the same: we first adjust the strains of the four third secondaries, esw, wri, 
i v k, k u p ; placing them at e,w,i, k , and p: then we transfer those thus accumulated 
at w and k, to e, i, and p ; and finally transfer them from i to e and p, at the ends of 
the rafter. Now, returning to Fig 20, we see that in addition to the original panel- 
i load of 4 tons at e, we have accumulated 6 tons of vert strain from the other panel¬ 
loads; and it is plain that the same process, performed along the other half truss 
ek I, Fig 21, would bring 2 + 4 = 6 tons more to e, as written in Fig 20. Thus it 
appears that we have 16 tons in all ate;f resting upon our supposed abut there. 
But as this abut has no existence, the 16 tons really come upon the rafters them¬ 
selves at e , and of course half of this wt (or 8 tons) is transferred by a rafter to each 
abut. Write down the 8 tons at a as in Fig 20. We thus have for the total vert 
pressure at a, 2 + 2 + 4 + 8 = 16 tons = half the total wt of truss and load, and 
this is a further proof of the correctness of the operation. 

Having thus finished our second process of finding the additional strains at the 
! several points of support produced by the original panel-loads on their way to 
the abutments, we have only by our third process, to complete the drawing, 
so that we may measure by scale the strains along all the members of the truss. To 
do this, from the tops of the struts draw vert lines bv, c v, dv to represent the total 
vert strains accumulated at those respective points.; namely, 4 tons at b, 4 at d, 8 at 
c. Draw v o,v o, v o parallel to the rafter a e. Then b o, co, do will give the strains 
along the struts; 3.6 tons on b m or d g; and 7.2 tons on c x ; and vo, v o, vo will 
give those produced directly by the final panel-loads from their points of support, 
upon the lower halves of the rafters of their respective secondary trusses. Thus the 
' 4 tons at d exert a pres there of 1.77 tons (as shown by the upper vo) upon the lower 
half dc of e c. The 4 tons at b exert an equal pres upon b a, and the 8 tons at c 
strain ca 3.54 tons. These strains v o of course become proportionally greater, and 
b o, c o, d o proportionally less, as the rise of the truss increases in proportion to the 
span. They will be referred to further on. 

Lay off m i, x i, g i respectively equal to bo,co,do\ draw ij, ij , i j ; and i y, i y, 
i y, parallel to the ties; thus completing the parallelograms of forces ij my, ijxy, 
i jg y. Draw the diags yj, yj, yj\ and the vert lines mu,xu,gu. Lay off the 
vert dist e f equal to the total vert strain (16 tons) at e ; make e z — to half of e /; 
draw z h hor; and h f. 

Now to y and mj give the strains (4 tons each) along the ties to or, m c, caused by 
the 4 tons at b; which strains extend from m to a and c.% In like manner g y and 


* When merely wishing to ascertain the strains along the members of such a truss, without caring 
to trace their progress, we may omit part of the following; and, after having made a correct diagram 
of the truss, we may at once write down the vert strains at the points of support, thus: At e (the 
apex or peak of the truss.) write one-Aa?/of the entire wt of a truss and its load, (for which, per sq ft, 
see Table 4, p 581) at the foot a of the rafter, one half; at the center strut c one fourth; at h and d, 
one eighth at each ; and when there are four intermediate subdivisions of the same kind, as along 
the rafter ep, Fig 19, one sixteenth of said entire weight at each of such additional points, &c. Thus 
the total load at any longer strut will be twice as great as that at any next shorter one. Then begin 
at “ Having thus finished our second process,” 

i This is precisely half the wt (32 tons) of the entire truss and its load; and as this will always be 
the case in trusses on this principle, it proves our work to be correct thus far. 

1 When the main tie a t is hor, as in Fig 20, these strains along the ties will be equal to those at 
the points of support, only where the height, of the truss is equal to y. of its span ; as in the case 
before us. When the height is less than y., the strains on the ties will be greater than those at the 
points of support; and vice versa. 









576 


TRUSSES. 


gj give the strains (4 tons each) extending from g to c and e. In Fig 21, the short 
ties, o a, o c, i c, i e, show this more distinctly.* 

Next xy and xj. Fig 20, give the strains (8 tons each) produced along x a and x t 
by the 8 tons at c. This also is shown more plainly in Fig 21, by the ties n a and 
n e. The hor line h z gives the strain (16 tons) produced along the entire hor tie 
a 1, Fig 21, by the 16 tons at e. Fig 20 may be considered one-half of Fig 21. 

We have for the total strains on the ties as follows: 


Along c m and c g, strain — mj or gy—i tons. 

Along x e, from x to g, strain — xj — 8 tons. 

Along i e, from g to e, strain — xj + gj — 8 -p 4 = 12 tons. 

Along t a, from t to x, strain = 4z=16 tons. 

Along t a, from x to m, strain =:A;z-|-xj/ — 16-f-8 — 24 tons. 

Along t a, from m to a, strain = t^-i-xj/ + OTjf=16-f-8-f-4 = 28 tons. 

The line fh or e h, Figs 20 and 21, gives the longitudinal pres (17.0 tons) brought! 
upon each rafter by the 16 tons vert load which our second process brought to e. 
This pres strains the entire rafter uniformly from end to end ; but the several por¬ 
tions of the rafter sustain, in additior , pressures arising more directly from the 
panel loads. For instance, the 4 ton load at d, produces at d, as already explained, 
a pres do =1.77 tons along d c. and one, do — 3.6 tons, along the strut d g ; and 
this last exerts pulls, y g and gj, = 4 tons each, along g c and g e. Now if on gj we 
draw the parallelogram g nj u, making j n and u g vert, and uj and g n parallel to 
the rafter, thenj n or u g will give the vert load of 2 tons which travels to e from 
the 4 tons at rf; while j u or g n will give the strain, = 2.7 tons, exerted along the 

second secondary rafter e c by the tie g e. Similarly the parallelogram y mv g gives 

the vert load y ?e or u g, = 2 tons, which travels from d to c, and the strain y u or 
w g, = 4.47 tons, exerted upon c e at c by the tie c g. The 4 tons at d therefore pro¬ 
duce a strain y tt = 4.47 tons along cd, and one, nj = 2.7 tons along de ; because uj 
= 2.7 tons exerted at e, and v o = 1.77 tons exerted at d (= 4.47 tons in r\ 1) hot/i press 
the lower part d c, and strain it against the equal pres of yu at c ; whereas this upward 
pres, = 4.47 tons, of y u, is diminished at d by the dowmvard pres, 1.77 tons, of r o, 
leaving an upward pres of 2.7 tons to strain d e against the equal downward pres uj 
at e. In the same way y u and uj of the lower parallelogram show the strains (4.47 
and 2.7 tons) brought upon a b and b c respectively by the 4 tons load at b; and y u, 
uj, of the middle parallelogram give the corresponding strains (8.94 and 5.4 tons) 
brought upon a c and c e respectively, by the 8 tons at c. 

The total strain upon any portion of the rafter is found by adding together the 
uniform strain e h or fh, common to all parts, and the strains peculiar to such por¬ 
tion, as shown by the lines y u and uj. Taking the part c d for instance; as the 
lower half of c e it sustains y u of the upper parallelogram, = 4.47 tons; as part of 
the upper half of a e. it sustains uj of the middle parallelogram, = 5.4 tons; and as 
part of the entire rafter it sustains e h = 17.9 tons; or, in all, 4.47 + 5.4 -|- 17.9 = 
27.77 tons. 

Thus we have, for tlio total strains along' a rafter. 


From e to d, strain = u j of upper parallelogram -(- uj of middle parallelogram -f- e h = 2.7 -f- 5.4 
-)- 17.9 = 2fi tons. 

From d to c, strain — upper y u -(- middle uj + e A =: 4.47 -f- 5.4 4- 17.9 — ‘27.77 tons. 

" c to h, strain — lower uj -f- middle y u e h ~ 2.7 —j— 8.94 - - 17.9 = 29.54 tons. 

‘‘ 6 to a, strain = lower yuf middle y u e h — 4.47 -j- 8.94 -(- 17.9 — 31.31 tons. 


The renter vert e. t may be omitted in short spans where the tie bar is hor, as 
a l Fig 21; since it then sustains nothing but the wt of the half (y y) of the central 
spread x x of the hor tie a l. 


Art 1<» A. If the main or primary tie is raised at its renter, 

ns p n. Fig 19, proceed as at Fig 16, and, after having found the vert strains 

at all the points of support, as before, add to that (16 tons) at e, an amount 


Said .. the vert ht t n, Fig 19, to which the 
16 tons * tiep n is raised above the horp t. 


the remaining ht, n e, of the truss. 

This additional amount is the strain on the cen vert rod e n, which is indispensa¬ 
ble in such cases. Then, as in Art 15, lay off the vert ef. Fig 20, equal to the 

total vert strain at e, thus found ; and after drawing f h parallel to the ralter, draw 


* In practice, the ties a o, a n, &c, Fig 21, of the secondaries, are not always made distinct from 
that (a f) of the primary truss a e l; but they are so represented in Fig 21, merely to show more 
plainly that the central portion x x of the primary tie a l needs only such dimensions as will enable 
it to sustain the thrust produced by the 16 ton strain at e: whereas, along its portions x m, x m it 
must be stout enough to bear, in addition , the pull along the first secondary ties, n a, k l\ while at 
its ends m a, m l it must resist not only the two preceding forces, but also those along the second 
secondary ties o a, o l. Likewise it is plain that the portion g e of the first secondary tie n e, must 
be stouter than the portion n g ; because g e has to bear also the pull along the second secondary tie 
i e. in Fig 20, those portions of the ties which are most strained are shown by stouter lines. 











TRUSSES. 


577 


h z parallel to the inclined tie, instead of hor. f h gives the pres throughout the 
rafter, due to the total vert load at e; and h z gives the pull throughout the raised 
tie-rod, due to the same load. Both /h and h z are of course greater than the cor¬ 
responding strains e h, h z, Fig 20. Like them, they are to he used in finding the 
total strains in the several parts of the rafter and tie-rod. 

If we omit the third secondary trusses e s w, &c, Fig 19, the strains on the struts, 
iv g % i y, k m , will be the same as those on the corresponding struts, d g, cx, bin , Fig 
20; but the strains on the rafter, corresponding to y u , uj, Fig 20, and those on all 
the ties, corresponding to in y, mj, &c, Fig 20, although found by the same process 
as in Fig 20, will be greater than in that case; in addition to which the uniform 
strains, fh, liz , exerted throughout the entire rafter and tie-rod respectively by the 
final vert load at e , will ateo be greater, as explained above. 

Art 16 IS. If the main tie is raised only part way, asp y , Fig 19, 
and then continued hor, as y o; draw it as if it extended to n, as in Art 16 A, and 
use the same hts t n, n e, for finding the additional vert pres at e. Find fh and h z 
as in Art 16 A. On p n lay off by scale the pull h z ou p y found as above; and on 
this as a diag draw a parallelogram with sides parallel respectively to ye and y o. 
The latter (hor) give the total strain on y o, and the former give an additional stiain 
on y e, to be added to those found for each part of that member as directed in Art 16 A. 

In this case, as in Fig 20, the cen vert e o sustains only the wt of half the hor 
stretch of the tie bar, and may be omitted in short spans. 

Rem. 1. It is not necessary actually to draw all three of the parallelograms, as in Fig ‘20. The large 
or center one alone will suffice ; for we need only div the several strains measured along the strut cn. 
Fig ‘21 ; and along the ties n a, n e, by ‘2, to get'those aloug the struts 4 o and 4 i ; and along the ties 
ao, co, ci, ei. And these, in turn, div by 2, will give those along the smaller subdivisions shown 
between e and p, Fig 10, if there are such'; and so on with any number of still smaller ones. 


88 









578 


TRUSSES. 


( 


H' 

r» 

di 

ft 

m 

d 

k 

tli 

an 

IT? 

fi 

Ti 


Art. 16C. Remarks on king- and queen : and on Fink trusses, 

lor roofs. The following comparison is founded upon total spans, or lengths - 
ot truss, of 154 ft. Rise 30.8 ft; or ^ of the total span. Trusses 7 ft apart from cen¬ 
ter to center. Each rafter 83 ft long. Total load, including the truss itself. 40 
lbs per sq ft of roof; or 20.8 tons to each truss. There are seventeen points ot sup¬ 
port in each truss; consequently a full panel-load (Art 11) is — = 1.3 tons. Irusses 

as shown, half of each, in Figs 47A and 48, p 606. The strain in tons (calculated in 
as if all the weight of truss and load were on tqe rafters) is marked on each 
member. The assumed coefficient of safety for ties is 3. Iron is supposed to be used 
that will not break with a less pull than 20 tons per sq inch; the assumed safe allow- 

20 jMii 

able pull being therefore here taken at j = 6% tons per sq inch. The safe pressure 

along the rafters is taken at 3^ tons per sq inch. The struts are assumed to be wrought 
cylindrical tubes, with an outer diam equal to l of their length; and of such 
thickness as will give them a metal area of 1 sq inch for each 2 tons of strain. The 
rafters are in the present case supposed to be 9-inch rolled l’hcenix beams; 7% sq ins^ 
transverse area; weighing 25 lbs per foot run. The ends of all ties are supposed 
to be enlarged, or upset; so that the cutting of the screw-threads shall not diminish 
their effective area. The purlins are supposed to be at or near the “points of sup¬ 
port,” so as to produce no cross-strains on the rafters. k 

Table 1. Weight of the Fink truss, of which Fig 47A shows 

one-half. 


Length 154 ft. Rise ^ length. Trasses 7 ft apart. Load 40 Q>s per sq ft of roof; including truss. 

_ (Original.) 


Name of part. 

Number 

of 

parts. 

Area of 
each part, 
sq ins. 

Lbs. per 
foot run 
of 

each part. 

Total 
weight of 
all the 
parts. 
Lbs. 

Lbs. 


2 






(n . 

2 

1.95 


430) 



•w . 1 o . 

2 

9.77 



Mam tie. ^ n . 

2 

3 41 



1446 

lr 

2 

*4 RR 


It 1 L 




4 

n 





\t . 

4 






o_.•_ ! m . 

2 






Secondary t . 

2 

1.47 

1 71 



[■ 

676 

| *. 

2 





L . ::::: 

8 

2 

0.25 

2.40 

1.20 

0.60 





(j . 

8.0 

4.0 

<OJ 



Struts. <w . 

4 




/* .. 

$ 


: 


Center vertical y . 

say 

say 

i 


40 

400 

400 


40 

400 

400 

Joint and splicing-pieces, nuts, &c, &c... 





Shoes at ends of rafters, say. 










Wt of purlins not included. Total wt of t 

russ 




=7588 

7588 






















































TRUSSES. 


579 


With the »ame total load per sq ft, including the trusses, (with trusses 7 ft apart; 
rise £ span) the area of each part, its wt per ft run, and the strain upon it, are 
as the spans; but the total wts of the trusses alone are as the squares of the spans, 
lienee, it is easy to deduce from the table the areas reqd for smaller spans. The 
rafters for small spans are frequently made of round iron rods from to ins 
diam ; or of ordinary flat bars. Tubes with the same area of metal, would be better. 
For trusses also of different spans, and rise of \ the span, 7 ft apart, in which the 
rafters and struts are of wood, with ties of iron, the strains may be deduced quite 
closely from those in Figs 47A and 48 . They will, however, be somewhat greater, 
because wooden struts, not being hollow like our assumed iron ones, must be heavier 
than the latter to prevent bending. The weight of the load, however, is generally so 
much greater than that of the truss, that this consideration of the strut is not very 
material; so that a roof partly of wood may be assumed in practice to weigh, together 
with its load, but little more than an iron one; and the strains on the several parts 
will be nearly the same in both cases. 


:Sf 

4 

iice 

It 

:a 

il&t 

ea 

!'«■ 

ills 





.«! 

11® 

lie 

!uf« 


shot 




1 



Tabic 2. Weight of the king- and queen truss, of which 
Fig 48 show’s one-half. 

Length 154 ft. Rise ^ length. Trusses 7 ft apart. Load 40 lbs per sq ft of roof; including truss. 

(Original.) 


Name of part. 

t 

Number 

of 

parts. 

Area of 
each part, 
sq. ins. 

Lbs per 
foot ruu 
of 

each part. 

Total 
weight of 
all the 
parts. 
Lbs. 


Rafter. 


2 

7.5 

25 

4150 


4150 00 


f H . 

2 

2.2 

7.33 

146 701 




G. 

2 

2.44 

8.14 

162.80 




F . 

2 

2.68 

8.94 

178.80 




E . 

2 

2.92 

9.74 

194.80 


1606.30 

Main tie. + 

D . 

2 

3.16 

10.54 

210.80 

- 



C . 

2 

3.41 

11.34 

226.80 




B . 

2 

3.65 

12.14 

242.80 




.A . 

2 

3.65 

12.14 

242.80. 












r i . 

2 

.0 

.0 

.0 1 




j . 

2 

.1 

.33 

5 34 




k . 

2 

.2 

.67 

16.00 




1 . 

2 

.3 

1 00 

32.00 1 

298 66 


m . 

2 

.4 

1.33 

53.33 




n . 

2 

.5 

1.67 

80.00 




lo . 

2 

.6 

2.00 

112.00, 



Center vertical... 

p . 

1 

1.4 

4.67 

150.00 


150.00 


[q . 

2 

.88 

2.92 

64.20'] 




r . 

2 

1.05 

3.50 

91.00 




8 . 

2 

1.28 

4.25 

136.00 




t . 

2 

1.55 

5.17 

196.37 


1587.50 


u . 

2 

1.82 

6.08 

267.63 




V . 

2 

2.12 

7.08 

368.30 



• 

tv . 

2 

2.40 

8.00 

464 00 > 







400.00 


400.00 





400 00 


400.00 






8592.46 


8592.46 

Wt of purlins not included. 







im 


i-i 


w 


Hence, the w r t of the king and queen truss in this instance is equal to 


8592.46 
~758fT" 

= 1.132 times (say li times) that of the equally strong Fink; or the Fink is about 
part lighter than the K andQ. Theoretically the diff would be less, because the rafters 
of the K and Q truss being so much less strained at top than at foot, may be diminished 
toward their upper ends, instead of being proportioned throughout with reference to 
the max strain at their feet. If the theoretical diminution toward the tops of the 
rafters, were made in both cases, the wts of the two forms of truss would be nearly 
equal. But in practice, on the score of inconvenience, this is rarely done in roofs 
of moderate span : say less than about 100 ft. No such diminution, or but very slight, 
would be admissible even theoretically, when the purlins are not placed at the points 
of support only. Willi same total load per sq ft, including 
trusses themselves at samedist apart, the total wts of trusses 
are as the squares of their spans; but their wts per ft of span, 
as well as the cross areas, wts per ft run, and strains along 








































































580 


TRUSSES. 


individual members, are directly as thespans. 

When the dist apart of the trusses is 7 ft from center to center; the 
rise ^ of the span; assumed load, including the wt of the trusses themselves, 40 | 
lbs per s(j ft of roof covering; and the various parts proportioned for the several 
strains per sq inch assumed in Tables 1 and 2; the weight of a properly con¬ 
structed Fink truss will be approximately as follows: 

Total wt in lbs of square of span in ft 

a Fink roof-truss " 3.1; 


and the wt in lbs per ft of span = 


span in feet 
3 7l 


A total K. and Q truss, will be about ^ part more; or 

^ span in feet 

Or per foot of span, = 


square of span in ft. 
2.7~ 


2.7. 


| ^ 23710 

These rules give ~ = 7650 lbs, for the foregoing Fink ; and = 8784 lbs, 

for the K and Q truss. From these rules we have drawn up the following 


Table 3. Approximate weights of roof-trusses of the Fink 

system. (Original.) 


Rise span. Trusses 7 ft apart. Load 40 Its per sq ft of roof, including truss. 


Total 

Span. 

Total wt of 
a Truss. 

Wt per ft. 
of Span. 

Wt per sq ft 
of ground 
covered. 

Total 

Span. 

Total wt of 
a Truss. 

W t per ft. 
of Span. 

Wt per sq ft 
of ground 
covered. 

Feet. 

Lbs. 

Lbs. 

Lbs. 

Feet. 

Lbs. 

Lbs. 

Lbs. 

20 

129 

6.46 

.92 

100 

3228 

32.3 

4 60 

25 

202 

8 08 

1.15 

105 

3557 

33.9 

4.83 

30 

290 

9.67 

1.38 

110 

3904 

35.5 

5.06 

35 

396 

11.3 

1 61 

115 

4267 

37.1 

5.29 

40 

516 

12 9 

1.84 

120 

4640 

88.7 

5 52 

45 

654 

14.5 

2.07 

125 

5041 

40.4 

5.75 

50 

807 

16.1 

2.30 

130 

5452 

42.0 

5.98 

55 

976 

17.8 

2.33 

135 

5880 

43.6 

6.21 

60 

1160 

19.4 

2.76 

140 

6336 

45.2 

6.44 

65 

1363 

210 

2 99 

145 

6782 

46.8 

6.67 

70 

1584 

22.6 

3.22 

150 

7260 

48.4 

6.90 

75 

1815 

24.2 

3.45 

155 

7750 

50.0 

7.13 

80 

2064 

25.8 

3.68 

160 

8256 

51.6 

7.36 

85 

2331 

27.5 

3.91 

165 

8782 

53.3 

7.59 

90 

2616 

29.1 

4.14 

170 

9324 

54.9 

7.82 

95 

2912 

30.7 

4.37 

175 

9879 

56.5 

8.05 


For king and queen trusses add ^ part to the tabular wts; when the rafters are 
as usual of the same size throughout. 

The wts in the 4tli column will remain nearly the same, whatever may 
be the dist apart. For if this be increased say to 14 ft, each truss will sustain twice 
as many sq ft of roof; and must itself be at least twice as strong and heavy, in order 
t<> do so. We say “ at least," because if the dist apart is increased, the wt of the pur¬ 
lins will generally increase more rapidly than said dist. Thus, if the dist be doubled, 
the purlins will not only be doubled in length, which alone would double their wt; 
but they must also be deeper. In practice, however, long purlins are usually pre¬ 
vented front becoming very heavy, by trussing them, as at 7, Figs 21%, 

The cost, at shop', of trusses alone for iron roofs and bridges, gener¬ 
ally varies between 2 and 2% times the cost of ordinary “ refined ” bar iron. The 
putting up alone from % to % the cost of the iron. With roof trusses 7 ft apart, iron 
purlins will weigh about 2 tbs per sq ft of ground covered by the roof. Therefore to 
any wt in the 4th col add 2 lbs. Add for covering with tin, slate, or corrugated iron 
See pp 404, 418, 429. 

Rem. 1. As to the proper total weight, or load, per sq ft 

of roof, (including snow and wind,) that shouldbe assumed to be sustained by the trusses, engineers 
differ considerably. The French appear to consider SO lbs as sufficient; while the English use 40. 
Since roofs are not subject to violent vibrations like bridges, they do uot require so high a coefficient : 
of safety; this should uot, however, in our opinion, be taken at less than 3; and this we consider 
sufficient. The load is evidently influenced by the character of the roof-covering. Withiu ordinary 
limits, for spans not exceeding about 75 ft, and with trusses 7 ft apart, the total load per sq ft, includ¬ 
ing the truss itself, purlins, &c, complete, may be safely taken as follows ; 






























TRUSSES 


581 


Table 4. 


Span 75 ft or less. 


Roof covered with oorrugated iron, unboarded,f 
If plastered below the rafters, 

“ corrugated iron, on boards, 

If plastered below the rafters, 

“ slate, unboarded, or on laths, 

“ on boards, 1J4 ins thick, 

“ if plastered below the rafters, 
“ “ ,l shingles on laths. 



Wind 

and Snow.* 

Total. 

8 lbs. 

20 lbs. 

28 lbs. 

18 “ 

20 " 

38 •• 

11 “ 

20 “ 

31 “ 

21 “ 

20 “ 

41 “ 

13 “ 

20 “ 

33 “ 

16 “ 

20 “ 

35 “ 

26 “ 

20 “ 

46 “ 

10 “ 

20 

30 “ 

20 “ 

20 “ 

40 “ 


Example of use of foregoing- tables. A Fink roof 60 feet span; rise i *, 
trusses 14 it apart; and to be covered with slate, on boards 1% inch thick. Here we 
see at once from Table 3 that at 7 ft apart, its wt would be about 1160 fts ; therefore, 
at 14 ft apart, it would be 2320 ft>s. But our table is for 40 lbs per sq ft of roof: while 
for slate on boards, 35 lbs, or % part less, is sufficient. Therefore, we may reduce the 
weight of the truss JX part, making it only 2030 lbs. 

Ex. 2. Roof as before, 60 span ; trusses only 7 ft apart. Turn to Table 1, where the 

pn 

areas are given for a total length or span of 154 ft. But 60 ft is the — = say the 


.4 part of 154 ft; therefore, the areas, and the wts per font run of each member of 
the 60 ft span, will be .4 of those of the 154 ft one. Thus, the area of a rafter will 
be 7.5 X -4 = 3. sq ins; which corresponds with a rolled T iron of 3 X 3}^ ins, and 
inch average thickness. Its wt per foot run will be 25 X -4 = 10 lbs. The area 


A 


of the part n of the main tie will be 1.95 X -4 = .78 sq inch, which we see at once 
from a table of circular areas, is equal to a round rod very nearly 1 inch diam. Its 
wt per ft run = 6.5 X -4 = 2.6 lbs ; and so with all the other members. But the total 
wts wi21 be as the squares of the span. The square of 154 is 23716; and that of 60 is 

And --!- == .152; therefore, the total wts will be .152 of those in Table 1. 


3600. 


Thus, the two rafters will weigh 4150 X -152 == 631 lbs. The main tie, 1446 X -152 = 
220 lbs, &c. Lastly, if for 35 lbs per sq ft, reduce each area and wt part. 


Since the rafters are generally made of T or I iron, a pattern precisely adapted to the calculated 
strains, will not always be procurable ; and in that case we may either take the nearest one in excess : 
or change the dist apart of the truss to suit the pattern on hand. Owing to the variety of modes of 
arranging the details of the junctions, &c, an exact coincidence between the calculated and the actual 
wts, is not to be expected; but we suspect that in properly proportioned roofs, the discrepancy will 
rarely be fouDd to vary more than about 5 per ct from the results of our rules. 


It might be supposed that with iron of a tensile quality considerably higher than 
our assumption of 20 tons per sq inch; as say of 25 to 30 tons, the truss might be 
made much lighter. But this is not the case: because the superiority would affect 
the ties only; inasmuch as the compressive strength of iron does not increase with 
its tensile strength ; but to a certain extent rather the reverse. Now, by Table 1, it 

appears that the ties in a Fink roof-truss, constitute less than yy of its entire wt. 
Therefore, iron of even 30 tons, would reduce the weight of the truss less than 
part of y 3 y part; or yg-; and 25 ton iron, about yy part. 


Short spans need not have as many subdivisions, or “points of support, ’ as a large one; and this 
will lessen the number of parts of the truss; but inasmuch as the remaining parts will require to be 
proportionally stronger, this consideration will not materially affect the wts. While on this subject, 
we will remark that too few points of support are probably used at times: owing to either an under¬ 
valuation. or an ignorance of the effect of the transverse strains produced by the load on the parts 
of the rafter between said points. These parts must be regarded as so many separate beams sup¬ 
ported at both ends: or rather, as firmly fixed at both ends, when the pieces composing a rafter are, 
as usual, strongly connected together; in which case the beam is about twice as strong as when 
merely supported. If the separate parts be trussed, like the purlin at 7, page 583, to neutralize this 
transverse action, it must be remembered that additional compression will be thereby produced 
lengthwise along the rafter. The best practice is, as far as practicable, to increase the number of 
points of support, so that the purlins may rest upon them alone, or near them ; and thus relieve the 
rafters entirely, or in part, from transverse strain. 

Rem. 2. As to the effect produced on the weight of a truss, by 
changing its rise, no short correct rule cau be laid down. Although as a 
roof becomes flatter, its area becomes less, so that each truss has less total wt of roof- 
covering, snow, and wind, to sustain, still the strains on most of its members become 
greater; requiring greater wt of truss. To find this increase with accuracy, it is 


* See Snow and Wind, p 216, 221. 

t The corrugated iron itself will weigh from 1X to 2 !bs per sq ft on the roof. If not plastered under¬ 
neath, the condensed moisture of the air, especially from crowded rooms, will fall from the iron into 
the rooms below. Mere boarding will not prevent this, even if tongued and grooved, unless the circu¬ 
lation of air against the under side of the iron is effectually cut off. 







582 


TRUSSES. 


necessary to make a diagram, and perform all the calculations. The strains on a 
Fink rafter become more nearly uniform throughout its length, as the pitch of the 
roof becomes less; while, with a rise of % span, the strain at its foot is about 
times that at its head. On the contrary, the strains on its struts remain nearly the 
same in amount for all ordinary rises. 

In the king and queen truss the strains at the heads and feet of the rafters retain the same pro- 
nortions to each other, at all rises; the strains on the verticals become less as the roof becomes flatter; 
while those on the obliques vary according to their several obliquities. Uuder these irregularities, 
which affect the K and Q, much more than the Kink, we can do nothing more than say that when it 
is merely wished at the moment to form a rough idea of the effect of changing the rise, we may 
assume the weight of a Fink truss to increase about in the same proportion as we diminish the rise; 
or to diminish as we increase the rise. Thus, if we increase the rise of the roof in Table 1, one-fourth 

part, so as to make it equal to .25 or >4 of the span, instead of .2or^ of the span, we may diminish its 

wt y K part; making it about 6000 lbs, instead of 8000. Or if we reduce the rise from i to yU, making 
it only half as great, we shall double its weight, making it 16000 tt>s; as rude approximations. 


Figs 21^2 show a few of the ninny forms of the detail* 
of iron roof*. Every maker has his own modifications of them. Most of the 
figs explain themselves. They will serve as hiuts. 

It and P stand for rafter and purlin. In small roofs, with the trusses only 3 or 4 
ft apart, the purlins may, as at 6, be simple x / 2 inch or % round rods, about 9 ins 
apart; and the slates may rest immediately on them, being tied to them by iron 
wire. They may be bent down at their ends, and riveted to the rafters. As the dist 
between the trusses increases, these purlins may be made of flat iron, from 1 to 3 ins 
deep, and }4 inch thick: or of light T iron, &c; and may be trussed, as at 7. so as to 
admit of being placed several feet apart. When, however, they have to bear great 
weight, the mode at c, Fig 7, of confining their ends to the rafters, will be too weak. 
Sometimes they may be arranged as at y. Or the purlins, of either iron or wood, 
may rest on top of the rafters, as at 1 and 5; or their ends may rest in a kind of 
stirrup, as at t , Fig 2; and at P, Fig 4; in castings placed at the “points of sup¬ 
port” of the truss; or they may be confined to the sides of the rafters by two angle- 
irons, as at 1*, Fig 9. Purlin* should, alien practicable, be sup. 
ported only at or near tile *• point* of support” of the truss; and 
as a general rule, it will be expedient to arrange the number of these points with 
reference to this particular. The rafters are then relieved from transverse strains; 
and may be proportioned with regard only to the compressive strain in the direction 
of their length. Too little attention is sometimes given to this point, and the trans¬ 
verse strain is overlooked, to the serious injury of the roof. It is well, however, to 
bear in mind that thin deep rafters are liable to 3 r ield by buckling sideways; and 
that this tendency is diminished by purlins well secured to them between the “ points 
of support.” Sometimes castings similar to 2, are used at the heads; and 3, at the 
feet, of the struts and vert ties; which last have their ends cut into right and left 
hand screws, for insertion into corresponding female screws cut in the castings. 
At 3, 11 is the main tie passing loosely through the lower opening through the cast¬ 
ing. Below it, is seen the head of a small set-screw 7 , for tightening together the 
casting and the tie; to prevent the former from slipping out of place. There must 
be different patterns of these castings, to suit the obliquities of the several obliques - 
or, in small roofs, the parts a a may be made w ith hinges, for the same purpose. 


TRUSSES. 


583 


At 4 nnd 5 are east-iron shoes for supporting the ends of the trusses 
upon the walls. With the exception, perhaps, of these shoes, it is better that the 
details generally should be of wrought iron. 

At 8 is a mode of confining thin metal roof-covering i, to 

•h® purlins P, by means of short (about an inch) XJ-shaped pieces (c c 11 is one 
of them) of the same metal; to which % is riveted by an y R inch rivet through each 
flange 11. iliis may be adopted with corrugated iron covering, which, by its strength 
allows the purlins to be placed several feet apart. See Corrugated Iron. 1'lat 
sheets require boards beneath them. 

At 10 is a mode of confining a wooden purlin P on top of an iron 
one p, by means of a crooked spike ns; which, after being driven from below, is 
clinched or bent on top. Wooden purlins are sometimes thus required, for nailing 
6lates or plain sheet metal. At 11, c, is a stick of timber inserted between an iron 



"Firjs 21 } 


purlin P and the corrugated roof-covering a a. To such sticks plastering-laths may 
be nailed, when the roof is to be plastered beneath, to avoid condensed moisture. 
There is room for much ingenuity in all these details. Fig 12 is a rafter made of two 
channel-bars riveted together; with a web member c c between them. Two angle- 
bars are often thus riveted together for a rafter. 

Fig 13 shows a tlirnbliekle or arm swivel tb, for shortening a tie-bar 
made in two lengths. If tb is made of a pipe or solid bar it is called a double 
nut or pipe swivel, and, at least for a part of its length, it is made square or 
hexagonal, so that it can be turned with a wrench. In either case a female screw is 
tapped in each end of the nut, right and left hand respectively; and corresponding 
screws are cut on the ends of the rods. W r hen the swivel is revolved, the two ends 
of the rods are drawu nearer together. In the arm swivel, one rod-end may be left 
plain and round, as in fig, and furnished with a head c. 

Fig 14 is a mode of tighten ing foil r lengths of tie-bars crossing 
each other, by means of a ring. The ends inside of the ring are cut into screws, and 
provided with tightening nuts, as in the fig. The rings are usually % to inch 
thick ; 3 to 5 deep ; and 7 to 10 diam. 








































584 


TRUSSES. 


Horizontal Fink Truss, 

uniformly loaded. 

Strains on posts = final 

panel loads : at c — * i — half the total wt 
of truss aud load ; at b~ mi, and at d ~ g i, 
each = half* i. Strains on ties: 
on m c aud g c, each ~ms~gv\ on * m 
aud * g, each =: x o = x n ; on m a aud g e, 
each — * o + 771 y — X 71 + gw. Strain 



on Ckor<l , t e (uniform throughout) = half on-)- half y s = half o n + half v w. 


In the Fink Irnss, the effects of a moving- load may be 

calculated as for a full uniform maximum load from end to end. Thus assuming 
at first, as in the preceding cases, that everything is borne by one truss only; then, 
when the load is upon the top chord of the truss, each vert post may in practice be 
regarded as upholding one-lialf of that portion of the entire wt of bridge aud dis¬ 
tributed load which is between the two extreme ends of the two obliques which 
uphold said post. Thus, in Fig 46, the half-way post d c, bears half of all between 
a and b. The post m g, half of all between a and d ; the post h o, half of all between 
m and d. This is equivalent to saying that the half-way post bears half the entire 
wt of the bridge and load: each quarter-way post, one-quarter; each eighth-way 
post, one-eighth, &c, <tc, of this same entire wt of bridge and load; and these consti¬ 
tute theoretically the strains on the several posts. But after having got thus far, it 
is necessary to examine whether some of the smaller ones may not have to be in¬ 
creased, for the following plain reason : Suppose we have assumed our max load to be 
a string of engines, weighing 1 ton per foot run; or, including the wt of the 

bridge itself, say 1.4 ton per ft; and suppose our posts to be as close together as 5 
ft; then the least loaded posts would each bear 5 X 1-4 = 7 tons. But we know that 

from 16 to 20 tons may 

lxo AAnonntrafA/1 tifif 1 



be concentrated within a 
length of 5 ft, on four 
drivers of an engine; and 
half of it will have to be 
supported by each post in 
succession as a train passes. 
When we thus find by trial, 
which posts will be more 
strained by an engine than 
by our assumed max per ft 
run of the whole truss, wo 
In the Fink, and Bollmar, 


must increase the load first found, correspondingly, 
trusses, the verts are always struts or posts. Having fixed upon the load for each 
post, as p o, Fig 21 h, then for the strain which said load will produce upon each of 
the obliques, or ties, p c, p h, upholding said post, take any distp d on the post, to 
represent the load by scale; and draw d w, d n , parallel to the ties; then p w,p n, 
measured by the same scale, will respectively give the strains on each ; whether they 
be equally inclined as usual, or not. The two hor lines na,w a, by the same scale, 
give the two hor forces which the load at the top o of the post, acting through the 
ties, produces upon the chord at c and h; which two equal and opposing/era s pro¬ 
duce along the intermediate stretch c h of chord, a strain equal to one of them. In 
other words, either n a, or w a, gives the hor strain produced along c h, by the load 
at o only. See “ Strain,” Art 2, of Force in Rigid Bodies. 

Strain on the chord. This, from a uniformly distributed load, is the same 
throughout the entire length of a Fink chord. To find it, observe which obliques, 
(as m e,ge,o e, Fig 21 1 ',) of one-half of the truss, terminate at one. end , c, of the chord. 
Then, having previously found the loads on the posts, c o, u g, t m, w hich pertain to 


f 
















TRUSSES. 


585 



those obliques, ascertain by the pro¬ 
cess in Fig 21 A, the hor force n a, 

(in both figs,) which each of those 
loads produces on one oblique. Add 
together these forces n a, (there 
will be but three of them in Fig 21 i, 
as marked by the dark lines ;) their 
sum will lie the strain along the en¬ 
tire chord. The obliques m u, u r , 
r c, g c, do not terminate at e; and 
are, therefore, omitted in finding 
the chord strain. The process is the same whether the verts are all of the same 
length or not. 

Or the hor chord-strain produced by each of the loads on the posts c o, u g, t m , 
may be calculated thus, and added together. 


Horizontal 

strain 


entire load v hor dist from post 
— on post to end e of chord 

twice the length of the post. 













586 


TRUSSES. 


Art. 17. Fig 22 represents a suspension truss on the Boll- 

inan plan ;* the whole weight supposed to be along the top a p. 

In this, the strain from each, 
panel-load, as for instance that 
at d, passes down to the foot of 
its supporting post dj ; and fror 
there is transferred to the tw 
ends a and p of the chord, by 
means of two ties, as j a, j p , 
upon which the post stands. In 
this manner the vert strain from 
each panel-load is separately 
sustained; and transferred di¬ 
rectly as a lior strain to the ends 
of the chords, by its own post 
and pair of ties; without pro¬ 
ducing, as in the foregoing cases, 
an additional vert strain at 
the points of support of the 
other panel-loads. So omit 



* * 
^52 

§ ft 
ft® 


3 

o 

O. 


>4-. 


Cd 


00 


t© 


N 




o 

'"3 

o 

CO 


proof that the strains have been drawn and 
weight of truss and load is carried in the same way 
the ties yp, xp, Ip, Ac. These wts cause the following strains : 


our 2nd process; and 

having divided the uniform wt 
of the truss and its load, among 
the several points of support a, 
b, c, d, e, Ac, as before, wo pro¬ 
ceed at once to draw the parallel¬ 
ograms of forces v u 1 g,v u k g, 
Ac, for measuring the strains. 
To do this we have only to set 
up the equal vert dists l v, k v, 
j v, Ac, each to represent by 
scale the 4 ton panel-loads on 
top of the respective posts; then 
complete each parallelogram by 
drawing v u, v g parallel to the 
two ties which support each 
post. Then the lines l u, l g ; 
k u, k g, Ac. give by scale the 
strains along the respective ties. 
The end a of the hor chord is 
pressed hor by tlife seven hor 
forces u o, v o„ u o u , u n in , Ac., 
equal to 1.75 + 3 -f 3.75 + 4-f 
3.75-1-3-1-1.75 = 21 tons; and 
the other end p is in like man¬ 
ner pressed by the seven corres¬ 
ponding forces not shown ; and 
these two sets of equal op¬ 
posing forces produce a strain 
equal to one of them ; or to 
21 tons, uniform throughout the 
entire chord. The tie la car¬ 
ries to a so much of the weight 
of the 4 tons at b as is rep¬ 
resented by l o, or 3.5 tons; 
k a carries to a a weight equal 
to k Oj, or 3 tons; j a carries j o t/ , 
= 2.5 tons ; i a carries i o n , = 2 
tons; w a, w o = 1.5 ; x a, xo — 
1; aud y a, yo = .b ton. All 
these amount to 14 tons ; which, 
with the 2 tons of half panel¬ 
load at a, give 16 tons ; or half 
the entire weight (32 tons) of 
the truss and its load. This is a 
measured correctly. The other half 
to the other end p, by means of 


* Invented by Mr. Wendel Bollman, C. K. 































TRUSSES. 


587 


The strain l u — 3.91 tons. 

ku = 4.25 “ 

** ju — 4.52 “ 

i u = 4.47 


The strain 1g = 1.82 tons. 
“ k g = 3.17 “ 

“ j g = 4.05 “ 

ig= 4.47 


Each post or vert is of course strained to the amount of a full panel-load, when 
the whole wt is supposed to be on top of the truss. 

In the Bollinttn, for a moving; load, having first pre¬ 

pared the working diagram, determine the max weight thatcan come upon a post. 
This will be the same for each post. If the moving load is on top of the truss, this 
load on each post will consist of the greatest wt of engine that can stand upon one 
panel-length of truss; together with (approximately enough for practice)one panel- 
length of floor: and the half of a panel-length of truss. If the load is at the bottom 
of the truss, the posts bear no part of either the moving load, or of the floor; but 
each of them will be strained to the amount of the wt of half a panel of truss. 

The loads on the posts may then be written upon the diagram. 


, Tlie obliques or ties, however, when the load is at the bottom, bear (as in the 
i Fink) the same amount of strain from the moving load and floor, as when it is on top. 

Therefore, when it is at the bottom, each pair of ties sustains not only the load rest- 
t! ing on the post which they uphold; but the wt of one panel-length of floor, and the 
1 max panel-weight of engine. In other words, whether the toad be on tip, or at bottom , 
t the two ties at the foot of each post, sustain a wt equal to a full panel-length of truss 
; and floor; together with the max panel-wt of engine. Having added these wts to- 
, gether, lay off their sum by scale at each post, as shown at / v. k v,j v, i r. Fig 22; com- 
- plete /. g v u, k g v u, &c; and measure the strains l u, l g ,k u, k g, «fec, along the ties. 

The strains oil any pair of ties, may also be calculated thus; having 
■ the load they sustain. 


i 

r 

i 

i 

r 


i! 

; 


Strain on 
short tie 


Strain on 
long tie 


; \y h° r f rom P os t f° 

° a X farthest end of chord 
total length of truss 

j j ss h° r dwt from post to 
oaa X nearest end of chord 

total length of truss 


X 


X 


length of 
short tie 

length of 
post. 

length of 
long tie. 

length of 
post. 


The hor strain on the chord will be uniform throughout, as in the Fink 
truss ; and will depend upon the max uniform load that can cover the whole bridge; 
and not, as in the case of the ties, upon the greatest load which each pair of ties may 
have to sustain in succession; unless we assume our max uniform load to be a string 
of engines which may bring a max panel-wt of engine upon every pair of ties at 
once. ^In that case we have only to measure upon one-half of our working diagram, 
the several hor lines corresponding to u o, <fcc, in Fig 22; and their sum will be the 
reqd hor strain on the chord. But if we take our max uniform load on the whole 
truss to be a string of cars, we must diminish the chord-strain thus found, in this 
manner: Add together a full panel-weight of truss, floor, and cars , then, as the full 
nanel-wt of truss, floor, and engine, (which we before assumed as the straining load 
. of each pair of ties,) is to the panel-wt of truss, floor, and cars, just found, so is the 
hor chord-strain before found, to the one reqd. , , , 

We will repeat, that chords must be strong enough to bear not only the hor pull 
or push to which they are exposed ; but also to sustain safely, as beams, the trans¬ 
verse strains from the floor, and from the moving load, when these rest upon them 











588 


TRUSSES. 


BOWSTRING, AND CRESCENT TRUSSES, UNIFORMLY 

LOADED. 

Art. 18. Before attempting to find tbe strains on either a uniformly loaded 
bowstring or a crescent truss, Figs 23 , 23 6, 23 c, by means of a diagram, the student should familiar¬ 
ize himself with the following remarks: 

Rem. 1. Tlie basis of the entire process is that at every point of sup¬ 
port, beginning at an abut as the first one, we have acting one or more known forces, balanced or 
held in equilibrium by either one or two unknown ones; and the object at each point is first, by 
means of the parallelogram of forces, to find the resultant of the known ones ; and second, by the 
same principle, to resolve this resultaut into two components in the directions of the unknown ones. 

Tliis is all that, is required in either the bowstring or the 
crescent truss.* 

Rem. 2. While more than two unknown forces exist at any 

point of support, their amounts cannot be found. If one force is known, 
ami two unknown, the three balancing each other, draw a line by scale to 
represent the amount and direction of the known one ; and, considering it as one side of a triangle, 
from its two ends draw lines parallel to the two unknown ones, to meet each other, thus completing 
the triangle. Then these last two will by the scale give the two unknown oues ; because wheu three 
forces meeting at one point, balance each other, three lines representing them both in amount and 
in direction, will form a triangle, t 

If there itre two or more known forces, first find the single re¬ 

sultant of them all and, taking it as one side of the triangle, find the other two sides 

(that is, the two unknown forces) as before. After a little practice the student will tiud it unneces¬ 
sary to draw more than half the sides of the parallelogram of forces. 

Rem. 3. The bow is to be considered straight from apex to 
apex. If actually curved it will be much weakened. 

Rem. i. At each point of support, or apex, consider every member that meets there, to be a force 
either pushing towards said point, if along a strut; or pulling from 
it, if along a tie. This is shown by the arrows in Figs 23, 23 a, &c.; thus iu 
Fig 23, at o, the forces along the struts co, so, and 70 , push towards o; qo also 
pushes towards q ; while the force along the tie ro pulls from o , as also from r. All 
loads on the bow are forces pushing vert downwards. 

Rem. 5. If the known forces are not all alike at any point, (that 

is, neither all pulls uor all pushes,) then while constructing the parallelograms of forces, one or more 
of the forces must be changed, and be regarded as acting at the opposite side of said point, but iu the 
same direction as before: so as to make 'hem all alike; otherwise the parallelogram will not give the 
oorrect resultant. For instance, in Fig 23 a, we have (as will be seen hereafter) 

three known forces acting at c, namely e c pushing towards c: a load (uot shown) on the bow at c, 
pushing vert downwards towards c ; and pc pulling from c. In this case we must, while drawiug 
the sides of the parallelograms, either consider the pull p c to be changed to a push in the direction 

_______I 

* The same process applies equally to all our figs from 10 
to 10 also; whether uniformly loaded on one chord or on both ; also to the in- 

verted bowstring, and to Fig 23 d. In this last, as in the others, the lower member (tttiH) is a tie 
which prevents the truss from spreading, aud thus causes it to exert only vert pres against the abuts. 
This is a distinctive character of all so-called truss girders, including all our 
figs back to Fig 5. But when nn n is converted into an arch for sustaining 
compression, it ceases to be a tie. and the truss then exerts hor as well as vert 
force against the abuts; and becomes what is called a braced arch. The pro¬ 
cess requires a slight modification before it can be used for such, as shown in Art 20. Al¬ 

though strictly speaking the process does not apply to the Fink truss. Figs 19, 20, 21, because we 
there encounter three unknown forces at any point where three web members as c.m } cx, co Fig 20 
meet at a rafter as at c, still as we can readily determine the strain on one of these, c x, by means of 
the entire vert strain at such point, (which strain can be found in a Fink truss bv a mere mental 
calculation) we thus reduce the unknown forces to two, and therefore may employ the process even 
for such trusses. The student would do well to test Figs 10 to 16 by this process. 

t Any one of the three forces is then the anti-resultant, or bal-l 

ancer, or opposer, of the other two; and if three arrow-heads 

showing their directions be added to the sides of the triangle as in this Fig they will.I 

/ A as it; were, chase each other around the triangle ; that is the head 
of any one of them will touch the butt end of the one next to it; or no two arrow-heads 
will meet. I»iit tins is not' so when three sides ot a triangle represent two 
forces and their resultant, or equivalent in effect ; for the arrow-head of the re¬ 
sultant will then meet that of one of the other forces. 









TRUSSES. 


589 

























590 


TRUSSES. 


I 


from t towards c; or the other two to be similarly changed to pulls. The first of course is the easier, i 
According as the known forces (after one or more are so changed, if necessary) are pulls or pushes, | 
the diagoual or resultant will be the same. 

Rem. 6 . To decide whether an unknown force is a puli or 
a push ; that is whether the member along which said iorce acts is a tie or a 
strut. Having fouud the resultant of the known forces, add to it an arrow-head to show its direc 
tiou. Then having on this resultaut completed the triangle by means of the two unknow n forces, 
add arrow heads to them also, placiug them so that the three arrows shall chase each other around I 
the triangle. Then imagine euch arrow of the unknown forces to be placed one at a time without 
changing its direction, upon the line representing its respective web member. If, in this position, : 
the arrow pushes toward the given apex, the member is a strut. If the arrow pulls au ay from, the y 
apex, the member is a tie. 

When there is but one unknown force, and it is found to form a 

straight line with the resultaut of the known ones, then its arrow must poiut in the opposite direc¬ 
tion from that of the resultaut. We may add that of the two unknown forces at any apex, one is al¬ 
ways along either the bow or the string ; and must plainly be a strut in the first case, and a tie in the 1 
last one. For the present we will call it the chord-force. The other unknown force will be along a 
web member. Now it can always be seen at once that theresultaut and the chord force together tend 
to displace the apex at which they act, by moving it either outwards or inwards, from or towards the 
truss ; and if we simply consider that it is the duty of the web-member to counteract this displacing 
tendency, we shall have uo trouble in deciding whether it must for that purpose pull or push at the 
apex, or in other words be a tie or a strut. 

Rem. 7. After having found the resultant of the known forces, said known forces themselves must 
be considered as no longer existing. 

Rem. 8 . To find Ihe strains eorreefly requires great eare and 

attention. Plain as the foregoing remarks are, the student will in his first attempts probably com- | 
mit many errors. A little practice however w ill rectify this. 

A good metallic parallel ruler on rollers, and about 18 inches long, ' 
is almost indispensable. Paper ruled in squares facilitates the work. The lead-pencil must be kept j 
sharp, and the lines should be drawn lightly. A scale of from to ^ of an inch to a foot or ton w ill 1 
generally be found convenient. It will be difficult to find all the 
strains to within tlie nearest .1 or .2 of a ton. or even more: be¬ 
cause, as the work progresses errors which are inappreciable at the start may insensibly enlarge 
themselves. It will be seen from our table of bowstrings of 80 ft span, p 592, that the strains on some 
ot the web members are less than .1 of a ton ; so that the diagram may even w ith great care mi-lead 
us to the extent of 100 per cent, or more, in these small strains. Fortunately this is a matter ol little 
importance, for these members are so small that a liberal allowance for errors involves but a trifling 
w aste of material. In the larger strains errors of .1 or .2 of a ton arc of no consequence. Never con- ( 
sider the work of a diagram complete, however, until after testing it by some of the proofs in Art 19. f 

Art. 19. Example. We will now apply the f«re$:oing re- l 
marks to tlie bowstring truss. Fig 2R. Its span is 80 ft; its rise in j 

ft. The bow is divided into 8 equal parts ; and the lower apices are horizontally half-way between I 
the upper ones. The trusses are assumed to be 7 ft apart from center to center. 'I he total wt of the | 
truss and its load is supposed to be equally distributed along the bow only, and to amount to 10.4 I 
tons, which corresponds to 40 lbs per square ft of roof covering. This gives 1.3 tons for a full panel J 
load ; and half as much, or .65 of a ton resting directly (or without passing along the web members) I 
ou each abut; and the finding of these, and figuring them on the diagram as shown, constitute our I 
*• first process," Art 11.* 

This done, draw at the abut a a vert line av equal by scale to half the entire wt of the truss and I 
load, minus the part panel load which rests directly upon one abut; or to 5.2 — .65 = 4.55 tons. This I 
line represents the vert upward reaction of the abut, against that portion of the wt ot the half truss 
and load that causes the strains which we are about to seek.t This reacting force, which is a known I 
one, balances the two unknown forces, ae along the bow, and ap along the string. To find these we 
have only (Rem 2) to consider a v as one side of a triangle, and from its two ends to draw two meet¬ 
ing lines an and vn parallel to, or in the same directions as, the unknown forces. Then will t> n 
give by the same scale 9.94 tons strain along the hor string ; and a n 10.93 tous, along the bow. '1 he 
arrow on an pushes toward a. Hence ae is a strut (Rem 6). But the arrow ou v n, if placed on 
ap. pulls away from a. Heuce ap is a tie.t .- 

Now let us go to the apexp. There we find that we have one known force, (the 9.94 tons along 
up) balancing three unknown ones, namely p q, pc. and p e. lienee (Rem 2) we i 
cannot now find these last; therefore we leave them for the present, and try at the 
apex e. Here we have two known forces, namely the 10.fi3 tons 

pushing along a e, and the 1.3 tons of load which of course push vert downwards. Both these being 
pushes, neither ot them requires to be reversed ; and they balance two unknown 
forces, namely ec along the bow; and ep along the oblique. Hence we have only to draw fe (see 
Fig W) to represent a c ; and xe to represent the 1.3 ton load, aud completing the parallelogram fe. | 
id. draw its dingoual d e, which is the resultant of /e and i e ; or it alone would balance the two j 
unknown forces. By measuring d e by scale it becomes a known force. Therefore takiug it as one 1 
side of a triangle, from its tw r o ends draw ds parallel to the bow ec, and es parallel to the oblique ' 


* Remember that in a truss of any form it is only when the stretches along which a load is uni- i 
formly distributed are equal, that the panel loads are also equal; or that the portion which rests di- I 
rectly on an abut is just half a panel load. 

t Each abut of course reacts vert upwards against the entire half wt of the truss and its load ; but 
inasmuch as that portion of said wt which rests directly on an abut, does not reach the abut by way 
of the web members, and therefore has nothing to do with their strains, it is omitted from the press¬ 
ure set up at the abut for ascertaining said strains. 

7 When, as in Fig 5J><j, p 552, the load is uniformly distributed along the rafter, and the latter is 
supported only at its ends, it will not do to assume that the load is concentrated at said ends. Such 
a rafter presses, at its foot, in the direction H a, and not in the direction of its length o a. There¬ 
fore, if a i in that Fig be drawn to represent the upward reaction of the abut, we must draw the tri¬ 
angle a H x (not a ox)-, and H x (not ox) gives the pull on the tie beam. Butin such simple trusses 
the strains are more readily fouud by the method given in connection with Figs 5, 5>$ and 6. 








TRUSSES, 


591 


HQSt 


ep f thus completing the triangle des. Then ds gives by scale the strain 10.56 along ec; and ee 
gives .14 of a ton along ep .* 

Now going again to the apexp, we fiud that we have two known forces pa, p e. both pulls, balanc* 
ing two unKiiown forces p q f p c. Therefore as at Fig Y. take p f and p s to represent the two known 
forces ; complete the parallelogram f p 8 d ; draw its diagonal p d ; and taking it as one side of a tri- 
augle, draw d o parallel top q, and po parallel tope. Then is do by scale 10.01 tons strain aloug 

pq; aml^o is .10 ot a ton along pc. Going to c, we have three known 
forces, ce, cp, and the 1.3 ton panel load, all of them pushes, so that none of 
u > them need be reversed; and balancing co, cq, both unknown. In this case, as shown at Fig V, we 
on must draw two parallelograms, beginning with any two of the known forces. Say 

A\e begin with cx representing the 1.3 ton load, and ce, representing the 10.56 tons. On these two 
diaw the parallelogram cetx , and its diagonal ct. Then draw ca to represent the thirdkuowu force 
cp of .10 ot a tou ; and on it and the diagonal ct draw the second parallelogram ctsa, and its di¬ 
agonal c 8. 1 his last diagonal is the resultant or single force which would balance the two unknown 

forces co, cq; therefore take it as one side of a triangle, and from its two euds draw two meeting 
lines parallel to said unknown forces, and measure them by scale to obtain the amounts of those 
forces. On account of the smallness of our scale we have not shown these two lines. In practice, in 
order to prevent coufusion, it is well to rub out the sides of the parallelogram after having found the 
first diagonal. In this manner proceed until the final strains on os, sr, and rt complete the whole. 

II the entire uniform wt of truss and load is assumed to 
be on the string or lower chord, as in Fig 23a, then all the web mem¬ 
bers become ties; but the process remains unchanged. Therefore first distribute the entire wt 
among the lower apices and abuts by our “ first process.” Then, as in Fig 23, draw the vert line a v 
(—half the enure wt, minus what rests directly on one abut) and from it find the strains on ae and 
ap. 1 hen going to e we have one known force ae balancing the two uukuown ones ec aud ep. 

11 After finding these go to p. Here we have three known forces, p e, p a, and the panel load ; the first 
two of which are pulls, while the third (resting on top of the string) is a push vert downwards. 
Therefore we will reverse this last, and consider it as being a vert pull pi; aud draw the first par¬ 
allelogram on p i and p a. After finding the resultant of the three pulls, and by it the two forces p q, 
»iii pc, we go to c. Here of the two known forces, ec, being a push; and pea pull, we must reverse 
one of them, say p c ; representing it by an arrow t c; and complete the parallelogram ou t c aud c e. 
We have now had an instance of all the cases that occur, and have shown how to manage them. 

tr Proofs of accuracy of the work. The resultant of the strains on 

lii-i those members (os and rs), Fig 23, of the half truss that meet at the center s, of the bow, and of half 
end; the pauel load, 8, at the center, should come out to be a hor line, and equal to the hor strain aloug 
the center stretch, r t, of the string. With all the care, however, that can be takeu, it will be very 
difficult to make the coincidence exact. In some trusses the half truss will have three members 
meeting at the center of the bow ; one of them (a center vert one) belonging partly to each half of 
the truss. Then only half the strain on this one, as well as half the center panel load, is to be used 
in the proof. All this applies to any form of truss however loaded. 

Again, if all the uniform wt of the truss and its load is as¬ 
sumed to be on the hor string, and if the string is divided into equal, 
or nearly equal, parts by the web members, the lior strain at the center 
should be equal, or about equal, to 

Wt of half truss and load X .25 of the span 

Depth of truss. 

But with very unequal divisions of the string, such as will rarely occur, this formula is not even 
roughly approx. 

With all the wt uniformly on the how, unequal divisions of the 
string have no effect on the center hor strain ; and if the rise does not exceed about one tenth to one 
eighth of the span, and the bow is about evenly divided, the above formula will be nearly as approx 
us when the load is on the striug. For greater rises, however, multiply the span by the following 
multipliers (instead of by the .25 of the above formula), when the load is ou the bow ; and the half bow 
about equally divided into at least two parts. 


.1 or loss 

1 | 

Rise, 

1 .2 | .25 

in Parts of the Span. 

1 .3 | .333 | .35 

1 -4 

(Original.) 

| .45 | .5 

.246 

1 ’ 243 1 

.239 | .233 

Multipliers. 

| .226 j .222 [ .219 | 

.211 

.202 | .192 


w Rem. The multipliers for intermediate rises may be taken in simple proportion. When multiplied 
' by the span they give the hor dist from the abut to the center of gravity of one half the loaded 
ir bow, (as the .25 or the formula gives that of one half the loaded string,) assuming the load to be 
,e concentrated at the points of support on the abut, and at the apices. It is this 
center of gravity of half load so concentrated that must be 
used for finding the strains in Hie truss, and not that of half load as 

actually evenly distributed; for these two centers of gravity under these two aspects may differ 
greatlv from each other, not only in the evenly loaded bow, but in the string, if divided into very un¬ 
it equal parts by the web members. Kven the number of divisions of the loaded bow makes some dif- 
, lerence in this respect, hut not so great but that the multipliers in the above table will be correct to 
; within about three per ct at most in any case iu wtiich the entire bow lias at least four nearly equal 
divisions. 

* It is usual and far more convenient, to draw all such lines at the apices of the diagram itself: but 
on account of the smallness of the scale of Figs 23, and 23 a, we have drawn them at W, T, aud V. to 
prevent confusion. Also our lines are not drawn to scale because some of the forces are too small 
to bo appreciable with so small a scale. 














592 


TRUSSES. 


If both the bow and the string 1 are uniformly loaded, it is 

plain that the multiplier for any given rise must be somewhere between the .25 of the formula, and 
the decimal for that rise in our table; and this furnishes an easy method of finding, approx enough 
for practice, the hor dist from the abut to the cen of grav of a half truss thus loaded. Thus after 
allotting to the bow and string their respective proportions of the entire wt of the truss and load, 
find the hor dist for each of the two separately ; aud then combine them. 

1abl« of approximate strains in tons on liowst ring trusses 

of 80 ft span. Trusses 7 ft apart from Center to center. Load (including wt of trusses) 40 lbs pet- 
square ft of roof covering; all assumed to be uniformly distributed on the bow. Bow divided into 8 
equal parts; like Fig 25; aud straight from apex to apex. Lower apices half-way hor between the| 
upper ones. Each column of the table commences near an abut, or end of truss. The first or end 
web member in the columns is a tie, the next one a strut, aud so on alternately towards the center, 
as iu Fig 25. Below the table is given the wt of each eulire truss aud its load. 


Rise 20 ft, or % Span. 


Rise 13% ft, or % Span. 


Rise 10 ft, or % Span. 


Rise 8 ft, or Span 

Bow 

Tie. 

Web. 


Bow. 

Tie. 

Web. 


Bow. 

Tie. 

Web. 


Bow. 

Tie. 

Web. 

Tons. 

Tons. 

Tons. 


Tons. 

Tons. 

Tous. 


Tons. 

Tons. 

Tons. 


Tons. 

Tons. 

Tons. 

7.01 

4.85 

0.58 


8.80 

7.50 

0.22 


10.9 

9.94 

0.14 


18.3 

12.5 

0.07 

6.02 

5.18 

0.55 


8.50 

7 72 

0.18 


10.6 

10.01 

0.10 


13.0 

12.6 

0.05 

5.56 

5.54 

0.25 


8.05 

7.78 

0.15 


10.4 

10.16 

0.07 


12.8 

12.7 

0.07 

5.54 

5.58 

0.25 


7.90 

7.88 

0.15 


10.5 

10.22 

0.07 


12.7 

12.7 

0.07 



0.10 




0.10 




0.05 




0.03 



0.10 




0.10 




0.04 




0.03 

Total wt 11.6 tons. 


Total wt — 10.75 tons. 


Total wt = 10.375 tons. 


Total wt = 10.25 tons. 


Art. 20 . The braced arch. Fig 23 W. Find the center of gravity of 


m w l n 



either half, as o n c, of the truss and its load. Then 

Weight of said half v Hor dist of said cen of grav 
truss and load, in tons from the nearest abut 

Vert dist c s from abut c to cen o of truss 


Having thus found the hor strain ft c at the center, and knowing the wt ti c of the half truss anc ; 
load minus the part panel-load that rests directly on the abut, draw them as shown, and fiud their I 
resultant r c. Then use this resultant precisely as the vert Hue a v. Figs 23 and 25a, is used;! 
namely, by drawing from its euds two lines for finding the first two uuknown forces; then proceed 
precisely as in those figs. 

Proof of accuracy of the work. If all has been done correctly, tliel 
last resultant near the center of the truss will be iu a straight line with the last member; the strain 
along which it represents. Also, the resultant of those members of the half truss which meet at the 
center of the truss, and of half the center panel load, should come out a hor line equal to the calcu¬ 
lated hor strain. But as in the bowstring, &c.. it is almost impossible to secure a perfect agreement. 
It would be well to test the strains near the center, especially the very small ones, by the principle 
indicated b.v the following remark. 

Rem. it is not necessary to begin at an abut in order to 
work out the strains: for after having disposed the load properly among 

the apices, and calculated the hor strain at the center of a truss, we may employ said strain, and one 
half or the panel load at the center, as two known forces acting against the hair truss at the center- 
find their resultant; and resolve it along the two members of said half truss that meet there; and 
thus work on down to an abut. The tine representing the hor strain must evidently be drawn’as if 
pushing against the half truss whose strains are sought. This remark ap¬ 
plies* also to bowstring trusses, and others, as Figs 10 and 11, Ac., 

in which we have two or more kuown forces acting against our half truss at the center: and not 
more than two unknown forces of our half truss meeting there also. In an actual bridge there would 
he a hor member w l, supporting the flooring, and a short vert one extending upward from o. aud 
helping to support wl. But these do uot form part of tne truss proper. 


Ilor pres at the 
cen o of the truss. 








































TRUSSES. 


593 


Cantilevers. Suppose the half o, n, c , of the braced arch, Fig 23 W, to be a 
trussed cantilever, with n and c firmly built into, or attached to a wall; and loaded either along the 
top o n, or otherwise. Then to calculate the strains, begin with only the load concentrated at the 
outer end or apex o, as a given known force, and resolve said force along o l and op ; and so on to n 
and c, taking in on the way the loads at the other apices as before. The upper 
chord will be ill tension; the lower in compression. A revolving? 
truss drawbridge, when open, assimilates to two cantilevers joined back to 
back. 

Art. 21. This fig represents an . a 

opened swing-bridge support¬ 
ed on rollers on the pierp, and by tie- 

rods a o, at, Ac. All the wt of oe is up- / I \ 'SOW 

held by ao; that of e s by a i, &c : and 

that of s t directly by the rollers. Draw d/' / j IC ,S \'t je\|0 

o b and i k vertical to represent the wts 

of oe and es; and draw bm, kn hori- jPi 

zontal. Then will o m and i n give the 

strains along a o and a i. Also b m will give a hor compressive strain reaching from 
o to c: and k n one reaching from ttoc. 


39 












594 


TRUSSES, 




Art. 22. 28, shows the general arrangement of a small 

wooden Howe bridge-trnss; Fig 29, some of its details; and Fig 30, those 
of an iron truss. High trusses are sometimes made as in Fig 1. The top and bottom 
«hords of the wooden one are 
each made up of three or more 
parallel timbers c c c, placed a 
small dist apart, to let the vert 
tie-rods r r pass between them. 

The main braces, o o, are in pfiirs 
or in threes. The pieces com¬ 
posing them, abut at top and bot¬ 
tom, against triangular angle 
blocks, s; which if of hard 
wood, are solid; and if of cast- 
iron, hollow; as shown at T, 

Figs 30; strengthened by inner ribs. 

These extend entirely across the three 
or more chord-pieces. Against their 
centers, abut also the counterbraces e. 




3 , 


These are single pieces in small bridges; 
or in pairs, in large ones; and pass be¬ 
tween the pieces which compose a main 
brace. Where the wooden braces and 
counters cross each other, they are 
bolted together. For wooden chords, 
the angle-blocks are cast,* as at T. The 
dotted lines show the strengthening 

ribs; and x serves to keep the block in place. The vert tie-rods r r, of iron,are in 
pairs, threes, or fours, Ac, according to size of bridge; with a screw and nut at each 
end. The heads and feet of the braces aud counters, butt square against the angle- 
blocks: and are kept in place only by the tightening of the screws of the vert ties. 
\N hen the floor is below, as in Fig 28, the end posts p d ; and the ends g i and w b, of 
the upper chord, may be omitted; also i c aud b y ; but it is seldom done. 



In Figs 30, of an iron Howe truss, the top chord P, M, and W, is cast in one piece 
transversely, as at P. Its separate lengths are connected together by flanges and 
bolts, somewhat as shown at W ; where a a are cast longitudinal flanges for strength- 
ruing the transverse bolting-flanges g. Instead of separate angle-blocks at the upper 
chord, solid ones may be cast in the same piece with the chord itself, as shown at M. 
The lower chord usually consists, as in other iron bridges, of four or more flat bars of 
rolled iron, c, placed on edge: and some dist apart, as at K. On top of them rest the 
lower angle-blocks s, which have shallow channels below, for receiving the chord 
pieces; and thus securing them from lateral motion. A cast washer, a, below the 
chords, is provided with similar channels on top, for the same purpose. The braces 


•X- In large spans, to prevent the pressure of the heads and feet of the obliques from crushing 
the chords, the angle-blocks are cast with deep projecting flanges under their bases; and which, pass¬ 
ing between the pieces which compose a chord, extend to the opposite face of the chord. There ths 
flanges hear upon broad washers at the ends of the vert rods. Ry this means the vert components 
of the strains along the obliques are transferred directly to the verts, without at all affecting the 
chords. Angle blocks of course have openings for the passage of the vert rods. 




































































TRUSSES. 


595 


i and counters, o, e, in moderate spans are usually cast in a star-shape, as at j.* The 
t following table gives dimensions sufficient for a strong Howe bridge; although in 
« wooden bridges it is customary to add arches when the span exceeds about 150 feet. 

Dimensions for each of two trusses of a How e bridge for a 
Single-track railway. Timber not to be strained more than 800 lbs per sq 
j inch ; nor iron more than 5 tons per sq inch. Iron supposed to be of rather superior 
quality, requiring from 25 to 27 tons (604&0 lbs) per sq inch to break it. The rods 
to be upset at their screw-ends. To each of the two sides of each lower chord is sup- 
: posed to be added, and firmly connected, a piece at least half as thick as one of the 
j chord-pieces; and as long as three panels; at the center of the span. 


Clear Span. 
Feet. 

Rise. 

Feet. 

No. of Panels. 

An upper 
Chord. 

A lower 
Chord. 

An End 
Brace. 

A Center 
Brace. 

A Counter. 

End Rod. 

Center 

Rod. 

No. of 
Pieces. 

j Size. 

No. of 
Pieces. 

Size. 

No. of 
Pieces. 

Size. 

No. of 

Pieces. 

Size. 

No. of 

Pieces. 

Size. 

No. of 

Rods. 

Diam. 

«4- . 

O CO 

. 'O 
o O 

a 

<3 

s 

! 




Ins. 


Ins. 


Ins. 


Ins. 


Ins. 


Ins. 



?5 

6 

8 

3 

5X 6 

3 

5X12 

2 

5X 8 

2 

5X 6 

1 

5X 6 

2 

l 'A 

2 

1 

50 

9 

9 

3 

OX 9 

3 

6X14 

2 

6 X 9 

2 

5X 8 

1 

5X 8 

2 

l % 

2 

i y< 

75 

12 

10 

3 

6 X12 

3 

6X14 

2 

6X11 

2 

6 X 8 

1 

6 X 8 

2 

2 M 

2 

ik 

mo 

15 

11 

3 

6X14 

3 

6X16 

2 

8X12 

2 

6X10 

1 

6 X10 

2 

m 

2 

i y 2 

125 

IS 

12 

4 

6X14 

4 

6X16 

2 

9X14 

2 

6X12 

1 

6X12 

2 

» 

2 

i% 

150 

21 

13 

4 

8X14 

4 

8X18 

3 

8X14 

3 

6X10 

2 

6X10 

3 

2 X 

3 

i% 

175 

24 

14 

4 

10X16 

4 

10X20 

3 

8X15 

3 

8 X10 

2 

8X10 

3 

3 

3 

ik 

200 

27 

15 

4 

12X16 

4 

12X20 

3 

9X16 

3 

8X14 

2 

8X14 

3 

3K 

3 

ik 


The same dimensions will serve for a double road for common travel. 

For bridges of iron, assuming the safe strain for iron to be 5 tons per sq inch, or 
14 times as great as the 800 lbs assumed for wood; the areas of the cross-sections 
of the individual members will as a general rude approximation, be about one- 
fourteenth part as great as those of wooden ones. Equally strong wooden, 
and iron bridges of the same span, will not differ very materially in weight. 

Art. 23. Fig. 31, 

y t 7 3C _ shows in like manner, 

lH a wooden Pratt 
truss: and Fig 32,some 
details of a small iron 
one.* After the forego¬ 
ing, they do not need 
much explanation. Since 
the angle-blocks support 
tension rods instead of 
struts, they are placed 
above the top chord, and 
below the bottom one. 
The main obliques are 
in pairs: and the smaller 
single counters pass be¬ 
tween them, as in the 
Howe. In large bridges 
they are in threes, fours, 
&c. The vertical posts, 
which, when of iron, are 
hollow,*are retained in 
their positions both by 
the strains on the 
obliques, which termi¬ 
nate above and below 
their ends ; and by being 
let into the chords. In 
large spans, the details 
generally vary more or 
less from those in the Figs. In Fig 31, cccare the main braces ; andooo the counters. 



i 

j 

i 


* Cast iron is now rarely used in trusses, except for short blocks sustaining compression, such as 
angle-blocks etc. Rolled iron and rolled steel have taken its place. 















































































































596 


TRUSSES. 


When the roadway is below, as in Fig 31, the ends, rb,y t, of the upper chord; the i 
end verticals p and u\ and the two tension obliques in each end panel, may be omit¬ 
ted ; and two diagonal struts from b and y must then be substituted, extending to 
the abutments, for upholding the upper chord, Ac. Ill the Pratt the chords I 
may be of the same dimensions as in the foregoing table for Howe's. The posts may 
have about ^th less area than the main braces of the Howe. The main brace rods, 
and others, (of the same number as the main brace pieces of the Howe,) may have 
the following diams in ins; allowing the safe strain to be five tons per sq inch. 


For each Truss of a Pratt Bridge. 


Spans. 


25 Ft. 

50 Ft. 

75 Ft. 

100 Ft. j 

125 Ft. 

150 Ft. 

175 Ft. | 

200 Ft. 




End Main 

-brace Rods. 



Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

2 of iy g 

2 of 2*4 

2 of 2*4 

2 of 3 

2 of 3 

3 of 3 

3 of 3>4 

3 of 3% 



Center Main-brace Rods. 



Ins. 

Ins. ! 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

2 of 1 *4 

2 of 1 % 

2 of 1*4 

2 of \% 

2 of \V A 

3 of 1*4 

3 of IX 

3 of IX 


Counter Rods at Center. 


Ins. 

Ins. | 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

Ins. 

1 of \% 

1 of \% 

1 of 2X 

1 of 2>4 

1 of 2J4 

2 of IX 

2 of 2 

2 of 2*4 





In Pratt’s truss the directions of the main braces and counters are respectively 
the reverse of the Howe. Many of the remarks in the preceding Art, apply equally 
in this. Neither the Howe nor the Pratt possesses any special advantage over the 
other as regards ease of adjustment, Ac. In both trusses, arches are frequently added 
in wooden railroad bridges when the span exceeds about 150 ft. 

Art. 24. Town's lattice truss, Fig33, as originally introduced, and very 
extensively employed, was of extremely simple construction; being composed en¬ 
tirely of planks from 2 to 3 ins 
thick; and from 9 to 12 w ide; de- 
pending on the span. Two sets 
of these were placed crossing 
each other at angles of about 
90°; and were connected to¬ 
gether at their intersections by 
either 2 or 4 treenails of locust, 
or other hard wood, about 2 
ins diam. At the top and bot¬ 
tom, similar planks, a a, c c, 
were treenailed hor, to form the 
chords; or in large spans, (many 
exceeded 150 ft,) there were two 
upper and two lower chords, (sometimes of timber 6 ins thick,) as shown in the fig, 
by n n, o o. The transverse section A shows the two upper chords on a larger scale ; 
each chord consisting of two planks; one on each side of the lattices. Two trusses 
of this kind, with a depth equal to % or ^ of any clear span not exceeding about 175 
ft; planks of 3 XI - white pine; the open squares 2% ft on a side, in the clear, were con¬ 
sidered sufficient for a common-road bridge 20 ft wfde. Many of these bridges warped 
sideways very badly; and when applied to railroad purposes, failed entirely. In 
some cases the better mode of three lattices was employed; two of them running in 
one direction ; and the third in the other direction, passing between them. A funda¬ 
mental defect was that the parts were of equal size throughout the span ; w hereas the 
chords should be stoutest at the center; and the lattices, near the ends. These de¬ 
fects caused the lattice to fall into unmerited neglect, in the United States ; whereas 


l t 11 v 


? WV W7 


I K\//\\//\-J^/AVAVAVAV/ 












































TRUSSES. 


597 


iu Europe it is, when properly proportioned, highly esteemed; especially for iron 

bridges; some of which, on this principle, have been 
constructed of more than BOO feet span. The tendency 
to warp, owing to the thinness of the trusses, is obviated 
in the large bridges referred to, by placing a double 
truss, Fig B4, (instead of a single one,) at each side of 
the bridge. The trusses T, D, composing a double one, 
are placed a foot or more apart; and are connected to¬ 
gether at proper intervals, by short pieces riveted to 
each one, for stiffening them. At T and D are seen the 
three lattices or lattice-bars, of each truss; two of 
which, on the outside, constitute the main braces; while the center one is the coun¬ 
terbrace. Such a double truss bears some resemblance to the Fairbairn box-girder; 
the diff being chiefly that in the former the sides are composed of lattice-bars; and 
in the latter of solid plates. 




*1 

n 

1 

w 

T 


1 

D 


Fig 34 


















598 


TRUSSES. 


I 

P 

O' 

ii 

o: 

m 

ti 


The Bowstring 1 truss, Fig 35, is an excellent one as regards strength, and 
economy of construction. It has, however, the disadvantage, in large spans, of a diffi¬ 
culty in connecting together overhead the two trusses of a span, so as to be as free 

from lateral vibration as a bridge 

4 


•77 






y 


Fig35 

BOWSTRING 








with parallel upper and lower 
chords. In the latter, this con¬ 
nection can be made from end to 
end of the span ; but in the Bow¬ 
string it can be done only for 
some distance each way from the 
center; from want of headway 
near the ends. In short spans 
with low trusses, this defect is 
not felt. For a graphical method 
of finding the strains in bow¬ 
string and similar trusses, see pp 
588 to 592. 


6 

r 

6 

1 

c 

i 





































TRUSSES. 


599 


Tlie Lock Ken viaduct, England, of 130 ft clear span, and 18 ft high, 
has two trusses, 13 ft 8 ins apart clear, for single-track railroad, on the Rowstring 
principle, Fig 35, omitting only the verts; which, however, affects the strains on the 
obliques. For convenience of construction, it was built chiefly of rolled channel- 
iron, (see t <,) of 8 ins by 4, by 4, by % inch. At Fig 35, d shows a transverse section 
of the arch or bow; which is uniform throughout. That of the uniform chord or 
string is of the same fig and size as the arch. Each has an area of 33 sq ins in each 
truss. The top strip, a, a, of rolled iron, is 24 ins by and is riveted to the upper 
flanges of the two channel-irons 11 ; which are 8 ins apart, so as just to allow the passage 
between them of the obliques d ; which also are of the same channel-iron. The panels 
are about 12 ft long near the center of a truss; and 8 ft at its ends. The wt of the 
two trusses alone, without the roadway, is very nearly 50 tons. The wt was increased 
beyond the theoretical requirements, to save the trouble and expense of preparing 
and fitting together many pieces of diff dimensions; yet, although the bridge is a 
strong one, the trusses alone, weigh together but .37 of a ton per foot run. Where 
two obliques cross, one is in two pieces, riveted to the other by straps. 

In the State of New York are many much-used bridges for common 
travel, by Mr. Whipple, of 100 feet span, and 12% ft rise; with two trusses 19 
ft apart from center to center ; for two roadways ; and having two outside footways, 
j each 6 ft wide. Each truss has 9 panels, braced altogether by vert and oblique tie- 
rods (no struts) arranged as in Fig 35. The verts next the center of each truss, consist 
each of two rods of 1% ins diam, welded together at top ; and straddling 2 ft at bottom. 
The other verts are single, and 2 ins diam. The obliques are all single, and 1% ins diam. 
The arches are of cast-iron. The transverse area of metal of each arch is 18 sq ins at tlio 
j crown ; and 21 sq ins at the spring. The shape of arch transversely resembles a channel- 
j iron with its back upward; the total depth of flange 7 ins; the width of arch on top, 11 
j ins at center of span; but increasing uniformly by means of wide open-work on top, 
to 3 ft at springs. Each consists of 9 straight segments, held together at their but¬ 
ting flanges, by the verts themselves; which pass through them, and have screws and 
nuts at their ends. The screw-ends are not upset. The thickness of metal in the 
arches nowhere varies much from % inch. Under the floor, and between the trusses, 
j are horizontal diagonal braces of rods % inch diam ; two of them to each panel; each 
of them with a tightening swivel. The chord of each arch consists of 4 rods of 2 ins 
i diam. In the same State, are also many similar common road bridges of 72 ft span. 

Rise 9 ft; two trusses, 19 ft apart from center to center; * and two outside footways 
• of 6 ft each in addition. Each truss has 7 panels, with vert and oblique ties, as in Fig 35. 
| Each cast-iron arch is in 7 straight segments, of the same shape as the foregoing; 
with a cross-area of metal of about 12 and 15 sq ins. Its width at center of truss 10 
ins; at springs, 30 ins. The two verts next the center of each truss, consist each of 2 
) rods of 1% diam; the other verts are single, each 1% diam. The obliques are all 
j single, 1 inch diam. The chord or string of each arch, is 4 rods of 1 % inch diam. 
Horizontal diag bracing of 9^ inch rods under the floor, as in the foregoing. 

Some cast-iron bridges of the Severn Valley Railroad, 
England, of 200 ft clear span, consist of arches rising 20 ft, and supporting the 
railroad on a level with the tops of the arches, instead of above them as in Fig 35%. 
There are no diags between the arches and the roadway, as in that fig; but cast- 
iron verts only, placed 4 ft apart. The railroad is double track ; and there are four 
arches, one under each line of rails. The transverse section of an arch is I; each 
flange is 15% ins wide, by 2 ins deep ; the web is 2 ins thick. Total depth at center 
of span, 4 ft; and at the skewbacks, 4 ft 9 ins Transverse area of each rib at crown, 
150 sq ins. Each arch is cast in 9 segments of equal length. 

The cast-iron bridge across the Schuylkill at Chestnut St, 

Pliila, Strickland Kneass, Esq, Engineer, roadway on top, has two arches of 185 ft 
clear span each, and 20 ft rise. Clear width, 42 ft. Each arch has 6 ribs, about 8 ft 
apart in the clear; and of the uniform depth of 4 feet, including a hor top rib 8 ins 
wide; and a similar one at the bottom. Thickness everywhere 2% ins ; thus giving to 
each rib a transverse area of 147% sq ins. The standards are vert, with ornamenta¬ 
tion. It is a city street bridge. The roadway consists of cast-iron plates, which sup¬ 
port a pavement of cubical blocks of granite, laid in gravel. The arches are cast in 
segments 12 ft 10 ins long; each with end flanges 12 ins wide, for bolting them to¬ 
gether with four 1% inch diam screw-bolts at each end. For a change of tempera¬ 
ture from 12° to 99° Fah, the crowns of the arches rise 2y g ins. Under a uniform 
extraneous load of 100 ibs per sq foot, the greatest pres on the arches is but 3600 lbs 


* With only two trusses, the width between them, in the clear, should not be less than 16 ft, to 
allow two ordinary vehicles to pass each other readily ; but 18 or 20 ft is still better; more would be 
unnecessary when there are outside footways. The headway should not be less than 13 ft. 








600 


TRUSSES. 


rJt-, 


per sq inch of their cross-section; or not more than 2*8 the ultimate crushing 
strength of average cast iron, in short blocks. 

Tile Moseley Bridge. Figs 35%, by Thos. W, H. Moseley, of Kentucky, is 
essentially a wrouglit-iron Bowstring, with a hollow plate-iron arch of triangular 
cross-section, apex up; and formed of three plates riveted together; the two side- 
plates, a b, ac, having their top and bottom 
edges bent to form flanges for this purpose. 

The chords oe ; the verts ; and the two counter¬ 
arches tt; are also of iron. These counter¬ 
arches are intended as a substitute for the ob¬ 
liques of Fig 35. Each of them consists of two 
angle-irons, back to back, riveted together, 
and to the verticals, which pass between them. 

Each of them has a sectional area equal to 
half that of the main arch. The verticals are 
placed about 2 ft apart. They are flat (not 
square) bars, for convenience of riveting. They 
pass through holes in the bottom-plate of the 
main arch, (see dotted line of top Fig,) and are 
fastened at a by the same rivets which connect 
the upper flanges of the two side-plates, ab, ac. The chords, ne , are also flat bars; 
and have a transverse area half as great as that of the main arches. At their ends 
they are attached to strong wrought-iron shoes upon which the feet of the main 
arches rest and abut. The rise of the main arches, measured to the bottom of the 


A 


w 



arch, is ^ or y 1 ^ of the clear span. 

The following are the principal dimensions of a single track bridge of 93 ft clear 
span,(97 ft from out to out of arch,) carrying the “Iron Railroad” at Ironton, Ohio. 
It was built in 1860, and is traversed by heavy engines with trains of pig iron, coal, 
Ac. Rise to bottom of arch 10% ft; to middle of arch 11 ft. Bottom-plate s s of arch, 
16% ins, by .44 inch. Side-plates ab, a c, in clear of flanges, 14% ins, by .29 inch. 
Top”flanges at a, each 3 ins. Bottom flanges b and c, each 1.88 ins. Vert rods, Bins, 
by % inch; and 2 ft apart from center to center. The chord of each arch consists of 
two flat bars, each of 4% sq ins of cross-section. The bridge was tested for three 
weeks by a dead load of % a ton; and a rolling load of 1 ton at the same time, per 
foot run; and deflected only % inch. With a load of 1 ton per foot, the pressure at 
the center of the arches would be about 5% tons per sq inch of metal; and a trifle 
more at the feet. The sectional area of metal in each main arch is 18% sq ins. These 
bridges are of easy construction, and consequently cheap. For long spans, vert 
diagonal bracing (see Fig 35) would probably be essential for preserving the form 
of the arches under heavy moving loads.* 

An iron nrch roof in Philadelphia, clear span 80 ft, rise 16 ft, con¬ 
sists of a uniform arch of single 7-inch Phoenix beam of 6 sq ins sectional area; weigh¬ 
ing 20 lbs per foot. This rests on cast-iron shoes on the walls. The lior chord or tie 
at the feet, is of two rods of 1% diam. At the center of this tie is an arrangement 
similar to No. 16, of page 583, from which diverge upward, to the arch, a central vert 
rod 1 inch diam ; two struts of 6-incli Phoenix beam, 20 ft apart at the arch ; and two 
ties each 1% diam, reaching the arch at half-way between the struts and the feet of 
the arch. There are 10 such trusses, each of which by itself weighs about 3900 tbs; 
they are placed 16 ft apart; and by means of purlins resting upon them, support the 
entire weight of the roof, which is of inch boards, covered by thin sheet-iron. 

The iron roof of a rolling-mill near Boston, Mass, of about 80 

ft span, and 16 ft rise, has arches of the Moseley section a, b, c. Fig 35%; but without 
counter arches. The trusses are 12 ft apart. Sides of the arches, clear of the flanges, 
7 ins; upper flanges, 2 ins; lower ones, 1 inch ; total of each side, lOins, by .19 inch thick. 
Bottom plate 8% inch by% inch thick. Total sectional area of an arch, 5.925 sq ins. 
There are besides, a chord; and 24 vert suspending rods; but no obliques. The roof 
Is covered with corrugated iron, on purlins. When required, the heavy iron rolls of 
the mill are lifted by tackle supported by a roof-truss. 


Fig'S 36, represent tlie Burr truss; which was formerly more used 


* Although this bridge seems to have stood very well for several years, the writer would prefer not 
to exceed two tons of compressive strain per sq inch on plate-iron in such structures. In several 
bridges, General Moseley has used a continuous web of % inch iron, instead of the vert suspenders. 
But in such cases the triangular tube is not applicable for the arch ; and he substitutes two Z bars, 
riveted together, and to the web, to, which passes between them, as in the foregoing tig. The counter 
arches being here unnecessary, are omitted. This web would be objectionable in large spans, espe¬ 
cially of draw-bridges, on account of the wind. More recently still, he has also introduced 
lattices, instead of vert bars, in some of his bridges; together with many innovations on the arrange¬ 
ment first described. At 1 ton per ft run, the pull on the chord above,=11 tons per sq in. 


hai 

(0 

ion 

in 

let* 

vrt 

>!' 

tort 
























TRUSSES. 


601 


tit than any other in the United States. It is at present regarded with disfavor by some, 
' because many early ones failed under railroad traffic, in consequence of bad propor¬ 
tions, and tlin absence (as in our Fig 36) of counterbracing. When properly con¬ 
structed it makes an excellent bridge. The common objection to it, and not without 
reason, is that a truss and an arch cannot be so combined as to act entirely in con¬ 
cert; yet, as soon as any ordinary truss begins to fail, the almost invariable remedy 
is to add an arch. When, however, the two are to be united, it is better to so pro¬ 
portion the arch as to be capable by itself of safely sustaining the max load at rest; 

ll 


; y, 

"Jlli 

«■ 


It 



c# and to confine the duty of the truss to preventing the arch from changing its form 
iicij under a moving load. Counterbracing may be effected by strapping the heads and 
. feet of the braces to the chords; or by iron rods parallel to the braces; two to a 
jin brace; with screws and nuts, as at v f. Or by similar rods across the other diags of 
the panels. The following- dimensions answer for a single-track R R 
bridge of about 150 ft span. Rise from out to out of chords one-eighth of the span; 
about fourteen or sixteen panels. Width in clear of arches, 14 ft. Six arch-pieces 
t, of 10" X 13" each, to each truss. Upper chord c, 14" X 16". Lower chord a «, 
two pieces each 10" X 15". Posts p, 14" transversely of the bridge, as in the right- 
hand fig; by 10"; except at the heads and feet, where enlarged to receive the ends 
of the braces. Braces 10" deep, by 13" wide. Floor girders o, 10" X 16"; and 2% 
to 3 ft apart from center to center. Suspending rods (shown at s; and dotted in x) 
1% diam. Counterbrace rods in pairs parallel to braces, about 1}^" diam. Bolts 
for arches, lower chords, &c, 1diam. Theoretically, the posts, braces, and arches 
should gradually diminish from the ends, toward the center of the truss; while the 
chords should increase; but in practice, the additional labor of getting out and fitting 
pieces of (liff sizes, frequently makes it more economical to use uniform sizes.* The 
s line amount of arch would answer also, if trussed, as in Fig 35; and the arch by 
itself with a full max load, would be strained less than 800 lbs per sq inch, at its feet. 
For a span of 200 ft, with the same proportion of rise, the transverse areas of the 
several arch and truss pieces should be increased 33 per cent; and for one of 100 ft, 
they may be diminished the same. None of these dimensions are the result of close 
calculation. The dimensions just given will answer for common travel , for a span of 
200 ft; with a depth from out to out of chords, of ]/^ the span ; panels 10 to 12 ft long. 
Many such spans have been built with timbers of about ^ less transverse section; 
and without counterbracing. The heads of the posts are notched about 2" to 3" into 
the bottom of the upper chords; and are moreover tenoned into it some ins further: 
with two wooden pins through the tenon; see n, Figs 36. Their feet are notched 
both into and upon the lower chords, so as to leave the two chord-pieces a a only 
about 2" apart. Through these and the post pass two bolts of about 1" to 1% diam. 

Since the upper chord resists compression only, its pieces may come together with 
a plain butt joint, d. To this may be added fishes e d, of stout plank, on the sides 
of the chord, bolted through by 4 or 8 bolts. 

The lower chords resist pull; and the pieces composing each lower chord must 
therefore be joined together These pieces should be as 


* It wilt be borne in mind that our examples are not intended to illustrate perfectly proportioned 
structures. None of them would endure strict criticism. There is more waste of timber in an arch 
built with a uniform trausverse section throughout, than in the straight upper chord of a Howe or 
Pratt similarly built; for both must be proportioned to the greatest strain. This is at the center of 
the two last; but while that at the center of the arch is as great as in these, that at its feet is muoA 
greater. See Example 2, Art 38, of Force in Rigid Bodies. 
































































602 


TRUSSES. 


long as possible, and should never be jointed opposite to each other, but one opposite ; ; 
the middle of the other. 

The braces are merely cut to fit to the heads and feet of the posts, after these bust, 
have been fixed in their places ; and usually have no other connection to them than 
one or two spikes at each end, for small bridges; or screw-bolts for large ones. 

The ends of the timbers composing the arches, butt full square against each other; 
and may also have a wooden dowel. The joints should occur at the posts, as shown 


III 


at W. The arches are screw-bolted to the posts, as shown in the figs, by bolts of 1 to 
Vy£ ins diam. Where the arches pass the lower chord, both are notched, and well 
bolted together. The feet of the arches abut against cast-iron plates. 

When suspension rods (dotted in Fig 36) are used for assisting to support the road¬ 
way, they are placed as shown at s s; m being a strong block of wood slightly notched ' 


P 


on top of the upper arch-pieces. The rods are suspended by a washer and nut on top 
of the block; and after passing down between the arches and chord, have a similar 
arrangement, but inverted, below the last; as shown at g. 


P 


- 


- 

; | 


: 


t 


tit 

lied 

j1*n° 

M 9dj 









TRUSSES. 


603 





If a long beam, n h, Figs 45 and 47, requires to be strengthened, 

this may be done by adding a vert post <Zc; and 2 inclined tie-rods oa, cb. And if, 
after this, the two halves, d a , d b, of the beam still are found to be too weak, addi¬ 
tional intermediate posts, oo, may be introduced; with other ties, ij, to sustain them. 

In Figs 45 and 47, the roadway 
is at the chord a b ; no parallel 
lower chord being necessary, it 
may be omitted. The inclined 
ties act as substitutes for it. But 
if the bridge is so near the water 
as not to allow the posts and ties 
to be placed beneath the roadway 
a b, we may raise the entire truss 
upon two posts or piers s s, Figs 
44, 46 ; and place the roadway 
n n,at the lower ends of the posts 
and ties; instead of letting it rest on top of the chord, as in Figs 45 and 47. In Figs 
44 and 46, the truss and its load do not then rest directly upon the abuts y y , but 

upon the tops of the posts 
8, s; and the only part that 
does rest directly on the 
abuts, is nne-lialf of that 
small portion of the road¬ 
way comprised at each end, 
between e and n; in other 
words, only one half the wt 
of the roadway of the end 
panels; the other half being 
sustained by the inclined 
ties which meet at e. 

When the tie-rods all pass 
from the feet of the posts to the 
ends of the chords, as in Figs 
44 and 45, we have the Boll- 
iimn truss. And when, as 
in Figs 46 and 47, only those 
which sustain the center post 
d c, both pass to the ends of 
the chord, while the others 
are disposed as in said figs, the 


BOLLMAN 

Fig. 45. 


Fink truss is the lesult. 



Vi 


t' 
































































604 


TRUSSES. 


The following: dimensions for single-track Fink bridges 

with chords and posts of wood; and iron suspension bars; are on the assumptioi 
that all the bars deflect ^ of the span ; that the road is on top; that the bars slial 
not be strained more than 10000 lbs, or 4% tons per sq inch, under a weight of bridgt 
and load, amounting in all to two tons per running foot. Assumed w r t on eacl 
driving-wheel of engine, 5 tons. Screw ends upset. 

Dimensions for one truss only, of a single-track Fink bridge 

The spans are in feet; the other dimensions are square inches of cross-section o 
each member. J 


Areas in sq ins. 

Areas in sq ius. 

■4J 


*d 

o 

■d 

i 

<n 

O 

*3 

'd 

»d 

© 

*d 

•d 

i 


a 

u 



05 

0, 

a 

u 

o 

« 


os 

05 

0. 

a 

£3 

■4a 

*d 

»d 

♦a 

M 

Q. 

£3 

to 

•d 

*d 

£3 


m 

o 


!N 

CO 

rH 





CO 



35 

275 

8% 

2% 

, , , 

56 

90 

470 

21 

5% 

1% 


r 

l4t> 

40 

300 

6% 

2% 

.. • 

64 

100 

505 

23% 

6% 

2 

• • • 

169 

45 

322 

io% 

2% 

•. • 

72 

125 

580 

29% 

8 

2% 

i% 

225 

50 

340 

n% 

3% 

•.. 

81 

150 

650 

35 

9% 

3 


280 

60 

375 

14 

3% 


95 

175 

730 

41 

11 

3% 

i% 

335 

70 

410 

16% 

•«% 


110 

200 

800 

46% 

12% 

4 

i% 

39. 

80 

440 

18% 

5 


126 









"We have given the area of the 1st post only. For that of the 2.1, we may take % of the first; fot 
the 3d, % of the second ; and for the 4th, % of the third; without pretending to any great accuracy 
The iron rods may be flat, square, or round, so that the proper area be maintained. It will usuall; 
be best to have them flat. 


A width of 18 feet is necessary for allowing two ordinary vehicles to past 
each other readily. It should never be less than 18 ft; and nothing is trained bv 
exceeding 20 ft. J 

It will of course be understood that each member, especially in large spans, will consist of two 
more pieces, side by side. Thus, in a span of 200 ft, the S00 sq inches of each chord will proba' 
consist of Tour beams of about 10" X 20", or 12" X 16", placed side by side ; but with sufficient in 
vals betweeu them to allow the several oblique bars to pass. Or it may consist of six beams of sma i 
size. So also, the 1st, or main rod, of 46% sq ius, will probably be made up of from 4 to 8 bars, of II 
or 5.85 sq ius each, placed side by side; occasionally some inches apart. And so with the others 
The reet of the opposite posts of the two trusses of a Fink span, are connected together by ties of woo " 
or irou, to prevent lateral motiou ; and for the same purpose, diagonals are carried from each uppe 
chord to the foot of the opposite post; as is usually done in top-road bridges of any kind These h 
better be tie-struts. 

' 

The weights of bridges of the same span, designed by different persons, 
vary considerably, from several causes; such as the form of truss; quality of iron • 
coefficients adopted for safety, and for strength of materials; whether the roadway i ’ 
on the top chord, or on the bottom one, &c, Ac. 

For a mere approximate weight to be assumed as a preliminary in 
calculating the strength and proportioning the parts of a bridge, or for forming som 
rude idea of the quantity of iron required, we suggest the following purely empirics 
formula;. They give the weight in pounds per foot run of span, of only the tw. 
trusses or main girders together and their lateral bracing, for a single-track railroa. 
bridge of standard gauge (4 ft S]/ 2 ius). The weight of cross-beams, flooring rails 
etc, is not included. 


For spans not exceeding 75 feet. 

(Plate girders.) 


Weight per foot run (see above) = 5 X span in feet + 50 X V span in feet. 

For spans of from 75 to 250 feet. 

(Pratt, Whipple, or Warren trusses.) 


Weight per foot run (see above) = 4.5 X span in feet + 22 X 1/ span in feet. 
For spans exceeding 250 feet.* 

(Trusses of various designs.) 

span 2 in feet 


Weight per foot run (see above) == 


60 


+ 500. 


* In spans longer than about 300 feet, the differences in truss designs are so ereat 
that the actual weights may differ widely from those obtained by simple practical 
formula; of any kind. y F 





































TRUSSES. 


605 


n ’able of approximate weights of single-track iron railroad 
bridges of standard (4 ft 8 1-2 inches) gauge. 


st 

:1 


* 

» Iti 


Span. 

Weight of the two trusses, or main girders, and their lateral bracing. 

Feet. 

Lbs per foot run of span. 

Lbs total. 

20 

325 

6500] 


30 

425 

13000 


40 

50 

520 

600 

21000 

30000 

- Plate girders. 

60 

690 

41000 


70 

770 

54000 


80 

• 550 

44000 ] 


100 

150 

670 

940 

67000 

141000 

Pratt, Whipple, or Warren 
trusses. 

200 

1200 

240000 

250 

1500 

375000 


300 

2000 

6000001 


400* 

500* 

3200 

4700 

1280000 

2350000 

Trusses of various designs. 

600* 

6500 

3900000 



* 




For double track bridges add 80 per cent to the weight of single-track ones 
ts obtained by the formula?. 

For narrow gauge bridges take 75 per cent of the weights for standard gauge 
>nes as given by the formulae. 


Iron floor systems, with two stringers under each track and adapted for 
leavy loads, such as A p 546, may be taken as weighing approximately as follows: 


O 

. 

Span, in feet. 

Weight of iron floor system, in pounds per foot run. 

•? 

t; J { 


Single track. 

Double track. 

. 1 . 1 

20 to 100 

200 tCL.275 

550 to 700 


100 to 250 

250 to 350 

700 to 850 

It 

250 to 300 

325 to 400 

750 to 900 

bu 

300 to 400 

375 to 450 

il 


400 to 500 

425 to 500 

it 


500 to 600 

475 to 575 

800 to 1000 


Two iron safety stringers will together weigh about 150 lbs per foot run. 

For woo«len floor systems, we may, as a rude average, set down for spans 
6 iot exceeding about 200 feet, cross floor girders of about 7" X 15", and about 2)4 to 3 
teet apart, from center to center; clear span 14 feet; together with substantial string- 
pieces of about 10" X 12", for supporting the rails; the rails themselves; and a plank 
pathway between the rails, all complete, at about .14 ton, or 314 lbs, per foot of span ; 
>r with a full floor of 3" plank, 14 feet wide, about .2 ton, or 448 lbs. For greater 
Spans, with the trusses farther apart, increase this to .25 ton, up to 300 feet; .3 ton, 
p 400 feet; .35 ton, to 500 feet; and .4 ton, to 600 feet. 

Two safety stringers will together weigh about 100 to 150 lbs per foot run. 

When a bridge is to be roofed and weather-boarded, an addition must, of course, 
be made to our weights. 

Wooden bridges weigh about the same as iron ones of equal strength. 

In the Northern Pacific R R bridge over the Missouri River, at Bismarck, Dak, 
built 1881-2, the two inverted bowstring trusses of the 115-foot approach spans, weigh 
^together) .38 ton per foot run; two Pratt trusses of the 400-foot channel spans, 1.09 

*1S 

wo trusses of the Newark Dyke bridge, England, Warren girder, 240)^ foot span, 
weigh 1 02 tons The single-track tubes of the Victoria bridge at Montreal, 244 feet 

’.an 1.14 tons • the 330 foot span, 2 tons. The single-track Britannia tube, Eng, 460 
f span 3 43 tons. Two trusses alone, of the Penna R R, at Philadelphia, 180 foot 
n Pratt’s system,) by Mr Linville, .52 ton. The fine 320 foot span across the Ohio 
Steubenville, (Pratt,)'also by Mr Linville, 1.6 tons per foot. All these are of iron; 
;le track. 


* See foot-note (*), page 604. 



























606 


TRUSSES, 



Our table on page 
605 makes sufficient al- ^3 
lowance for the greater r$ 
loads per ft run that may 
probably come upon small • 
spans. The diffs between 
our 4th and 6th cols will ^ 
give our assumed max 
loads. 

On very small spang 
the loads cannot be as¬ 
sumed to be equally dis¬ 
tributed. On bridges for 
turnpikes and common 
roads, no probable contin¬ 
gency could crowd people © 
upon them to such an ex¬ 
tent as to weigh more than K 
60 fl>s per sq ft of floor. “ 
The French standard in- 
deed is but half of this, 
or 4a lbs per sq ft; and 
is sufficient for probabil¬ 
ity, but not for possibility. 
The latter may increase it 
to 80 lbs; and this may 
safely be taken as the 
maximum load on spans 
of 20 or more feet. To 
compensate, however, 
for momentum, we re¬ 
commend to adopt 100 lbs, 
or .045 of a ton, as the 
limit for crowds, t A 
bridge for a single-track 
carriage-way; with room 
between the trusses for a 
footway also, should not 
be less than 12 ft wide in 
the clear; in which case , , 
its greatest load at 100 ^ 
lbs per sq ft, would be % H* 
a ton per ft run; or if 24 ** 
ft wide, with but two trus¬ 
ses, the load would be full 
1 ton per ft run. 


But in a com¬ 
mon bridge also, the great¬ 
est load per ft run, on a 
very short span, will be 
greater than in a long 
one; as in the case of two 
wheels of a truck hauling 
a large block of stone, &c ; 
and this must, be taken 
into consideration in 
building such. 


The greatest load that can come upon a bridge. 


If for a single-track rail¬ 
road, can scarcely exceed 
that of a string of heavy 
locomotives coupled to¬ 
gether, without their ten¬ 
ders. Such engines will 
weigh from one to two 
tons per foot of their ex¬ 
treme length but a long 
striug of such, without 
their tenders, is hardly 
probable. 


nr, 


o Go 


t The engineers of the Chelsea bridge, London, packed picked men upon the platform of a weigh¬ 
bridge; with a result of 84 lbs per sq ft. Mr Nash, architect of Buckingham Palace, experimenting 
with referenoe to fire-in-oof floors for that building, wedged men together as closely as they could pos¬ 
sibly stand upon an area of 20 ft diam ; the last man being lowered down from above, among the 
others. Result, 120 lbs per sq ft. See foot note p 623 


















Qi iErN Twmr s 


TRUSSES. 


C07 


It must also be remembered that each transverse floor-girder must bear at feast all 
the weight resting upon two wheels; no matter how close together the girders may 
be placed. If they are farther apart than the dist between two axles of a vehicle, 
they will have to bear more than the load on one pair of wheels. 

The allowance for safety in a truss bridge. 

I the result of a long-continued series of deflections applied to an experimental plate-iron girder 

i of 20 ft span, Mr Kairbairn concludes that a bridge subject to 100 deflections per day, each equal to 
'that produced by % of its extraneous breaking load, would probably break down in about 8 years; 
while, with 100 daily deflections equal to that arising from but x /\ of its breaking load, it would last 
fully 300 years. We are of the opinion that a bridge should not have a safety of less than 4 for its 
max extraneous load, and Its own weight, combined; nor do we see any use in exceeding 6. From 
4 to o may be used in temporary structures, or in those rarely exposed to maximum strains ; and 6 in 
more important ones frequently so exposed. The last will (roughly speaking) generally give a safety 
of about 2 against reaching the elastic strength, which is the true guide in such matters. But 4, 6, 
&c, usually refer to the ultimate or breaking-down strength; so that a truss with such a safety 
of 2 would in fact be very unsafe. 

Oil t he camber of truss bridges. In practice, the upper and 
lower chords of bridges are not made perfectly straight, but are curved slightly up¬ 
ward ; and this curve is called the camber of the truss or bridge. Its object is to 
prevent the truss from bending down below a hor line when heavily loaded. A cam¬ 
bered chord is of course longer than a straight line uniting its ends ; but in practice 
the camber is so small that this diff is inappreciable, and may be entirely neglected. 
But when the chords are cambered, (see y s and c d, Fig 51,) they become concentric 
arcs of two large circles, of which the center is at t; and the upper one plainly be¬ 
comes longer than the lower, to an extent which, although much exaggerated in our 
fig, cannot be overlooked in practice. The verticals, instead of remaining truly vert, 
become portions of radii of the aforesaid large circles; and although their lengths 
remain the same, yet their tops become a little farther apart than their feet; and 
this renders it necessary to lengthen the obliques or diags a trifle. Therefore, we 
must find how great is this increase of length of the upper chord beyond the lower 
one; and divide it equally among all the panels, along said chord; otherwise the 
several parts of the truss will not fit accurately together. 

1st. To find the amount of camber of the lower chord. Di¬ 
vide the span in feet, (measured from center to center of the outer panel-points,) 
by 50. The quot will be a sufficient camber, in inches; as shown in the Allowing 


Table of cambers for bridge trusses. 


Span. 

Feet. 

Camber. 

Ins. 


Span. 

Feet. 

Camber. 

Ins. 


Span. 

Feet. 

Camber. 

Ins. 

25 

0 5 


100 

2.0 


250 

5.0 

50 

1.0 


150 

3.0 


300 

6.0 

75 

1.5 


200 

4.0 


350 

7.0 


Rem. 1. It is by no means necessary to adhere strictly to this rule; and the 
camber by experienced builders of iron bridges is often but one-half the 
above, or 1 inch per 100 ft of span. 

Rem. 2. A well built bridge of good design should not, under its greatest 
load, deflect more than about 1 inch for each 100 feet of its span. The deflec¬ 
tion is frequently much less than this. 

2d. To find the increase of length in the npper chord. 

beyond the lower one, having the span; the depth of truss; and the camber; (all 
in feet, or all in ins.) 

With any camber not exceeding of the span ; (which, however, is about 7 times as great as is 

usually given to trusses;) mult together the depth of truss, the camber, and the number 8; div the 
prod by the spaD. The quot will be the increase, in ft, or in ins, as the case may be. Or as a formula. 

Increase in ft, — X ca mber X 8 M {n j^et; or 

or in ins, span, all in inches. 

This rule may be considered practically perfect with any camber not exceeding 
I' of the span. Based upon this principle, wo have prepared the following table, which 
may be used instead of making the above calculations. 






















608 


TRUSSES. 




Table for Gliding increase of length of upper chord beyond 

lower one. 



Depth 

of 

Truss. 

Mult 

Camber 

by 

Depth 

of 

Truss. 

Mult 

Camber 

by 

Depth 

of 

Truss. 

Mult 

Camber 

by 

Depth 

of 

Truss. 

Mult 

Camber 

by 

y± span. 
1-5 *• 
hi “ 

1-7 “ 

2.00 

1.60 

1.33 

1.15 

% span. 
l-» “ 

1-10 “ 

1-11 “ 

1.00 

.888 

.800 

.727 

1-12 span. 
1-13 *• 

1-14 “ 
1-15 “ 

.666 

.614 

.571 

.533 

I- 16 span. 

1 17 “ 

1-18 “ 

II- 20 “ 

.500 

.470 

.444 

.400 


can:" 

twfc 

As 

m ■ 

or l« 
es 

h 

bt.:- 

Mi 

tr: ' 
to 


Ex. How much longer is the upper chord than the lower one, when the depth ol 
the truss is w of the span; and the camber 5 ins? Here in the table, and oppositej 
span, we find the multiplier L15. Therefore, 5 ins X 1.15 = 5-75 ins, Ans. If tin 

truss has say 8 panels, then ^4- = -72 inch of this increase must be given to eact 

panel, along the upper chord. 


8 


The length of a diag, or oblique, b c, Fig 50, may readily be fount 

thus: Let as«c in this fig represent a panel when there 
is no camber; theu ob n c will represent a panel when 
there is a camber; and o a and s b together are the 
portion of the increased length of upper chord given to 
each panel; but to an exaggerated scale. Now, to find 
b c, we have the right-angled triangle a b c, in which we 
know the side a c, (the depth of truss;) and the side a b, 

(equal to the panel width cn on the lower chord; added 
to s b, or half the portion of the increased length of 
upper chord given to one panel.) Hence, we have only 
to square each of those two sides; add the two squares 
together; and take the sq rt of the sum. This sq rt is b c. 

Example. Span 200 ft. Height (a c) of truss % of the 
span, or 25 ft, or 300 ins. Camber 5 ins; 10 panels each 
20 ft, or 210 ins, (c «,) measured on the lower chord or 
span. Now. the height being % of the span, the increase 
of length of upper chord will be equal to the camber, 5 ins; and this divided among 
5 



10 panels, will be — = .5 inch to each panel; or o b will be .5 inch longer than cn 


and o a and s b will each be .25 inch. Hence, in the right-angled triangle a b c, we 
have a b — 240.25 ins; and a c = 300 ins. Hence, 


be = v /a6 3 -+-ac a = v / 57720.0625 + 90000 = y/ 147720.0625 = 384.344 ins; 
or 32 ft, .344 ins. Without any camber, b c would be in the position s c, which is 32 


ft, .187 ins long; or .157 of an inch (about % inch) shorter than 6 c. 


An error to this extent would prove seriously inconvenient if the oblique were a cast-iron stru 
with carefully planed ends, inteuded to fit closely between planed bearings at the chords ; or a bai 
with a drilled hole at each end for fitting over pins whose position was fixed and unalterable. In 
many cases, as when the obliques are merely rods with screw-ends, it is only necessary to be sure 
that they are long enough; because their exact length can then be adjusted when put iuto place, by 
means of the nuts on their ends. So also when the obliques or other pieces are flat bars inteuded to lie 
bolted or riveted to the sides of the chords; for the final rivet-holes may be made when the pieces 
come to be finally fitted in place. In the case of wooden obliques, &c, if too long, they can readily be 
reduced by the saw or chisel. I 


When the panels are all of one size, as is generally the case, it is usual for builders 
to draw one of them full size on a board platform or floor, to guide in fitting the 
parts together. 

In raiNing- a truss, or in other words, when putting its parts together in 
their proper position on the abutments and piers, a scaftold or false-works, 
must first be erected for sustaining the parts until they are joined together so as to; 
form the complete self-sustaining truss. Upon the false works the bottom chords; 
are first laid as nearly level as may be : and the top chords are then raised upon tem¬ 
porary supports which foot upon the one that carries the lower chord. The upper! 
chords are at first placed a few inches higher than their final position, or than the 
true height of the truss, in order that the obliques and verts may be readily slipped 
into place. After this is done, the top chords are gradually let down until all rests 
upon the lower chords. The screws are then gradually tightened to bring all the 
surfaces of the joints into their proper contact; and by this operation (the upper 
chord being supposed to have the increased length given by the foregoing rule) the 





































TRUSSES. 


609 


camber, as it were, forms itself; and lifts the lower chords clear off from their false¬ 
works ; leaving the truss resting only upon the abuts or piers, as the case may be. 

As a support for the falseworks themselves on soft bottoms piles 
may be driven, to which the uprights of the falseworks may be notched and bolted 
or banded. In some cases, as of rock bottom in a strong current, it may become 
expedient to sink cribs filled with stone, as a support for the falseworks. 

The falseworks should be well protected by fender-piles or otherwise from passing 
boats, ice and other floating bodies, especially in positions liable to sudden floods; 
and numerous accidents have shown the expediency of guarding the unfinished 
truss itself against higrli winds. This last remark applies as well to roofs as to 
bridges ; and is too frequently neglected. 


I To prevent an overturning: tendency in a whole truss when it is 
not high enough to admit of being horizontally braced overhead, we may introduce 
wooden knees, or short straight struts or ties, of either wood or iron ; which may 
foot upon the cross-girders of the floor; and head against either some of the web 
members, or the upper chord. These braces or ties may be placed either between 
the two trusses of a span ; or outside of them; or both. When outside, some of the 
J floor-girders may be lengthened out a few feet beyond the lower chord, for receiving 
the feet of the braces or ties. 

The clear distance apart of the trusses in railroad bridges for 4 ft 8% 
inch gauge, is generally made not less than 14 or 15 ft, and 26 or 28 ft, respectively, 
for single and double track, in through bridges; and 11 or 12 ft, and 16 ft, respec¬ 
tively, in deck bridges. For lateral stability, in long: spans, the distance be- 
j tween centers of trusses is generally made not less than one-twentietli of the span. 
A headway of 18 to 20 ft should be allowed for clearing smoke-stacks, &c. 








610 


TRUSSES 


Floor-girders not exceeding 14 ft clear span, may lie 8 ins, by 15 ins deep; 
and placed not more than about 2% ft apart from center to center. Upon them 
should be notched and spiked stout string-pieces, say 12" wide, by 9" deep, to carry 
the rails; and to distribute the pressure of the load. 

Ilor <liag bracing for diminishing lateral motion, can be used only under 
the floors of low bridges; but in high ones it is introduced also at the top of the 
trusses. When of timber, these braces are about 4 to 6 inches thick ; by 6 to 9 deep; 
and form a hor cross between each two opposite panels of the two trusses. If the 
bridge is roofed, and has girders r, Figs 36, 56%, upon and well secured to the upper 
chords c c, the upper lateral bracing may consist simply of 4 iron rods n n, passing 
through the chords about midway of their depth ; and having heads and washers on 
their outer sides. At the center of the cross the rods terminate in an adjusting-ring; 
see No 14, of page 583. In a bridge of 150 ft span, these rods need not exceed 
inch diam at the center panel, and 1% at the end ones. If the bridge is high, and 
not roofed, but open at top, then cross-struts r r, Figs 66%, must be inserted pur¬ 
posely, when this rod-bracing is used. If it is also used at the lower chords, the floor- 
girders perform the duty of these struts. Iron-bracing is not liable to catch tire 
from the locomotives. 

A favorite mode of lateral bracing. W, resembles a Howe 

truss laid flat on its side. In it the diags of the cross are struts of timber; and the 
pieces r r are round rods. One of the struts is whole, with the exception of a slight 
mortice on each vert side, at its center, for receiving tenons cut on the inner ends of 
the two pieces which compose the other ding. At the sides of the chords, the ends 
of the diags rest upon a ledge, (shown by 
the dotted line i i,) about 1% ins wide, cast 
at the bottom of the cast-iron angle-block. 

The tie-rod rr, passing through the chords 
of both trusses, being tightened by means 
of the nut s, holds the diags firmly in place; 
and in case of their shrinking a little in 
time, can be again tightened up by the same 
means. 

Various modifications of these methods 
are in use; but we cannot afford them space 
here. The cast angle-block is as deep as a 

brace; its thickness need not exceed % inch, in a large bridge. The dark triangle is 
a top view of it. It has holes for the passage of the rod r r. 



Art. 25. Longtlioning-Kcarfs, splices, or joints. The lower chords 

of bridges, being exposed to great pulling strains, require much care in connecting 
together the ends of the several pieces of which they are composed. There is much 
uncertainty regarding the strength of the joint-fastenings in common use for this 
purpose. Experiments on the subject are much needed. When only two pieces, as 
t and y, Fig 57, or 58, are joined by any of the ordinary methods, it. is probably not 
safe to depend on their possessing more than % of the tensile strength of a single 
solid beam of equal cross-section. When the chord, as in Fig 59, is composed of two 
parallel parts a a, w n, made up of long pieces, breaking joint with each other, as at 
j jj, each of the two parts may be made somewhat stronger than either one of 
them would be by itself. This is owing to the opportunity afforded of connecting 

































TRUSSES. 


611 


them also by bolts bb, and packing-blocks, cc, of wood or iron, intermediate of the 
joints jj, &c. By this means the strength of the entire chord may probably be prac¬ 
tically rendered equal to one-half of what it would be if solid. If the chord con¬ 
sists of 3 or 4 parallel parts, of long pieces, breaking joint, and connected in the 
same way, it will probably have about % of the strength of the solid. Care must 
of course be taken that the serviceable area of the pieces shall not be reduced at any 
intermediate point, to less than it is at the joints. 



Fig. 57. SIDE 
a Fig. 59. J 


TOP 


SIDE Fig . 58 . 

a 



Tj b 1) 1 

top e 



SIDE 

Fig. 60. 


Fig. 60 a. 



Fig 58 is a simple and efficient form of scarf. Its length i i may be about 3 to 4 
times the greatest transverse dimension of the beam. At the center is a block t of 
hard wood, with a thickness equal to ^ that of the beam ; a width of 2 or 3 times its 
thickness; and a length just sufficient to reach entirely through the beam. The 
beams are connected by 4 screw-bolts nn; or by 8 of them, if the length requires it. 
Plates of stout rolled iron, aa,cc, with their ends bent down into the beams, are 
occasionally added. These require bolts o 0 , beyond the ends i i of the scarf. These 
bolts are not shown in the side view. 

Fig 57 is another excellent joint with splicivg-hlncks e e, instead of the block t of 
Fig 58. The indentations, v v, may each be about % as deep as the beam is thick. 
The length of each splice-block, about 6 times s From 4 to 8 screw-bolts, as the 
case may require. Length of each indent about % that of the block itself. 

Fig 60 is a joint fonned by two flat iron links or rings, 11, let flush into the tim¬ 
bers, and retained in place by spikes. The iron may vary from ^ to 1 inch in thick¬ 
ness ; from 1 to 4 or 5 ins in width; and 2 to 6 ft in length, as occasion may require. 

Fig 60 5, is a joint formed by two blocks, c c, of hard wood, passing through the 
timbers; and connected by bolts, a a, n v. 

In Fig60 a, s s are cast-iron packing-blocks, sometimes used instead of plain wooden 
ones, at points bb, Fig 59, intermediate of the joints jj. The openings in the centers 
of the blocks are needed only when vertical truss-rods have to pass through those 
points. At e, e, of the same fig, is shown another form, much used in chords composed 
of two or more parallel strings. Both these are as deep as the chord; and their 
cross-sections, or end views shown in the fig, may be from 4 to 10 ins long; 2 to 4 ins 
wide; and from to V/ 2 ins thick; according to size of bridge, <fec. 

Rem. In selecting hard wood for splicing-blocks, treenails, or for any part of a 
bridge, it is well to remember that the oaks when in contact with the pines, expedite 
the decay of the latter; therefore, it is generally better to employ the best southern 
yellow pine heart wood for such blocks, &c, or interpose sheet iron.* 

* The tendency of some kinds of timber to produce rapid decay when brought into close contact 





























































612 


TRUSSES. 


Eyc-Rnrs and Pins. The lower chords or iron bridges usually 

consist of flat links or bars c and o, W and H, Figs 61 on edge aud connected by tight-fitting wrought- 
irou pius 6 and P. After deciding on the size of the body W or H of the bars to bear safely the pull 
upon them, the proper proportioning of their heads or eyes and pius is an abstruse and difficult poiut 
upon which much has been written. It was formerly supposed that the diam of the pin should be 
governed by its resistance to shearing, but experience has shown that this was entirely insufficient. 



We give a table of practical conclusions arrived at by that accomplished expert, Chs. Shaler Smith, 
from ill experiments by himself on a working scale. The table shows some irregularities, for as Mr. 
Smith remarks “ the bars declined to break by formula.” The pin is more strained at the outer links 
o o than at the inner ones c c c, so that the latter would not require so large a diam. but that this 
must be uniform throughout in order to secure tight fitting for all of them. When web members as 
well as chords are held by the same pin the diam and head must be proportioned for that bar of them 
all which is most strained. When the heads are made by pressure iu one piece with the body, the 
metal vs and u x at the sides of the pin b must be wider than when the heads are first made iu sepa¬ 
rate pieces by hammering and then welded to the body. But the welded one W requires more iron 
back of the pin as shown at 11. This width 11 must be equal to the diam of the pin.* 

The links are supposed to be of uniform thickness. 

Having drawn a circle b for the pin, lay off on each side of it as at v s, v r, half 
the width of metal in the table for the head of W or H as the case may be. 
Then for forming the head of H use only the rad b s as shown. For the head of 
W lay off also 11 = diam of pin. Find by trial the rad g n or g t and use it, except 
for uniting the head to the body, where use a rad = 1.5 g n as shown. 


Width 
of bar. 

Thicks, 
of bar. 

Diam. 
of pin. 

Metal i 
acros. 

W. 

n head 
pin. 

H. 


Width 
of bar. 

Thicks, 
of bar. 

Diam. 
of pin. 

Metal i 
across 

W. 

n head 
pin. 

H. 

1 . 

.2 

.67 

1.33 

1.50 


1 . 

.55 

1.28 

1.50 

1.60 

1 . 

.25 

.77 

1.33 

1.50 


1 . 

.60 

1.36 

1.55 

1.72 

1 . 

.30 

.86 

1.40 

1.50 


1 . 

.65 

1.43 

1.60 

1.76 

1 . 

.35 

.95 

1.50 

1.50 


1 . 

.70 

1.50 

1.67 

1.85 

1 . 

.40 

1.04 

1.50 

1.50 


1 . 

.80 

1.64 

1.67 

1 95 

1 . 

.45 

1.12 

1.50 

1.53 


1 . 

.90 

177 

1 70 

2.05 

1 . 

.50 

1.20 

1.50 

1.56 


1 . 

1.00 

1.90 

1.76 

2.21 


Art. 26. Figs 62 exhibit joints adapted to most of the cases that 
occur in practice with wooden beams, Ac. They need but little explanation. Fig a 
is a good mode of splicing a post; in doing which the line n n should never be in¬ 
clined or sloped, but be made vert; otherwise, in case of shrinkage, or of great 
pressure, the parts on each side of it tend to slide along each other, and thus bring 
a great strain upon the bolts. "When greater strength is reqd, iron hoops may be 
used, as at b, h, and./, instead of bolts. Fig b, a post spliced by 4 fishing pieces: 
which may be fastened either by bolts, as in the upper part; or by hoops, as in the 
lower. The hoops may be tightened by flanges and screws, as at* *; or thin iron' 
wedges may be driven between them and the timbers, if necessary. Fig C shows a 
good strong arrangement for uniting a straining-beam A*, a rafter l, and a queen-post | 
u ; by letting k and 1 abut against each other, and confining them between a double 
queen-post 11 ; n n are. two blocks through which the bolts pass. A similar arrange¬ 
ment is equally good for uniting the tie-beam w , with the foot t>, of the queens; with 
the addition of a strap, as in the fig. Fig e is a method of framing one beam into 
another, at right angles to it. An Iron stirrup, as at /, may be used for the 
same purpose; and is stronger. Figs g A, i j are built beams*. When a beam 
or girder of great depth is required, if we obtain it by merely laying one beam flat 

- — ■ ‘ 1 
with other kinds, is a subject of great practical importance; but one which hitherto has received but 
little attention. Black walnut and cvpress are said to cause mutual rot within a year or two. Ots 
served oases of this kind should be reported to the leading professional journals. 

* The strength of a given hinged-end pillar (see p 439) is increased to an important extent by en¬ 
larging the diameter of the pin. 

















































TRUSSES. 


613 


5 

j| upon another, we secure only as much strength as the two beams would have if 
nt separate. But if we prevent them from sliding on one another, by inserting trans- 
11 verse blocks or keys, as at g\ or by indenting them into one another, as att j ; and 




































































































































































































614 


TRUSSES. 


then bolt or strap them firmly together to create friction ;we obtain nearly the strength 
of a solid beam of the total depth ; which strength is as the square of the depth. 

The strength of a built beam is increased by increasing its depth at its center, where it is most 
strained ; as in the upper chords of a bridge. This may be done by adding the triangular strip y y 
between the two beams. 

Tredgold directs that the combined thicknesses of all the keys be not less 

than 1.4 times the entire depth of the girder; or when indents are used, as in ij. that their combined 
depths be at least % that of the girder. If the girder ij be inverted, it will lose much of its strength. 

A piece of plate-iron may be placed at the joints of timbers when there is a great pressure ; which 
is thus more equalized over the entire area of the joint; or cast iron may be used. 

Frequently a simple strap will not suffice, when it is necessary to draw the two timbers very 
tightly together. In such cases, one end of each strap may, as at x, terminate as a screw ; and after 
passing through a cross-bar Z, all may be tightened up by a nut at x. Or the principle of the dou¬ 
ble key. shown at K, may be applied. Sometimes, as at A, the hole for the bolt is first bored ; then 
a hole is cut in one side of the timber, and reaching to the bolt hole, large enough to allow the screw 
nut to be inserted. This being done, the hole is refilled by a wooden plug, which holds the nut in 
place. Then the screw-bolt is inserted, passing through the nut. By turning the screw the timbers 
may then be tightened together. 

When the ends of beams, joists, &c, are inserted into walls in the usual square manner, there is 
danger that in case of being burnt in two, they may, in falling, overturn the wall. This may be 
avoided by cuttiug the ends into the shape shown at to. 

Wheu a strap o. Fig R, has to bear a strain so great as to endanger its crushing the timber p, on 
which it rests, a casting like v may be used under it. The strap will pass around the back r of the 
casting. The small projections in the bottom being notched into the timber, will prevent the casting 
from sliding under the oblique strain of the strap. The same may be used for oblique bolts, and 
below a timber as well as above it. When below, it may become necessary to bolt or spike the casting 
to the under side of the timber. Wheu the pull on a strap is at right angles to the timber, if then 
is much strain, a piece of plate-iron, instead of a casting, may be inserted between the strap and tht 
timber, to prevent the latter from being crushed or crippled ; see I and I. 




Art. 27. Expansion rollers, or planed iron Slides, or rock¬ 
ers. or snspension-links. must be provided wheu an iron span exceeds 
about 80 feet; in order to allow the trusses to contract and expand freely 
under changes of temperature, without undue strain upon some of its members. 
Figure 63 shows the general arrangement of roll¬ 
ers; which are cylinders of cast iron or steel, from 3 
to 6 ins diam; and 1 to 4 ft long; planed smooth. From 
4 to 8 or more of these are connected together by a 
kind of framing, n n ; and one such frame is placed 
under at least one end of the truss. The rollers rest 
upon a strong planed cast bed-plate no; bolted to the 
masonry below. Under the end of the truss is a sim¬ 
ilar plate s s, by which it rests on the rollers. Since 
a truss of even 200 ft span will scarcely change its 
length as much as 3 ins by extremes of temperature, 
the play of the rollers is but small. They are kept in 
line by flanges cast along the side of the bed-plate. Flanges should also project 
downward from s s, so as completely to protect 
the rollers from dust, rain, <fcc. 


In Fig 64, r r gives a general idea of a kocker ; and Fig 65. 
* s, of a suspension-link. U U in each tig is aside view of 
a cast-iron Fink upper chord, through each end of which 
passes a round pin o, which sustains the entire weight of the 
truss and its load; and which is sustained by from 4 to 6 
rockers or links, as the case may be. In a railroad bridge of 
205 ft span, across the MonoDgahela, the links are 3% ft long; 
and the pins 5 ins diam; and in others of the same size, over 
Barren and other rivers, the rockers are a foot wide from r 
to r; and about 5 ins wide transversely on the curved tread 
or rim. For the accommodation of these several links and 
rockers; as well as of the various bars ft ft, which constitute 
the oblique ties of a Fink truss; the ends of the octagonal 
cast-iron upper chords are widened out, as shown by Fig 66; 
which is a top view of a longitudinal section of such an end. 
The rockers, or links, and bars, ft, occupy the spaces w n, Ac, 
between the several partitions of the chord ; and the pin o o 
passes through them all, except when it is expedient to 
attach some of the bars to the sides, or to the top of the 
chord, as at t. These figs nre intended merely to illustrate 
the general principle, without regard to detail of construction. 

In some English bridges of considerable size, such as the 
Crumliu Viaduct, of 150 ft spans; and the Newark Dyke 
bridge, of 240 span; (both bf them Warren girders,) and 
sustained like the foregoing, by the ends of the upper 
chords ; no further precaution is taken with regard to expan¬ 
sion and contraction, than merely to rest the ends of said 
chords upon smoothly planed iron plates, upon which they 
may slide. So also several American bridges. 

































SUSPENSION BRIDGES 


615 


SUSPENSION BRIDGES. 


Art. 1 
chains 


• Table of data required for calculating the main 
or cables of suspension bridges. Original. 


Deflection 
in parts 
of the 
Chord. 

, Deflection 
in Deci¬ 
mals of 
the Chord. 

Length of 
Main Chains 
between Sus- 
J pension Piers, 
in parts of the 
Chord. 

Tension on all 
the Main 
Chains at 
either Suspen¬ 
sion Pier, in 
parts of the 
entire Sus¬ 
pended Wt. 
of the Bridge, 
and its Load. 

Tension at the 
Center of all 
the Main 
Chains; in 
parts of the 
entire Sus¬ 
pended Wt. 
of the Bridge, 
and its Load. 

Angle of 
Direc¬ 
tion of 
theChains 
at the 
Piers. 

Natural 
Sineof tl>e 
A ugle of 
Direction 
of the 
Chains, at 
the Piers. 

Natural 
Cosine of 
the Angl« 
of Direc¬ 
tion of the 
Chains at 
the Piers. 

1-40 

.025 

1.002 

5.03 

5.00 

Dog. Min. 
5 43 

.0995 

.9950 

1-35 

.0286 

1.002 

4.40 

4.37 

6 31 

.1135 

.9935 

1-30 

.0333 

1.003 

3.78 

3.75 

7 36 

.1322 

.9912 

1-25 

.04 

1.004 

3.16 

3.12 

9 6 

.1580 

.9874 

1-20 

.05 

1.006 

2.55 

2.51 

11 19 

.1961 

.9806 

1-19 

.0526 

1.007 

2.43 

2.38 

11 53 

.2060 

.9786 

1-18 

.0555 

1.008 

2.30 

2.25 

12 32 

.2169 

.9762 

1-17 

.0588 

1.009 

2.18 

2.12 

13 14 

.2290 

.9734 

1-16 

.0625 

1.010 

2.06 

2.00 

14 2 

•24*25 

.9701 

1-15 

.0667 

1.012 

1.94 

1.87 

14 55 

.2573 

.9633 

1-14 

.0714 

1.013 

1.82 

1.74 

15 57 

.2747 

.9615 

1-13 

.0769 

1.016 

1.70 

1.62 

17 6 

.2941 

.9558 

1-12 

.0833 

1.018 

1.57 

1.49 

18 33 

.3180 

.94,80 

in 

.0919 

1.022 

1.46 

1.37 

19 59 

.3418 

.9398 

1-10 

.1 

1.026 

1.35 

1.25 

21 48 

.3714 

.9285 

1-9 

.1111 

1.033 

1.23 

1.12 

23 58 

.4062 

.9138 


.125 

1.041 

1.12 

1.00 

26 33 

.4471 

.8945 

1-7 

.1429 

1.053 

1.01 

.881 

29 45 

.4961 

.8726 

3-20 

.15 

1.058 

.972 

.833 

30 58 

.5145 

.8574 


.1667 

1.070 

.901 

.750 

33 41 

.5547 

.8320 

1-5 

.2 

1.098 

.800 

.625 

38 40 

.6247 

.7803 

H 

•225 

1.122 

.747 

.555 

42 0 

.6690 

.7433 

.25 

1.149 

.707 

.500 

45 00 

.7071 

.7071 

.3 

.3 

1.205 

.651 

.417 

50 12 

.7682 

.6401 

Vi 

.3333 

1.247 

.625 

.375 

53 8 

.8000 

.6000 

.4 

.4 

1.332 

.589 

.312 

58 2 

.8483 

.5294 

9-20 

.45 

1.403 

.572 

.278 

60 57 

.8742 

.4855 

34 

.5 | 

1.480 

.559 

.250 

63 26 

.8944 

.4472 


These calculations are based on the assumption that the curve formed by the main chains is a 
parabola; which is not strictly correct. In a finished bridge, the curve is between a parabola and a 
catenary ; and is not susceptible of a rigorous determination. It may save SOI1IC trou¬ 
ble ill making the drawings of a suspension bridge, to remember that when the 
deflection does not exceed about y 1 ^ of the span, a segment of a circle may be used instead of the 
true curve; inasmuch as the two then coincide very closely ; and the more so as the deflection be¬ 
comes less than j-j. The dimensions taken from the drawing of a segment will answer all the pur¬ 
poses of estimating the quantities of materials. 

For some particulars respecting wire for cables, see pages 412 and 413. 

The deflection usually adopted by engineers for great spans is 

about to y*y the span. As much as yL. is generally confined to small spans. The bridge will 
be strouger, or will require less area of cable, if the defleetion is greater; but it then undulates more 
readily ; and as undulations tend to destroy the bridge by loosening the joints, and by increasing the 
momentum, they must be specially guarded against as much as possible. The usual mode of doing 
this is by trussing the hand-railing; which with this view may be made higher, and of stouter tim¬ 
bers than would otherwise be necessary. In large spans, indeed, it may be supplanted by regular 
bridge-trusses, sufficiently high to be braced together overhead, as in the Niagara Railroad bridge, 
where the trusses are 18 ft high ; supporting a single-track railroad on top; and a common roadway 
of 19 ft clear width, below.* 


* The writer believes himself to have been the first person to suggest the addition of very deep 
trusses braced together transversely, for large suspension bridges. Early in 1851. he designed such 
a bridge, with four spans of 1000 ft each : and two of 500; with wire cables; and trusses 20 ft high. 
It was intended for crossing the Delaware at Market Street, Philada. It was publicly exhibited for 
several months at the Franklin rnRtitute. and at the Merchants’ Exchange; and was finally stolen 
from the hall of the latter. Mr Roebling's Niagara bridge, of 800 ft span, with trasses 18 ft high, was 
not commenced until the latter part of 1852; or about 18 months after mine bad been publicly ex¬ 
hibited. 













































616 


SUSPENSION BRIDGES. 


Another very important aid is found in deep longitudinal floor timbers, firmly united where their 
ends meet each other. These assist by distributing among several suspeuder-rods, and by that 
means along a considerable length of maiu cable, the weight of heavy passing loads; and thus pre¬ 
vent the undue undulation that would take place if the load were concentrated upon only two opposite 
suspenders. With this view, the wooden stringers under the rails on the Niagara bridge are made 
virtually 4 ft deep. The same principle is evidently good for ordinary trussed bridges. 

Another mode of relieving the main cables is by means of cable-stays; which are bars of iron, or 
wire ropes, extending like c y, Fig 1, from the saddles at the points of suspension c, d, obliquely down 
to the floor, or to some part of the truss. In the Niagara bridge are 64 such stays, of wire ropes of 
inch diam; the longest of which reach more than quarter way across the span from each tower. 
They transfer much of the strain of the wt of the bridge and its load directly to the saddles at the top 
of the towers: thereby relieving every part of the main cable, and diminishing undulation. They 
end at cand d, where they are attached, not to the cables, but to the saddles. They of course do not 
relieve the back stays. 

The greatest danger arises from tlie action of strong: winds 
striking: below' the iloor, and either lifting the whole platform, and letting 
it fall suddenly ; or imparting to it violent wavelike undulations. The bridge of 1010 ft span across 
the Ohio at Wheeling, by Charles Ellet, Jr, was destroyed in this manner. It is said to have undu¬ 
lated 20 ft vertically before giving way. It had do effective guards against undulation ; for although 
its hand-railing was trussed, it was too low and slight to be of much service in so great a span. 
Many other bridges have been either destroyed or injured in the same way. When the height of the 
roadway above the water admits of it, the precaution may be adopted of tie-rods, or anchor rods, 
under the floor at different points along the 3pan, and carried from thence, inclining downward, to 
the abutments, to which they should be very strongly confined. In the Niagara Railroad bridge 56 
such ties, made of wire ropes 1 M inch diam, extend diagonally from the bottom of the bridge, to the 
rocks below. They, however, detract greatly from the dignity of a structure. 

Mr Brunei, in some cases, for checking undulations from violent winds striking beneath the plat¬ 
form, used also inverted or up-curving cables under the floor. Their ends were strongly confined to 
the abuts several ft below the platform; and the cables were connected at intervals, with the plat¬ 
form, so as to hold it down. 

Art. 2. The angle adg, or a c i, Fig 1, which a tang dg or ci to the curve at 
either point of suspension c or d, forms with the hor line cd or chord, is called the HIlgle of 
direction Of the main chtlillS, or cables, at those points. Frequently the ends 
c h, and dr, of the chains, called the backstays. are carried away from the suspension piers 
in straight lines; in which case the angles l dr, e ch, formed between the hor line e l and the chain 
itself, become the angles of direction of the backstays. 



Silie Of angle Of direction (id {t — _ Twice the deflection a b _ 

j/ (twice the deflection-(- (Half the chord)2 

Notb 1. The direction of the tang dg or ci, can be laid down on a drawing, thus: Continue the 
line a b. making it twice as long as ah; then lines drawn from d and c to its lower end, will be tanas 
to the parabolic curve at the points of suspension. 


Note 2. It the chord c d be not hor, as sometimes is the case, the angle 
must he measured from a hor line drawn Tor the purpose at each point of suspension; as the two 
angles will in that case be unequal, the piers being of unequal heights. 

Tension on all the main 


chains or cables, together, 
at either one of the piers, 
e or d f Fig 1. 


Half the entire suspended weight of the clear 
__span and its load 

Sine of angle of direction adg 


or 


lMM S pan) 2 + ( 2 Defl) 2 
2 Deflection 


X 


Half the entire sus¬ 
pended weight of 
the clear span and 
its load. 


Tension on all the main 
chains or cables, toget her, 
at the middle, h f of the = 
span. Fig 1. 


Half the entire suspended 
weight of the clear span 
and its load 


w Cosine of angle of 
direction adg 


Sine of aDgle of direction adg 


Half the entire suspended weight of v Half the 
or _ the clear span and its load x span 

Twice the deflection 

The diff between the tensions at the middle, and at the points of suspension, is so trifling with the 
proportion of chord and deflection commonly adopted In practice, viz, from about Jn to JL that it 
is usually neglected; inasmuch as the saving in the weight of metal would be fully compensated for 
by »hc increased labor of manufacture in gradually reducing the dimensions of the chains from the 
points of suspension toward the middle: and in preparing fittings for parts of many different sizes 
The red notion has, however, been made in some large bridges with wrought-iron main chains - but 
in none with wire cables. ' 


























*S4 -;- t -HF* *1T=Ss 1 i-ETf 


SUSPENSION BRIDGES 


617 


Art. 2A. As it is sometimes convenient to form a rough idea at the moment, of 
the size of cables required for a bridge, we suggest the following rule Tor finding approximately the 
area in sq ins of solid iron iu the wire required to sustain, with a safety of 3,# the weight of the bridge 
itself, together with an extraneous load of 1.205 tons per foot run of span ; which corresponds to 100 
lbs per sq ft of platform of 27 ft clear available width. This suffices for a double carriage-way. and 
two footways. The deflection is assumed at yyg- of the span ; and the wire to have an ultimate 
strength of 36 tons per solid square inch, as per table, page 412; and which can be procured without 

difficulty. For spans of lOO ft or more, 

Rulk. Mult the span in feet, by the square root of the span. Divide the prod by 100. To the 
quot add the sq rt of the span. Or, as a formula, 

Area of solid metal of all span X sq rt of span 

the cables ; in square ins ; —- -j- sq rt of span. 

for spans over 100 feet 100 

For spans less than 100 feet, proportion the area to that at 100 ft. 

If a defl of y-g- is adopted instead of yy, the area of the cables may be reduced very nearly y part- 

The following- table is drawn up from this rule. The 3d col 

gives the united areas of all the actual wire cables, when made up, including voids. (Original.) 


Spao 

in 

Feet. 

Solid Iron 
in all the 
Cables. 

Areas of 
all the 
Finished 
Cables. 

Span 

in 

Feet. 

Solid Iron 
in all the 
Cables. 

Areas of 
all the 
Finished 
Cables. 

Span 

in 

Feet. 

Solid Iron 
in all the 
Cables. 

Areas of 
all the 
Finished 
Cables. 


Sq. Ins. 

Sq. Ins. 


Sq. Ins. 

Sq. Ins. 


Sq. Ins. 

Sq. Ins. 

1000 

348 

446 

400 

100 

128 

150 

30.6 

39.2 

900 

300 

385 

350 

84 

108 

125 

25.2 

32.3 

800 

254 

326 

300 

69 

89 

100 

20 

25.6 

700 

212 

272 

250 

55 

71 

75 

15 

19.2 

600 

171 

219 

200 

42 

54 

50 

10 

12.8 

500 

134 

172 

175 

36.4 

46.7 

25 

5 

6.4 


Having the areas of all the actual cables, we can readily find their diam. Thus, suppose with i 

172 

Bpan of 500 ft, we intend to use four cables. Then the area of each of them will be — = 43 sq ins; 

4 

and from the table of ciroles, we see that the corresponding diam is full 1% ins. 

The above areas are supposed to allow for the increased wt of a depth of truss, and other additions 
necessary to secure the bridge from violent winds, and from undue vibrations from passing loads. 

When these considerations are neglected, and a less maximum load assumed, the following descrip¬ 
tions of the Wheeling and Freyburg bridges show what reductions are practicable. Weight, sutfl- 
siently provided for, is of great servioe in reducing undulation. 


'Of 

I 


it 


ie 

il 

or 

se 


■ ■ ■ ■ " ' ~ ' 

* The writer must not be understood to advocate a safety of 3 against 100 lbs per sq ft, in addition 
o the weight of the bridge, in all cases. He believes that limit to be about a sufficient one for a pro- 
>erly designed wire suspension bridge for ordinary travel ; but for an important railroad bridge, he 
vould (according to position, exposure, Ac) adopt a safety of at least from 4 to 6 against the greatest 
mssible load, added to the wt of the bridge. A train of cars opposes a great surface to the action of 
iide winds: and trains must rnn during violent storms, as well as during calms ; but a large open 
>ridge for common travel is not likely to be densely crowded with people during a severe storm. 





























618 


SUSPENSION BRrDOES 


nnd 


o Tension on the hack-stays, e /t an«I fir* 
»,r:!n*«n il..- ,.!•», or ,. Ta jr^Uljr,. II^ 

*S.T.V?£&X »*«* »• - '■“ 

upon, the tops of the piers. 


At 


fit 


kit 



A 


e- r<- 
a- .■ 


r 

Pci 


Art. 4. In Fijrs ,, 

2, ft and 4, the piers hi 

dnm are supposed to be im- rtt 
movable; and the cables lilt 
k d u, passing over them, 
rest immediately upon hori¬ 
zontal rollers, which have no 
other motion than that of re 
volving about their horizon 
tal axes ; the frame to which aik 
they are attached being bolt ^ 
ea to the top of the pier. On 
these rollers the cables slide, 
when changes of loading 
or or temperature produce K 1 
changes in their directions. 
fn this case the tCII 

sion on the back 

stays is equal to that on 
the main cable. See Funic < 
ular Machine. 

To find the direction audjjj 
amount of the |» r CSS nrC j(|| 

on the pier; from d.u 

Fig 2, 3 or 4, lay off <is aud nti 
d r, each equal, by scale, tof«. 
the tension, in tous, on the on 
main chain at d ; and from s ti 
and r lay it off to v. InotheijlH 
words, draw the parallelo- tl 


Ml 


ISC 

in: 


direction and amount of tht^i 
pressure upon the pier. lot 

When, as in Fig 2, the,,,,, 
angles a d g and i d it are bn 
equal, the pressure d v will , 
be vertical, and equal to thei 
entire weight of the clear 0 
span and its load. .. 

When, as in Figs 3 and 4,'* 
the angles a d g aud l d u are l 
unequal, the pressure d v | 
will uot be vertical, but willL 
incline from d toward the 
smaller augle. 

When, as in Fig 3 , a d g exceeds l d u, the pressure d v will be less than the entire weight of the 


f 


clear span and its load. 

When, as in Fig 4, l d u exceeds a d g, the pressure d v will be greater than the entire weight of the 
clear span and its load. 

If we suppose symmetrical piers, d n m, to be used in each case, tbe base to n of that in Fig 2, may 
be much uarrower than in the other two figs ; because, the direction of dv being vertical, the pressure 
has no tendency to overturn the pier. In Fig 2, the masonry of the pier should be laid in the usual , 
horizontal courses, in order that its bed joints may be at right angles to the pressure upon them. - 1 

But, in Figs 3 and 4, if the bases were made as narrow as in Fig 2, the lines of direction do, of the 
pressure, would fall outside of them ; and the piers would consequently be in danger of overturning. 
Also, the stones of the masonry, if laid in horizoutal courses, would have a tendency to slide on each 
other. To prevent this, the beds should be at right angles to dv. 

In Fig 3. the obliquity of the pressure would tend to slide the base of the pier outward as shown 
by the arrow ; but in Fig 4, inward. This tendency is produced by the horizontal component of the 
force d o. The amount of this may be found thus, in either fig: From d downward draw a vert line 
as in Fig 4; and from v a hor one. meeting it in z, then v z. measured by the same scale of tons as 
before, will give this horizontal force, aud d z will give the vertical component of the pressure d v. 
The effect upon the pier, of the one pressure dv is precisely the same as would be produced upon it by 
one vertical force equal to dz and a horizontal one equal to vz acting at the same time, as explained 
under Composition and Resolution of Forces. 

If, in either fig, we draw the vert lines s p and ro, see Fig 4, thend o, measd by the foregoing scale, 
will give the tons of horizontal pull, and r o the vertical pressure, produced on the pier by the back 
stay ; and p d and ps will, in like manner, give the corresponding forces produced by the main chain. 
If we add together ro and p s they will be found to be equal to d z. and if we subtract d o from p d 
their difference will equal v z. It is this difference only that tends to slide, or to upset, the pier ; tbe 
other portions of do and p d neutralizing each other in that respect. 

The foregoing strains may all be calculated, thus: 


Horizontal pull inward liy the main chain — Tension X Cosineof adg 

6* thll # UM> nil lk«r t Ik I... ^.1. . . A _ _ . . A. . . f . : 


Vertical pressn 

44 44 


outward by the buck-Ktuy = Te 
re by main chain - tci 

hack-stay 


— Tension X Cosineof Idu. 


— Tension X Sine of a d g. 

— Tension X Sine of Idu. 



















SUSPENSION BRIDGES 


619 


Art. 5. If the cables pass freely overa loose pin, d, Fig 4 A, supported bv a link h, 
hanging from the fixed pin z, and capable of moving freely about both 
of its pins ; the tension in the back-stay will, as before, be equal to 
that in the main cable; and the direction and amount of the straiu 
on the piers will be found in the same way as for Figs 2, 3 and 4; 
namely : lay otf d s and d r, each equal to the tension, and draw the 
parallelogram dsvr. Then will d v give the amount and direction of 
the straiu on the piers. This last will, of course, be transmitted through 
the pins and the link. The amouutof tension on the link will be given 
by the leugth of d v ; and the link (being free to move) will be in line 
with this tension. The shearing straiu on each piu is also given by d v. 



Art. 6. But if the ends of the cable and back-stay. 

Figs 4 B, 4 C and 4 D, at the top of the pier, be made fast to a truck 

or wagon which is supported by rollers on a smooth platform on top of the pier, the axles of the 
rollers being fixed in the truck ; then the strain on the back-stay will not be the same as that on the 
cable, unless the angles adg and l d u are equal, as in Fig 4 B. 

If a d g exceeds Id'u , as in Fig 4 C, the strain on the 
back-stay will be less than that on the cable, and vice versa 
(Fig 4 D). 

But, in either case, if the top of the pier is horizontal, 
as is usually the case, the horizontal components of the 
strains on the cables and on the back-stays, will be equal, 
and will thus counteract each other, aud there will conse¬ 
quently be no horizontal or oblique straiu on the pier. 

That is, the strain on the pier will be vertical. 



-^7/t 

i n 

$ 

V 


FiiX.4 B 


To find tlie amount of the ten¬ 
sion on the back-stay, and of the pres¬ 
sure on the pier; on dg in either Fig. 4 

B. 4 C or 4 D, lay off ds, equal, by scale, to the tension 
on the cable at d. Draw d v perpendicular to the surface 
mn on which the rollers rest. We assume that mn is 
horizontal, as is generally, but not necessarily, the case : 
and d v, therefore, vertical. Draw s v horizontal, or par¬ 
allel to mn.* 

Then s v will give the horizontal pull of the main cable 
on the wagon, and d v will give the vertical pressure of 
the wheel d on the tower (to which that of the wheel d' has 
yet to be added). From d' lay off d o horizontal, and equal 
to<»; and draw r o vertically. Then d’r will give the 
amountofthe pull on the back-stay; and ro will give the 
vertical pressure of the wheel a on the pier; which 
must be added to d v for the total vertical pressure. 

Or the various strains may be calculated , thus 

Horizontal pull 
ft v or Wo at tlie 
top of the pier 

Strain W r in back¬ 
stay 

l*res OI1 pier, perp ) /Tension ds 

rol- r = di>-f-ro=( 



y 



i n 

,'s 

V 


Fhir.4 C 


T-* 



}■ 


Fi^.4 D 


Horizontal pull at Tension ds in x Cosine of o d a. 
middle of span — cable at d 


t = Horizontal pull * v or d'o at top of + Cosine of j M . 
j pier, or at middle of span 


to surf on 
lers rest 


which the 


on main X 
cable at d 


Sine of \ 
adg J 


/Tension d' r 
Von back stay 


Sine of 
l d 


° fN ) 

« / 



Art. 7. When, as is sometimes the case in light 
bridges, the piers are posts, P Fig 4 E, of wood or iron, hinged at 
the bottom, and having the cables and back-stays firmly fixed to 
their tons • from d draw d s. equal, bv scale, to the tension on the main cable at d ; and d w toward 
the foot of’ the post. From s draw s'r parallel to the back-stay, and meeting d w in r. Then will 
s r give the strain in the back stay, and d r will give the amount and direction of the pressure upon 
the post. 


Art 8. As in the Niagara bridge, the cables often merely rest upon 

movable trucks or saddles, T Fig 4 F. curved on top to avoid sudden bends in the cables, and resting 
upon loose rollers which lie upon a thick horizontal iron plate bolted to the top of the pier, and are 
free to move horizontally. In such cases the angles adg and l d'v. are made equal; so that the pul.s 


* The lines a l and s v must be drawn parallel to the surface m n on which the wagon rests, whether 
said surface be horizontal or inclined. 














































620 




SUSPENSION BRIDGES. 


d s and d' r are equal, as are also their horizontal components p d and d'o • and the pressures on the 
pier are vertical; and if changes of temperature or of loading produce slight changes in the angles 
a d g and l d' it, the truck will (by reason of the inequality thus brought about between the hori- 
zoutal components) move far enough to restore the equality between the angles, and between the 
horizontal components, and consequently the pressure upon the pier will at all times be vertical. 

Art. 9. To find, npproximalely, the leng th of a main chain 

C 6 d | Fig. 1 ; having the span c d, aud the middle dell a b. See preceding table, Art 1. 


Half length of main chain = pi 4 (deli 2 ) -f- 1% chord; 2 . 

In Menal bridge the chord cd is 579.874 ft: aud the detl is 43 ft. 

According to the above formula, the eutire leugth is 5SM.3 feet. By actual measurement the chain 
is precisely 590 feet. The approximate rule below gives 559.764 ft. 

Noth. The lengths obtained by this rule are only approximate, because the calculation is based 
upon the supposition that the chaius form a parabolic curve : whereas, in fact, the curve of a finished 
bridge is neither precisely a parabola, nor a catenary, but intermediate of the two. 

The following simple rule by the writer is quite as approximate as the foregoing tedious one, 
when, as is generally the case, the detl is not greater than of the chord, or span. 

Length of main chain when detl does not exceed one-twelfth of the span = chord -f- .23 detl. 

Art. 10. To find, approximately, flic length of the vert 
suspending rods jc y, Ac, Fig 1; assuming the curve to 
he a parabola. 

Let *, Fig 1, be any point whatever in the curve; aud let x w be drawn perp to the chord c d : and 
*/perp to a h ; then in any parabola, as u c 2 : at t* :: a b : b f. Aud 6/thus found, added to 6 1, 
(which is supposed to be already known, being the length decided on for the middle suspending rod,) 
gives * y, the length of rod reqd at the point x; aud so at any other poiut. 

| f I) 

If b f thus found he taken from the middle deflection a b, 

it leaves #r ,r; and thus any deflect ion w x of the main chain or cable, may he 

found when we know its hor dist, a w, from the center, a, of the span. p 

In the foregoing rule, the floor of the bridge is supposed to be straight ; but generally it is raised p 
toward the center; and in that case, the rods must first be calculated as if the floor were straight, - 
and the requisite deductions be made afterward. When it rises in two straight lines meeting in the ‘ 
center, the method of doing this is obvious. When an arc of a circle is used, its ordinates may be 
calculated by the rule given on page 141, and deducted from the lengths obtained by this rule, t 
Or, having drawn the curve by the rule for drawing a parabola, the dimensions can be approx- i 
(mated to by a scale. The adjustments to the precise lengths must be made during the actual con- ’ll 
structiori of the bridge, by means of nuts on their lower screw-ends. The rods require, therefore, 
ouly to be made long enough at flrst. 

The towers, piers, or pillars, which uphold the chains or , 
cables, admit of an endless variety in design. According to cir¬ 
cumstances, they may consist each of a single vertical piece of timber, or a pillar of cast or wrought , 
iron; or of two or more such, placed obliquely, either with or without connecting pieces; like the 
bents of a trestle, Or they may be made (with any degree of or- t 

namentation) of cast-iron plates; as in iron house-fronts. Or they may be of masonry, brick, or \ 
concrete; or of any of these combined. 

Each of the suspending-rods, through which the floor of the bridge is ! 

upheld by the main chains, requires merely streugth suflicieut to support safely the greatest load ! 
that can come upon the interval between it and half-way to the nearest rod on each side of it; in¬ 
cluding the wt of the platform, &c, along the same interval. i 

In anchoring the backstays into the ground, it is necessary to 

secure for them a sufficiently safe resistance against a pull equal to the strain, upon the backstay. ' 

As to the anchorage of the cables below the surface of the ground, 

natural rock of firm character is the most favorable material that can present itself. When it is not 
present, serious expense in masonry must be incurred in large spans, in order to secure the necessary : 
weight to resist the pull of the cables. Our Figs 4tj give ideas of the modes most frequently adopted. 
For a very small bridge, such as a short foot-bridge, for instance, the backstays may simply be an¬ 
chored to large stones, t. Fig A, buried to a sufficient depth. Or, if the pull is too great for so simple 
a precaution, the block of masonry, mm, may be added, enclosing the backstay. A close covering 
of the mortar or cement of the masonry has a protecting effect upon the iron. 

To avoid the necessity for extending the backstays to so great a dist under ground, they are usually 
curved near where they descend below the surface, as shown at B, D, and E; so as sooner to reach 
the reqd depth. This curving, how f ever, gives rise to a new strain, in the direction shown by the 
arrows in Fig3 B and D. The nature of this strain, and the mode of finding its amount, (knowing 
the pull on the backstay,) are very simple; and fully explained uuder the head of Funicular Ma¬ 
chine. The masonry must be disposed with reference to resisting this strain, as well as 

that of the direct pull of the backstay. With this view, the blocks of stone on which the bend rests 
should be laid in the position shown in Fig D ; or by the single block in Fig B. Sometimes the bend 
is made over a cast-iron chair or standard, as at x. Fig F, firmly bolted to the masonry. 

Fig E shows the arrangement at the Niagara railway bridge of 821^ ft span. The wire backstays 
end at cc; aud from there down to their anchors, they consist of heavy chains; each link of which 
is composed of (alternately) 7 or 8 parallel bars of flat iron, with eve ends, through which pass bolts 
* Each of the 7 bars of each link is 1.4 ins thick, by 7 ins wide, near the 


* When chains of iron bars are used instead of wire cables, they are usually made as at p 012. 
Since bar iron has hut about half the tensile strength of wire, the chaius must have a sectioual area 
twice as great as that of a cable. 










SUSPENSION BRIDGES, 


621 


lowest part of the chain; but they gradually increase from thence upward, until at c, c, where they 
fci unite with the wire cable, the sectional area of each link is 93 sq ins. These chain backstays pass in 
'n '• curve through the massive approach walls. (28 ft high,) and descend vertically down shafts s, s 25 
tij ft deep in the solid rock. Here they pass through the cast-iron anchor-plates, to which they are con- 
M lined below by a bolt ins diam. The anchor-plates are 6% feet square, and V/c, ins thick ; except 
for a space of about 20 ins by 26 ins, at the center where the chains pass through, where they are 1 



foot thick. Through this thick part is a separate opening for each bar composing the lowest link. 

■ From this part also radiate to the outer edges of the lower face of the plate, eight ribs, 2}$ ins thick, 
t The shafts s, s, have rough sides, as they were blasted ; and average 3 ft by 7 ft across ; except at the 
bottom, where they are 8 ft square. They are completely filled with cement masonry, with dressed 
i beds, well in contact with the sides of the shafts; and thoroughly grouted, thus tightly enveloping 
the chains at every point; as does also the masonry Of the approach wall tv tv; which extends 28 ft 
i above ground; and is 6 ft- thick at top, and lO)^ ft thick at its base on the natural rock. 

I), Figs 4*^, shows a mode that may be used in most cases, for bridges of any span. The depth 
and the area of transverse section of the shaft, and consequently the quantity of masonry in it, will 
depend chiefly upon whether it is sunk through rock, or through earth. If through firm rock, then 
if its sides be made irregular, and the masonry made to fit securely into the irregularities, much re¬ 
liance may be placed upon it to assist the weight of the masonry in resisting the pull on the back¬ 
stays. Earth also assists materially in this respect. 

F is the arrangement in the Chelsea bridge of 333 feet span, across the Thames, at London ; Thos. 
' Page, eng. The space from one wall b b, to the opposite one, is 45 feet; and is built up solid with 
• brickwork and concrete; except a passage-way 4 ft wide, and 5 ft high, along the backstay ; and a 
1 small chamber behind the anchor-plates. It rests chiefly on piles. 

t The arrangement by Mr Brunei, in the Charing Cross bridge, London, is very similar. In it also 
the entire abutment rests on piles; and is 40 ft high, 30 ft thick, and solid, except a narrow passage- 
;| way along the chains. The backstays extend into it 60 ft. Span 676 feet. Defl 50 feet. 

G is intended merely as a general hint, which, variously modified, may find its application in the 
case of a small temporary, or even permanent bridge ; for the number of pieces, i, l, Ac, may be in* 

! creased to any necessary extent; and they may be made of iron or stone, instead of wood. 

Ill estimating* the action of the backstays upon the ma¬ 
sonry, «fcc, to which they are anchored, it is safest to consider the 

, tension along them to continue undiminished to their very ends ; although, when they are embedded 
in masonry, friction causes it to diminish; especially when they are curved, as in E, Figs 4^, in 
which case the friction is greatly increased, and the tension thereby materially reduced as the ends 
are approached. Frequently, however, they are not so embedded; for, although embedding preserves 
the iron, many engineers prefer to leave an open space around the entire length of the anchor-chains ; 
as well as around the anchor-plates; in order that they may be examined from time to time. To this 
end, the masses mm of masonry in Figs 4)^, may be made not solid, but to consist of two parallel 
walls, between which the backstay may pass; and the anchor-stones, or anchor-plates, will extend 
across the space between the walls, and have their bearing against the ends of the walls. In F, the 
cable may be supposed either to be tightly surrounded by the masonry, and grouted to it; or else to 
be surrounded by a cylindrical passage-way like a culvert, so as to be at all times accessible. And 
the same with regard to the cable in the vertical shafts at D or E. 

Art 35, Art 49, Ac, of Force in Rigid Bodies, will assist in calculating the resistance 

which the masses m m of anchorage masonry oppose to the pull of the backstays. Soft friable stone 
must be carefully excluded from such parts of these masses as are most directly opposed to this pull. 

If blocks of stone large enough for securing good bond are not procurable, heavy bars of iron, or 
I l>eams, may be advantageously introduced for that purpose. . 

The masses must be founded at such a depth as not to slide by the yielding of tne earth in front 


Experience shows that with due attention to periodical painting, and renewal of woodwork, a 
properly designed suspension bridge will be very durable, The transverse floor joists should be of 
wrought iron ; to prevent interruption to travel while putting in new wooden ones. 


Particular care should he bestowed upon the strength of 
the joints of the side parapets; for the undulations and lateral motions 
of the bridge expose them to violent deranging forces in every direction. The parapets should be 

high and stout; and not restricted to mere service as hand-rails, or guards. 

As a ru'e of thumb, one-half the sq rt of the span will be about a good height for them in ordinary 
oases, provided it is not less than a hand-rail requires. 

































622 


SUSPENSION BRIDGES, 


We do not think that diagonal horizontal bracing should, as is usual, be omitted under the floor. 

It may readily be effected by iron rods. 

All the cables need not be at the sides of the bridge. One or more of them may be over its axis; ■ 
especially in a wide bridge. One wide footpath iu the center may be used, instead of two narrow I* 
ones at the sides. i 11 

The platform or roadway should be slightly cambered, or curved upward, to the extent say of about 

^ O of the span. 

Art. 11. The Niagara suspension bridge, built in 1852-3,*John A 

Roebling, engineer, consists of a siugle span of 821 ft measured straight from center to center of 
towers ; and 800 ft of clear suspended length of roadway. It has two floors or roadways : the upper ki 
one, for a single-track railway, is 25)4 ft; and the lower one, for common travel, 24 *4 ft wide, out to out l e 
of everything. The lower one is 10 ft wide in the clear of everything. They are 17 ft apart verti- ,;t 
cally. The trusses are 18 ft total height, throughout. They are on the Pratt arrangement; j„ 

with verticals 5 ft apart from cen to cen; and single oblique iron rods, 1 inch square, running in , sf 
each direction across four of the 5 ft panels. Where these rods pass each other, they are tied together H 

by 10 or 12 turns of yhr inch wire. Each vertical consists of two pieces of 4*4 by 6*4 timber, placed 'sp 
4}^ ins apart, to allow the oblique rods to pass between them. Both upper and lower floor girders are 


1 !ue 

a. ''i 


iu two pieces, of 4 by 16 ins each. Pairs 5 ft apart. The tops and bottoms of the verticals pass be¬ 
tween the two pieces which form each floor girder. No tenons or mortises are used in the framing. 

There are four cables of iron wire; two on each side of the bridge. Each 

cable is 10 ins diatn. The wire is scant No. 9 of the Birmingham wire gauge, or scant .148 inch diatn. 
Sixty wires have a united transverse section equal to one square inch of solid iron. Each of the four isi 
cables contains 3640 wires, with a united cross-section of 60.4 sq ins of solid metal. Therefore, the i4 
area of solid metal in a section of all the four cables together is 241.6 sq ins, or 1.678 sq ft; weighing 
814 lbs per ft of span. The wires of each cable are first made up, in place, into 7 small strands: and 
these are firmly bound together throughout by a continuous close wrapping of wire. The strength of 
each individual wire is 1640 lbs, or .73214 of a ton. This is equal to 98400 lbs, or 43.93 tons per sq inch I 
of solid metal; or to 5943360 lbs, or 2653.3 tons per cable ; or to 10613.2 tons ultimate strength of the F ! 
four cables together. One cable on each side of the bridge deflects 54 ft; and the other 64 ft; average ■ 

deflection 59 ft, or about of the span. With this av defl the tension on the cables at the tops of the n 
towers averages 1.82 times the total suspended wt of the span and its load. See table, Art 1. The wt i;t 
of the suspended spau itself is about 900 tons; and if the greatest extraneous load on the two floors Tl 
together be taken at 1 *4 tons per ft run, we have the total suspended wt 900 -(- (800 X 1)4) = 1900 tons, uj 
And 1900 X 1.82 = 3458 tons tension at towers; or very nearly *4 of the ultimate strength of the cables, ; t 
without any allowance for momentum, or wind. But such loads, although possible, are not permitted fort 
to come upon the bridge. 

The wires were perfectly oiled before being made into strands; and when the strands were being ‘:k 
bound together to form a cable, the whole was agaiu thoroughly saturated with oil and paint. 

The cables do not hang vertically; but the two upper ones are about 37 ft apart from center to cen- i: 
ter, where they rest upon the towers, (where all four are on the same level;) and are drawn to within - js 
13 ft of each other at the center of the span; and at the level of the railway track on top of the •«! 
bridge; while the two lower ones are about 39 ft apart at the towers, and 25 ft at the center of the span, 
and at the level of about halfway between the two floors. 

This drawing-in of the cables contributes much to lateral stability; as do also the upper and lower t ■<_ 
floor of stout plank. There is no horizontal diagonal floor bracing. 

There are 624 suspenders of wire rope, 1 % ins diam, and 5 ft apart, or corresponding with the floor v 
girders, which they uphold; and with the wooden verticals of the trusses. They do not hang verti- 
cally; but incline inward. p. 

The masonry towers are all founded on rock. They are 78*4 ft high above the bottom of the bridge; m 
and 60*4 ft above the upper floor. The two at each end of the spau are 39 ft apart from center to cen- tlii 
ter. At the level of the lower floor they are 19 X 20 ft; and 21 ft apart in the clear. At the level of ! - 
the upper floor they are 15 ft square; and 24 ft apart in the clear. From there they taper regularly > lir 
to the top, where they are 8 ft square. They are built of limestone, in heavy dressed hor courses; U 
laid in cement; vertical joints grouted. The upper courses are dowelled. On top of each tower is a i 
cast-iron plate, 8 ft sq, and 2J4 ins thick, bedded in cement. Bart of the top of this plate is planed, Hit 
as upon it move the rollers which support the cast-iron saddles on which the cables rest. At each ( 1 * 
tower, each cable has its separate saddle and rollers. Each saddle rests on 10 cast-iron rollers 25*4 , J 
ins long, and 5 ins diam, carefully planed. See Fig 4 F. They lie loosely, and close together ; 

and are kept in place by side flanges nu the hed-plate. 


The cast saddles are each 5 ft long, by 25*4 ins wide. Their bottoms, which rest on the rollers, are 


flat, and planed. Their tops are curved to a rad of 6*4 ft; to suit the bend of the cables over the piers ; 
and each saddle has a longitudinal groove, in which the cable lies. The passage of the heaviest trains’ 
produces less than )4 an inch of movemeut in a saddle. 

The floors have a camber of 5 feet. 


A change of 100° Fah of temperature causes an average variation of about 2J4 ft in the deflection t» 
of the cables, or in the camber of the roadways ; and one of 150°, (about the extreme to which the 


bridge is exposed,) about 3 3 4 ft. The passage of a train weighiug 291 tons, and covering the entire >• 
length of the span, caused a deflection of 10 ins ; and an ordinary train deflects it only from 3 to 5 ; 


iuches. 


This bridge has, since the year 1853, demonstrated the applicability of the suspension principle to t' 
large span railway bridges. Its entire cost was not quite $400,000. 


Art. 12. The wire suspension bridge near Freybnrjj, Su it- * 
xorlanil, finished in 1S34, Mr. Chaley, engineer, and still in full service, is of * 


very simple construction, and has served as the prototype for several in this country. It is Tor 
common travel only ; and is narrow: its entire width of platform being but 21*4 ft; and its clear 
available width but 19 ft. The dist from cen to cen of its towers is 889 feet; and its clear span be¬ 
tween abutments 800 ft; or the same as the Niagara. There are 4 cables, each 5 ins diam Each of 
them consists of 1056 wires of No. 10, or full % inch diam, (or 71 wires to the sq inch of solid metal;) L 
arranged in 20 strands of about 53 wires each. The four cables, therefore, have a united area of but " 
60 sq ins of solid metal; weighing 202 lbs or .09 of a ton, per ft run of spau. All its suspeuders are f ! ! 

* In 1886 the wooden piers and trusses here described were replaced by iron ones. 






SUSPENSION BRIDGES, 


623 


vertical; about 5 ft apart; and each upholds one end of a transverse door girder. It has no sido 
trussing except the slight one of the wooden hand-railing, which is about 6 feet high; and conse¬ 
quently, with its great span it is quite flexible. The deflection of the cables is 0 p the span ; henco 
the strain upon them at the top of the towers at either end, is 1.82 times (see table p 615) the wt of 
the suspended span itself, and its extraneous load; and supposing the wire to be as good as that of 
the Niagara, the breaking strain of the four cables would be 60 X 44 — 2640 tons; and their safe 
strain cannot be taken at more than % as much, or 880 tons. The suspended weight reqd to produce 

gyQ 

this safe strain would of course be -— = 484 tons. The suspended weight of the span itself cannot 

1.82 

well bo less than .3 of a ton per ft run ; or 240 tons in all; * thus, leaving 484 — 240 — 244 tons for 
the maximum safe extraneous load. This amounts to .305 of a ton per ft run of span ; or 36 lbs por 
sq ft of its platform, 19 ft wide in the clear. The French allowance is 41 lbs per sq ft; t and since no 
allowance is here made for momentum or wind, it is plain that this celebrated bridge, on account of 
its slight cables, and its flexibility, is by no means a strong one. In that respect, as well as steadi¬ 
ness, it is much inferior to the one next spoken of. It is said, however, to have withstood very severe 
tempests; and also to have been occasionally completely covered by crowds of people. If so, their 
lives were not very secure. 

Art. 13. The wire suspension bridge across the Schuylkill 

at Philada, finished in 1842,JChas Ellet, Jr, engineer, is somewhat similar in character, and in the 
dimensions of its details, to the preceding; but being of much less span, is much stronger. Its span 
from cen to cen of towers is 358 ft; suspended platform between abuts 342 ft. It has ten cables of 3 
r ins diam; five on each side. Their united sections present 55 sq ins of solid iron ; or nearly as much 
6 as the preceding bridge of 800 ft clear span. The five cables on either side have different deflections, 

jj ranging between the an< * yyf of the span from tower to tower. The dist from cen to cen of 

■ towers at either end of the span is 35*^ ft; and on top of each tower the cables (considerably flattened 
. i at that poiut) lie side by side on a single roller about 30 ins long, and 6 ins smallest diam, which has 5 

’ grooves, for their reception. Each cable is drawn-In about 'A% ft at the center of the span. At in- 
, tervals of 20 ins the parallel wires of the cable have a close wrapping of finer wire for a distance of 

3 ins. 

* The suspenders are of wire; and are % inch diam ; and 4 ft apart. On any one cable they are 20 
ft apart. They all incline slightly inward. 

i The width of the platform from out to out is 27 ft; and in clear of hand-rails 25 ft. Inside of the 
. hand-rail is a footway, 4 ft 4 ins wide, on each side of the bridge. Tbe remaining 16 ft 4 ins serves 
for a double carriageWay, or double-track street railway. Figs 5 show the arrangement of the wood- 

1 work, on a scale of % _ , ' ... _ ,., , ■, _ , 

inch to a ft. The trussing 
; >f the parapets is on the 
: Iowe system, 

■ which does not appear to 
l ye as well adapted as the 
! ’ratt, to suspension 

bridges. The diagonals 
in the Fairmount bridge 
work themselves out of 
place laterally, by the 
vibrations of the bridge; 
and we have occasional¬ 
ly seen several of them 
almost on the point of 
falling out entirely. 

Being under municipal 
charge, it is of course 
neglected. The upper 

chords «, are 12 ins ; , , 

wide by 6 ins deep; the lower ones l, and the stringer c, below them, are each 12 wide, by 7 deep. 
The diagonals i are all 4 ins wide, by 5 deep. The angle-blocks at their ends are of cast iron, hollow, 
and about % inch thick. The vert iron rods v, (in pairs,) are % inch diam near the center of the 
span ; and 1 % at its ends. The top chords are spliced on each vert face by an iron bar, of 5 ft by 3 
ins, by V, inch ; with 4 bolts passing through them. The splice ot' the bottom chord has merely 2 
bolts, side by side ; (see Figs 5 ;) which (except S) are to a scale of % inch to a ft. The floor girders g, 

4 ft apart from cen to cen, are 6 by 14 ins at their ends; and 6 by 16 at center. 

The floor is of two thicknesses of 2-inch plank; except the footpaths, which are single thickness. 

The wires were well oiled when the cables were made; and afterward painted. 

At 8 is shown the mode of uniting a suspender with a cable, o, by means of a small cast-iron yoke 
o which straddles the cable; and on the back of which is a groove % of an inch wide, in which the 
suspender rests. Tbe metal of the yoke is about 'A inch thick. Since tbe lower ends of the wires 
which compose a suspender cannot themselves be formed into a screw-bolt, for upholding the floor 
girders they are passed through the eye of a screw-bolt of bar iron ; then doubled on themselves, and 
held by a wrapping of wire. It is well to introduce a yoke here also, to prevent the wear of the wires 
by friction. The small fig on the right of 8 is an edge view of the yoke g. 

* This is probably nearly its actual weight, as obtained by comparing it with the Fairmount bridge? 
which hv a careful estimate by the writer, weighs .375 of a ton per ft run ; but. is considerably wider 
than t’heVreyburg; and carries four lines of light street-rails. But if the Frey burg has longitudinal 

v joists, it will weigh about .03 ton more per ft run. ..... - , _ i„ . 

J t The greatest load that can come upon an ordinary bridge, is a dense crowd of people , and this 
the French engineers estimate at 41 Tbs per sq ft of platform. This is certainly as great as can well 
occur under ordinary circumstances; but it may be considerably exceeded. The French estimate 
moreover, includes no allowance for wind, or for the crowd being in motion. 

writer thinks that no suspension bridge should have a less safety than 3 against KKl lbs per sq ft, 
added to the weight of the bridge itself. A less coeffof safety is admissible in a wire bridge than 
in an iron trussed one. on account of the greater reliability of the material. See foot-note, p 606. 

1 Removed, 1873, and replaced by a truss bridge of 34» tt span, by Keystone Bridge Co. 






































624 


SUSPENSION BRIDGES. 


There is no transverse bracing under the floor; nor are there longitudinal floor joists resting on the (>r 
girders. Owing to the waut of the distributing effect of these; and to the use ot so many small cables . ( | 
instead of but 2 or 4 larger ones ; as well as to the inefficient trussing of the hand-railing or para¬ 
pets, the bridge is much less steady than it would otherwise be. 


With wire of the same quality as the Niagara, (or 44 tons per sq inch breaking strength,) the 1* air- 
mount bridge would, with a safetv of 3, (omitting momentum and wind,) sustain an extraneous load 
of 346 tons; which is equal to 1.01 ton per ft run of span; or 90 fts per sq ft of its clear platform. 
This last is 2.5 times as great as the strength of the Freyburg, with the same quality of wire. 1 he 
Fairmount is, however, we believe, built with w'ire of but 36 tons per sq inch ultimate strength. If 
so, its greatest extraneous load becomes reduced to 260 tons ; or .76 ton per ft run; or 6b u>s per sq ft 
of platform, or nearly twice that of the Freyburg. 

The towers are of cut granite, in heavy courses. They are 8)4 ft square at the ground line, or level 
or the floor; about 5 ft sq at the top; and about 30 ft high. The backstays have the same angle of 
direction as the main cables. • 


Art. 14. The Wheeling bridge across the Ohio at Wheeling, Vir¬ 
ginia, also by Mr Filet, had a span of 1010 ft between the towers; and 960 feet clear span between the 
abuts ; and w r as 26 ft wide from out to out. Its mode of construction w r as much the same even in de¬ 
tail as that of the Fairmount bridge; except in having 12 cables instead of 10. The 12 cables con¬ 
sisted of 6600 wires of No. 10 Birmingham gauge, presenting a sectional area of 93 sq ins of solid metal, 
weighing 313 lbs, or .14 of a ton. per foot of span. The weight of the woodwork was about the same 
per foot run of Rpan as in the Fairmount- Although its clear span was 2.8 times as great as the 
Fairmount, yet its cables had but 1.7 times as great area of solid metal. The entire suspended wt 
between towers, is stated at but 440 tons; therefore, with an average deflection of T3 of the span, 
for a safety of 3 against 100 R>s per sq rt of platform of 24 ft clear width ; or 1.07 tons per ft run of span, 
the area of solid metal in the cables should have been 175 sq ins, with 44 ton wire like that of the Ni¬ 
agara : or 214 sq ins, with 36 ton wire, which we believe was the quality actually used. 

Art. 15. The suspension canal aqueduct at Pittsburg. Penn, 

built in 1845, John A Roebling, Esq, engineer, has seven spans of 160 ft each. Deflection 14J4 ft; or 
about yy of the span. It has but two cables, each 7 ins diam. The two together contain 3800 No. 10 
wires, making 53 sq ins of solid metal section. Ultimate strength of each wire 1100 lbs ; equal to 35.2 
tons per sq inch of solid metal; aud making the ultimate strength of the two cables together 1866 tons. 

The prism of water in the wooden aqueduct is 4 ft deep ; bj 14 % ft average width; and weighs 265 
tons per span. The wt of one span of the structure itself is about 111 tons; making the total sus¬ 
pended wt at each span 376 tons. The tension on the two cables at either end of a span, with a deli 

of yy, is 1.46 times the total suspended weight; see table, p 615. Hence it is in this case 376 X 1-46 

1866 

= 549 tons ; and the strength of the cables is —- 3.4 times the constant strain upon them. 

549 

On one side of the water is a towpath for horses; and on the other a footpath ; each 7 ft clear width. 
With these occupied by horses and people, the foregoing safety would be reduced to about 3. The 
loaded boats do not add materially to the weight, inasmuch as they displace a bulk of water equal to 
their own wt; and but little of the displaced water remaius on a span at the same time with the boat. 

The great wt of the water prevents undulations; and the aqueduct is therefore very steady. On this 
account a less coeff of safety is admissible than on a common bridge. 

The aqueduct leaked badly along its lower corners. 


Art. 16. In 1796, Mr James Finley, of Fayette Comity, Penn, 
introduced suspension bridges in tlie U. S.; and built several with 

spans of 200 feet and less. Many of them were very primitive structures; but answered sufficiently 
well for the times. They had usually either two or four chains, composed of links from 7 to 10 feet 
long, formed by bending about 1^-inch square bars of iron, and welding their ends together. At 
each link-end, was a vertical suspender rod of 2 ins by % inch iron; which, at its lower end, was 
bent aud welded into a stirrup for upholding one end of a transverse floor beam. On these beams 

rested longitudinal joists supporting the floor plank. Finley used deflections as great as or even 
% of the span ; and his piers were frequently single wooden posts; the two at each end being braced 
together at top. Such were used in a span of 151)4 ft clear, across Will's Creek, Alleghany Co, Penn. 
It had two chains. The defl was % of the span. The double links of inch sq iron, were 10 feet 
long. The center link was horizontal, and at the level of the floor; and at its ends were stirruped 
the two central transverse girders. From the ends of this central link, the chains were carried in 
straight lines to the tops of the single posts, 25 ft high, which served as piers or towers. The back¬ 
stays were carried away straight, at the same angle as the cables; and each end was confined to four 
buried stones of about )4 a cub yard each. The floor was only wide enough for a single line of ve¬ 
hicles. All the transverse girders were 10 ft apart; and supported longitudinal joists, to which the 
floor was spiked. There were no restrictions as to travel ; but lines of carts and wagons in close suc¬ 
cession, and heavily loaded with coal, stone, iron. &c, crossed it almost daily ; together with droves 
of cattle in full run. The slight hand railing of iron was hinged, so as not to be bent by the undu • 
lations of the bridge. Six-horse wagons were frequently driven across in a rapid trot. It was built 
in 182°; and an observant engineer friend, who in 1838 took the sketch and measurements upon 
which this description is based, informed the writer that the iron was as perfect, and as sharp ou all 
its edges, as on the day it was built. The iron was the old-fashioned charcoal, of full 30 tons per sq 
inch ultimate strength. The united cross-section of the two double links was 7.56 sq ins: which, at 
30 tons per sq inch, gives 227 tons for their ultimate strength ; or say 76 tons, with a safetv of 3. Now, 
with a defl of % span, the tension on the cables (see table) is but .9 of the suspended total wt. 

A, c " alns P'sce would therefore sustain, with a safety of 3, a quiet suspended load of 76 X 
.9 - 68 tons; and ns the span itself did not weigh more than 15 tons, we have 53 tons for the safe ex¬ 
traneous weight, omitting all consideration of wind and momentum. This is equal to .35 ton per ft 
of span ; equal to .7 ton for a bridge wide enough for two vehicles to pass. This primitive 




SUSPENSION BRIDGES 


625 


bridge would therefore safely snstain a greater load per foot 
run of 8|)iin, than the Frey burg. 

These old bridges frequently failed by the rotting of the end posts ; or were carried away by fresh¬ 
ets; but we have never heard of a failure from the breaking of the chains. Many of them were built 
iu a much more perfect style than the oue just described; and on the most used' roads in the Union. 


626 


TEST BORINGS. 


tof 

m fa 



sj 



1 llv 

s f 


1 

1 





FlO 1. 


Fig 2. 


« 

Iru: 


Hi 


Pierce’s Well-borer. made by the Pierce Well-Excavator Co, of New York 
City and Long Island City, N. Y., is an excellent tool for boring into soils, 
C*lrty, sum I, or gravely eyen when quite indurated, or when frozen. It will 

not bore through hard rock, or through large boulders. It 
consists of two sheet-iron cylindrical segments S S. called | 0 » 
“pods,” having their lower or cutting edges shod with steel, ;t|.n 
These edges project (as shown in Fig 1) beyond the sides of j s |i 
the auger, and thus make the hole larger than it, so that it tJ 
cannot bind or stick. The two cutting edges are equidistant jv 
from the vert cen line of the tool, and this insures a straight^ 

! and vort hole. At a the auger is attached to the lower end,]),, 
of avert boring rod composed of a number of l^-inch square 
iron bars, or 2j^-inch iron tubes, about 10 to 15 it long.W 
jointed together at their ends by means of square socket-!® 
joints. At the top of this boring-rod is a swivel-hook, by f j 
means of which the entire apparatus is hung to the end of aj|, 
rope, which passes over a pulley at the top ot a derrick or;jjj| 
tripod, and down to a drum worked by a windlass and gear-^ # 
ing. By means of this drum and rope, the anger and boring-rod (which at first con-jj; 
sists of only one bar) are lifted, and suspended over the intended hole. The auger r 
is then lowered, and rotated lior by two men or one horse, working at the ends oi 
levers which grip the boring-rod a few ft above the ground. The swivel at the top 
of the boring-rod permits this rotation to take place without twisting the rope.E, 
The shape of the auger is such that its rotation feeds or screws it into the ground; 
and the man at the windlass has, during the boring, merely to keep the rope tight, 
so as to prevent the auger from boring too fast, and becoming clogged. In about 8i,| 

revolutions the auger fills with earth. By means of the windlass it is then raised r 

to about 2 ft above the ground; and by unkeying and removing the band b the augerk 
is opened like a pair of tongs, and the earth emptied into a wooden box which bask 
in the meantime been placed over the hole. The box is then removed and emptied,^ 
and the boring proceeds as before. When the boring lias reached a depth of about i_| 
10 ft, a second bar must be added to the top of the rod. For this purpose the rod ,, 
and auger are raised a few inches; a slight frame-work of hoards is placed on tlicL 
ground, close to the boring-rod and surrounding it; and a flange is clasped tightlyL 
to the rod just above, and close to, the framework. The framework and flange now 
support the rod and auger; the swivel-hook and rope are removed,and attached to , 
the upper end of the second bar, which is then raised, and its lower end is fastened J,| 
into the socket-joint upon the top of the first one. The rope is then drawn tight ;j) B , 
the flange removed; the auger lowered to the bottom of the hole; and the boring. 
resumed. Additional lengths of boring-rod are attached in the same way from timoL 
to time, as required by the descent of the auger. i (1( 

The l>orers are made from 6 to 18 ins diam, or larger to special order. If desired, u 
the boring may be made from 21 to 36 ins diam by attaching a reamer to the auger.: t 
This auger will bore to a depth of 100 ft or more at the rate of from 5 to 20 ft perk 
hour. It removes stones as large as half the diam of the hole. In dry soils a bucket- 
fill of water is poured into the hole each time the auger is raised. 

The Pierce well-borer may be advantageously used in boring the holes for snnfl- 
piles, p 650, and at times, instead of driving' wooden piles, it may , 
be better to plant them (butt down if preferred) in holes bored by this auger; ram¬ 
ming the earth well around them afterwards. This will save adjacent buildings 
from the jarring and injury done by a pile driver. 

If sand, mud, or loose gravel is reached in boring with this tool, 
the hole is reamed out 4 ins larger, and a tubing of inch boards is inserted into 
the hole, and driven into and through the sand or gravel, which is then removed 
from within the tubing by means of the sand-pump furnished with each machine. 
This consists of a hollow iron cylinder, about 5 ins diam X 30 ins long, with a valve 
at its foot, opening upward. It is lowered to the bottom of the hole; covered with 
water to a depth or 2 to 4 ft, and churned quickly up and down 4 to 6 ins, by hand, 
20 or 30 times, during which the sand fills the pump, which is then drawn up and 
emptied. From 10 to 20 ft in depth of sand, mud, Ac, per hour can thus be taken 

Ac. 


)ii 


from a 6 to 18-inch bole. This pump is also used for removing broken earth, Ac, 


from a hole bored in compact earth by the Pierce borer first described. 

Tl»e cost of a Pierce auger, with derrick, boring-rods, rope, sand-pump, Ac, Ac, 
complete, is (1886) about $150. The Huger weighs from 150 to 200 lbs,accord¬ 
ing to size. Boring-rod 1^ ins sq, 3j^ lbs per ft. Derrick, 150 lbs. 

Pierce’s Sand-borer, Figs Sand 4, like the sand-pump just described, is 
used inside of tubing, and for the same purpose. The hollow iron cylinder C, 10 ms 
diam X 30 ins long, slides vertically on the rod, but the screw is fast to the rod. 


While boring, the saml below and around the cyl keeps it in the position shown in 

















ARTESTAN WELL BORINO 


627 


Fis 3. Six revolutions of the rod and screw fill the cyl with sand. The rod is then 
lifted. This first draws the screw up into the cyl, as in Fig 4; and a valve at the 

4 



foot of the screw closes the bottom of the cyl, and prevents the sand 
from falling out when the borer is lifted from the hole. The rod is 
hollow, and open at top and bottom. This allows passage of the air, 
and thus prevents resistance from suction in withdrawing the borer. 

This tool is rotated and withdrawn in the same way as the earth borer 
first described. Price (1886), $32. 

The Pierce Co also furnish a steel prospecting auger, from 
2 to 4 ins diam, and 2 ft long, for boring holes from 2% to 6 ins diam, 
and to depths of 10 to 50 ft, into clay, sand, or fine gravel, of 
all of which it brings up samples. It is turned by wrenches, and by 
man or horse power, as is the well-borer; but requires no derrick, as 
it can be withdrawn by hand. Price (1886), of auger alone,$10 to $25; 
or. with boring-rods for a depth of 50 ft, levers. &c, complete, $50 to $75. 

Tlie boring tool shown in vert section by Fig tt, 
and in hor cross section by Fig 5, is very useful for boring snal- 
low holes* by hand through surface sails, clay, and gravel, and 
bringing up samples. The borer proper consists of a cylinder of 
spring steel, 3 or 4 ins diam, and 4 or 5 ins high, with sides 
inch thick, having a vert slit (see cross section) throughout its 
height, and beveled to a cutting edge all around its foot, as 
shown in the vert section. At its top it is riveted, as shown, or 
welded, to the inverted-r-shaped forging, which, by means of the 
socket at its top, is screwed to a length of gas-pipe which serves 
as a handle, and to which other pieces are joined by sockets as 
boring proceeds. 

The boring is done by two men, who grasp the handle, and, 
holding the tool vert, drive it into the ground by repeatedly 
lifting it and forcibly bringing it down upon the same spot. As 
the tool strikes the ground, the beveled shape of its cutting edge 
causes it to open slightly, and when the downward pres is re¬ 
lieved in lifting it. it springs back and grasps the earth which 
has entered it. It soon fills; and the men, finding that it ceases 
to penetrate readily, lift it to the surface and empty it. The 
character of its contents from different depths, measured along 
the handle, is noted from time to time. 

In six days of 8 hours each, three men (one 
resting at intervals) using one such auger 
between them, bored 20 holes,averaging 9% 
ft each, m loam, gravel, clay, and decom¬ 
posed mica schist, at a cost of 22 cts per foot. 

Wages of each man, $2 per day. 

For work in loam, clay, or non-running 
sand, an effective screw-auger can 
be made by any good blacksmith, by merely 
forming a one-inch sq bar of iron or steel 
into corkscrew shape about 2 ft long with 
6 complete turns 6 ins in diam ; its lower end sharpened to form a vertical cutting 
edge, which should project say .5 of an inch beyond the spiral of the screw, in order 
to diminish friction. It will bring up full samples. Requires a derrick, or some 
other simple mode of lifting, when the screw is full. 

Artesian Well Drilling'. Deep vert holes in earth and rock, 6 and 8 ins 
in diam, such as are reqd for artesian wells for water and oil, and for mining explora¬ 
tions, are drilled by repeatedly lifting and dropping, in the same vert line, a heavy 
iron bit, Figl,p 629, with a steel cutting-edge. The bit is partly revolved horizon¬ 
tally after each blow, to insure roundness of hole. The length of the cutting-edge 
of the bit is a little greater than the diam of the bit, and the hole is thus made suf¬ 
ficiently large to prevent the bit from binding in it. See also diamond drill, p 652. 

The bit is the lowest one of a series of iron and steel bfu-s, &c, Figs p 629, screwed 
together at their ends, and called a “string of tools.” The string of tools 
varies in length from 25 to 60 ft. according to the size and depth of the hole, and the 
hardness of the rock; and its diam throughout (above the cutting-edge) is an inch 
or two less than that of the hole. Its weight is from 8nO to 4000 lbs. Its upper 
member is always a “rope-socket,” Fig 4 (without a swivel), to which the lower end 
of the supporting rope cable is attached. This cable passes up out of the hole to 
a hor lever, which, by means of a horse-power or steam-engine, is kept con¬ 
stantly moving up and down with a see-saw motion. The string of tools, with the 
cutting-edge of the bit at its lower end, is thus alternately lifted from 2 to 4 ft, and 




Fig 5. 


Fig 6. 





























628 


ARTESIAN WELL BORING. 


let fall, from 30 to 50 times per minute, and so drills its way into the ro'k or earth. 1 


From 4 to 10 ft in depth of water are kept in the hole, to facilitate the drilling and " 


the removal of debris. After water is reached, the drilling may bo continued, even 
if the hole is full of water; but a great depth of water of course diminishes the force 

A suitable arrangement must be provided for pitying' 


of the blow's of the bit. 


Okit (he rope as the boring tool descends. A damp is attached to the cable 
and the man in charge, by turning the clamp, twists the rope, and thus turns 
the bit horizontally about one-fifth of a revolution after each stroke, until 
six or eight complete revolutions have been made in one direction, lie then re¬ 
verses the motion, and makes an equal number of turns, at the same late, in the 
opposite direction. 

After drilling a few feet, the string of tools is lifted out of the hole by means ot 

val of the debris w hich has accumulated in the 


’ not 


J1 

Hwoi 


(itlie 

Ufa 


the cable, to allow the reinov 


hole. This is done by means of a sand-pump. 


which is a sheet-iron cylinder, 


' tla 
•ilru 


iw 

din 

int 


say 4 ins diam, and 4 to 6 ft long, provided, at its foot, with a valve opening upward. 
The pump is lowered to the bottom of the hole, and filled with the mixed water and 
debris by churning it up and down a number of times. Sometimes, in addition to 
the valve, the pump is fitted with a plunger, which is at the foot of the pump when 
the latter is let down to the bottom of the hole. The plunger is then drawn up into 
the pump, and the debris follows it. In either case, the pump, when filled, is lifted 
out of the hole and emptied; the String of tools is again lowered into the hole, and * 
the drilling resumed. The debris must be removed after every 3 to 5 It of drilling. 
Otherwise it would interfere too greatly with the action of the bit. 

Wells are usually drilled from « to 8 ins diam. Fordiams less 
than (i ins, the tools are so slender that they are liable to be broken in a dee]) hole. 

The same apparatus is used for drilling* through the earth above 
the rocli, before the latter is reached. This is called “spudding.” In this case 
the sides of the hole must be prevented from caving in. For this purpose a wronght- 
iron pipe of such diam as to fit the hole closely, and x /± inch thick, is inserted into * 
the hole, and is driven down from time to time as the drilling proceeds. The pipe ™ 
is driven by means of a heavy maul of oak, or other hard wood, 14 to IS ins square, 
and 10 to 16 ft long. This maul is attached, by one end, to the lower end of the llr 
same cable which, during drilling, supports the string of tools. It is thus repeat¬ 
edly lifted, and dropped upon the head of the tube, which is protected by a cast-iron 
‘•driving-cap.” The foot of the tube is shod with a steel cutting-edge ring,or “steel 
shoe.” When the tube has been driven as far as it will readily go, the maul is re¬ 
moved from the end of the rope; the string of tools substituted; and the drilling 
resumed within the pipe. 

The pipe is put together in lengths of from 8 to is ft, and the drilling and pipe¬ 
driving proceed alternately until the rock is reached, and the foot of the pipe forced *'l 
into it to a depth of a few ins, or far enough to shut off quicksand or surface water. *■ 

If quicksaiul is encountered, the string of tools is removed, and the 
sand-pump is used inside of the pipe. 

For reaming out. or enlurging*, boles, or for straightening | 

crooked ones, &<*,, special tools, such as reamers, &c, are substituted in place of the 1,1 


boring bit 

.Special care must be taken to have all tlie rubbing surfaces thor¬ 
oughly lubricated. The pulley in the mast-head, and the pinion-wheels 
of the horse power (if such be used) should be well oiled every two or three hours. 

In very cold or wet weather, a shed of rough boards, or a cover¬ 
ing of canvas, about 8 ft high, should be erected, to protect the men ; and, if steam 
is used, 2 or 3 boards should be used as a covering for the belt, which will slip if wet. 

The following description is based upon the improved machines made by the 
Pierce Well Excavator Co, of New York City, and Long Island City, N Y, 
who make a specialty of artesian-well machinery, and of steam-engines and horse¬ 
powers for its operation. They also sink wells to any reqd depth, in any 
country. 

For holes from 200 to lOOO ft deep, this Co furnish portable drill¬ 
ing machines, to be worked by horse or steam power. In these machines, the 
drill-rope, extending from^he string of tools up out of the hole, passes over a sheave j 
at the top of a wooden mast; down to, and around, a pulley fast to the working 
lever; and thence, by way of a pulley fixed at the foot of the mast, to a drum upon 
which it is wound. To tiiis drum a friction and ratchet wheel is attached, for pay¬ 
ing out the cable as the tools descend. 

The mast is hinged six feet above its foot, so that its upper part may he 
laid hor when the machine is to be moved. When sit work, it is held in position by 
two timber struts or braces, bolted to it near its top, and having their lower ends 
fastened to the drill-jack,” which is a light and strong framework. 9 ft long, 
3 ft wide, and 4 ft high, at the foot of the mast, containing the working lever which 


it 










ARTESTAN WELL BORTNG. 


629 


Fig 4. 


raises the rope and lets it fall, the drum on which the rope is wound, the shaft and 
cam which work the lever, &c. The operator stands at the foot of the mast, and, 
by means of foot- and hand-levers within his reach, regulates 
all the movements of the machine. One of these governs 
the pawl and ratchet wheel regulating the paying out of the 
cable. By letting the ratchet-wheel of the drum move one 
notch, the bit is fed down quarter of an inch. 

The operator, by moving a slide with his foot, holds the 
working lever down, out of reach of the cam, thus stopping 
the up-and-down motion of the rope and tools. By means 
of another lever he can now put the rope-drum in gear with 
the main driving-shaft, so that the rope is wound up on the 
drum, and the tools drawn up out of the hole. Another 
lever controls the separate reel on which the light rope, car¬ 
rying the sand-pump, is wound. All these operations are 
performed by the same power (horse or steam, as the case 
may be), which works on without stopping; the various 
changes being made by merely throwing the different parts 
into, or out of, gear with the main driving-shaft. 

Bne of these portable machines requires 
two horses or a small steam-engine, a man to attend the 
same, and another man to operate the machine, empty the 
sand-pump, change the tools, &c. It can be transported on 
a farm wagon over any common road. Two men can unload 
it, set it up, and commence drilling, in two hours; and, un¬ 
less steam is preferred, the two horses used for its transpor¬ 
tation furnish the motive power. The machine can be taken 
down and reloaded in the wagon in two hours. 

Figs 1 to 4 show the tools used u itli these ma¬ 
chines. For the different sizes of machine they differ 
chiefly in their dimensions and weights. Fig 1 is the 
<1 rilling' bit, called a “ Z ” bit from the shape of its cut¬ 
ting-edge. This edge is 6 ins long. The bit is 30 to 36 ins 
long, and weighs about 100 lbs. Its top is screwed into the 
foot of the “auger-stem,” Fig *2, which is of 3-inch 
round iron, 12 ft long, and weighs 350 lbs. Its use is that 
of a weight, giving additional force to the blows of the bit. 

Its top is screwed into the foot of the drill-jars,” Fig 
3; and to the top of these is screwed the “ rope-socket,” 

Fig 4, to which the drilling cable is attached. If the bit, 
or auger-stem, becomes w edged in llie hole 
by any means, the operator stops the churning motion 
of the tools, and the rope is let out about 12 ins. This per¬ 
mits the upper link II of the drill-jars, Fig 3, to slide down 
about 12 ins in the slot S in their lower link. The churn¬ 
ing motion is then started again, and the upward jerk of 
the link U against the upper end of the slot looseus the 
tools. 

Four sizes of these machines are fur¬ 
nished, as follows : 


Fig 2. 


1 

Depth of hole, ft. 

| Letter. 

| Ht of mast, ft. 

Length of string 
of tools, ft. 

O 

H) . 

co -n 

o ° 
w *-> 

«-> 

Es 

Total wt of ma¬ 
chine, mast, tools, 
rope, &c, but ex¬ 
clusive of power, 
tbs. 

n 

CT 

4> 

u 

c r. 
Q> 
on 

*- 

O 

EE 

Horse power of 
engine reqd. 

Wt of horse pow¬ 
er, lbs. 

Wt of steam-en¬ 
gine, lbs. 

Cost (1886) of ma¬ 
chine complete, 
but exclusive of 
horse power * or 
steam-engiue, $. 

200 

A 

25 

23 

800 

1800 to 2500 

2 

3 

800 

1600 

800 

350 

B 

25 

23 

800 

1850 to 2500 

2 

3 

800 

1600 

875 

600 

C 

35 

30 

1200 

2800 to 3500 

S 

5 


2200 

1000 

1000 

D 

40 

35 

1800 

3800 to 4500 

f« 

10 


3600 

2200 


s 


* The cost of a horse power (1886) is $75. 

For wells from 1000 to 3000 ft deep, a sta- Fig 1. Fig 3. 

tionarv machine, with a walking-beam, is used, similar to 

those employed in the oil regions of Penua. A square pyramidal derrick is erected, 




















































































































630 


ARTESIAN WELL BORING. 


74 ft high, 20 ft sq at base, 4 ft sq at top. Each of its 4 corner legs is of 2 in X 8 ii 
and 2 in X 10 in planks, spiked together so as to form a 10 in X 10 >*> angle-piece | 

2 ins thick. The legs are braced together by hor and diag timbers. The walking 
beam is of timber, 26 ft long, 12 ins wide, and 26 ins deep at the middle of its length 
where it is pivoted to the top of a wooden post 18 ins sq and 12 ft high, called i 
44 Samson post.” This post, at its foot, is dovetailed into the main sill of the ma , 
chine, which is 18 ins wide X 24 ins deep. 

The motive power is a 15-hp steam-engine, which, by means of a belt and pulley | 
crank and pitman, working at one end of the walking-beam, gives to the latter it: 
see-saw motion. To the other end of the beam, and immediately over the well, ii 
suspended, by means of a hook, a “temper-screw.” This last is composed of tw< 
bars of iron, about % X 2 ins, 5 ft long, hung 2 ins apart, fastened together at theii ) 
top ends, at which point there is an eye, which is suspended on the walking-bean: 
hook. At the bottom of the two bars there is a sleeve-nut, and between the twi j 
bars and passing through the nut., is a screw 5 ft long,at the bottom of which thert ' 
is a head, which carries a swivel, set-screw, and a pair of clamps. These grasp the 
cable, 2 or 2^ ins diam, which carries at its lower eud the string of tools. 
This, for a 2000-ft hole, consists of a steel bit, 3 or 4 ft long, weighing 200 to 400 lbs: 
an auger-stem of 4 or 6-inch round iron, from 24 to 30 ft long, and weighing from 
1200 to 2100 lbs; steel-lined drill-jars 8 ft long, weighing 600 to700 lbs; a sinker-bar 
of round iron of same diam as the auger-stem, 12 to 15 ft long, and weighing from 60C 
to 1100 lbs; and a rope-socket, 2% ft long, weighing 200 lbs. Total length of string 
of tools, 50 to 60 ft, total weight, 3000 lbs ; or, for an 8-inch hole in the hardest rock, 
4000 lbs. The sinker-bar is added to give additional wt, and thus to assist in 
pulling the cable down through the water, either in lowering the string of tools or 
in working the drill-jars. The shapes of the other tools are given by Figs 1 to 4. 
Special tools are used for recovering articles that may be accidentally dropped 
into the hole. 

The drilling cable is wound on a drum, called a bull-wheel shaft, at the 
foot of, and inside of, the derrick. While drilling is going on, it passes from the 
bull-wheel shaft loosely over the sheave at the top of the derrick, and down to the 
clamps at the lower end of the temper-screw on the end of the walking-beam. As 
the drilling progresses, the temper screw is turned or fed out by the man in charge, i 
who also, by means of a clamp, twists the rope, so as to change the position of the 
bit after each stroke. 

When the tools are to be lifted out of the hole, the cable is disengaged from the 
clamps on the temper-screw, and is wound upon the bull-wheel shaft, which, for this 
purpose, is thrown into gear with the steam-engine; the pitman being at the same 
time removed from the crank-pin, so that the walking-beam is at rest. As in the 
portable machines, the sand-pump is also raised by the same power which does the 
drilling. 

About lOOOO ft b m of rough lumber are reqd for the derrick, walk¬ 
ing-beam, sills, Ac, and about 3000 ft more for sheds over the boiler, engine,and belt. 

In ordinary hard limestone rock, such a machine will drill about 1^ ft 
per hour under the most favorable circumstances. Tw o men are required ; 
one to attend to the boiler, sharpen the bits, Ac, and one to operate the machine. 
The cost of the apparatus, in 1886, is from $1800 to $2600. Lumber 
for the derrick, $350 to $400 more. The cost of drilling, in 1886, in lime¬ 
stone, is about $6 per ft run for 6-iuch holes; $8 for 8-inch. In granite and trap < 
rock, $10 and $12. 

For quantity of masonry in w alls of w ells, see p 158. 







COST OF DREDGING. 


631 


'f. 


'!» 

«• 

9. 

its 

it 

no 

sir 

10 

no 

« 

lie 

s. 

«i 

m 

w 

DO 

I 

k, 

in 

>r 

i, 

4 

it 

* < 
e 

: COST OF DREDGING, 

s Dredging is generally done by skilled contractors, who own the requisite machines, 
! scows or lighters, &c; and who make it a special’. It is necessary to specify whether 
! the dredged material is to be measured in place before it is loosened; or after being 
) deposited in the scow: because it occupies more bulk after being dredged. It was 
i found, in the extensive dredgings for deepening the River St Lawrence through the 
Lake of St Peter, that on an average a cub yd of tolerably stiff mud in place, makes 
1.4 yds in the scow ; or 1 in the scow, makes .715 in place. Also stipulate whether the 
removal of bowlders, sunken trees, Ac, is to constitute an extra. These often require 
sawing and blasting under water. The cost per cub yd for dredging varies much 
with the depth of water; the quantity and character of the material; the dist to which 
it has to be removed; whether it can be at once discharged from the machine by 
means of projecting side-shoots or slides; or must be discharged into scows, to be re¬ 
moved to a short dist by poling, or to a greater dist by steam tugs; whether it can be 
' dropped or dumped into deep water by means of flap or trap doors in the bottom of 
the hoppers of the scows; or must be shovelled from the scows into shallow water, (at 
say 4 to 8 cts per yd ;) or upon land , (at say from 6 to 10 or 20 cts for the shovelling 
alone, or shovelling and wheeling, as the case may be;) whether much time must be 
consumed in moving the machine forward frequently, as when the excavation is 
narrow, and of but little depth; as in deepening a canal, Ac; whether many bowl¬ 
ders and sunken trees are to be lifted ; whether interruptions may occur from waves 
in storms; whether fuel can be readily obtained, Ac, Ac. These considerations may 
make the cost per cub yd in one case from 2 to 4 times as great as in another. The 
actual cost of deepening a ship-channel through Lake St Peter, to 18 ft, from its orig¬ 
inal depth of 11 ft, for several miles through moderately stiff mud, was 14 cts per 
cub yd in place, or 10 cts in the scows ; including removing the material by steam 
tugs to a dist of about %a mile, and dropping it into deep water. This includes re¬ 
pairs of plant of all kinds, but no profit. It was a favorable case. When the buckets 
work in deep water they do wot become so well filled as when the water is shallower, 
because they have a more vertical movement, and, therefore, do not scrape along as 
great a distance of the bottom. Hence one reason why deep dredging costs more 
per yard ; in addition to having to be lifted through a greater height. Perhaps the 
following table is tolerably approximate for large works in ordinary mud, sand, or 
gravel; assuming the plant to have been paid for by the company; and that common 
labor costs $1 per day. 







632 


COST OF DREDGING. 


Table of actual cost of dredging on a large scale; Includ 
ing dropping the material into scows, alongside; or int< 
side-shoots, on board. Common labor $1 per day. Repair* 
of plant are included : but no profit to contractor. (Original. 


Depth 
in Ft. 

Cts per Yard, 
in place. 

Cts per Yard, 
in scow. 


Depth 
in Ft. 

Cts per Yard, 
in place. 

Cts per Yard 
in scow. 

Less than 10 

8.4 

6 


25 to 30 

18.2 

13 

10 to 15 

0.8 

7 


30 to 35 

25.2 

18 

15 to 20 

11.2 

8 


35 to 40 

35.0 

25 

20 to 25 

14.0 

10 






For towing of the scows by steam tugs to adist of 34 mile, and dropping the mud into deep water, adc 

4 cts per yard in the scow ; for Jsj mile, 6 cts ; for % mile, 8 cts ; for 1 mile, 10 cts Add profit to con 
tractor. On a small scale work is done to a less advantage; and a corresponding increase must be madi 
in these prices. Also, if the contractor himself furnishes the dredgers and plaut, a still further addi 
tion must be made. It is evident that the subject admits of no great precision. Small jobs, even it 
favorable material, but in inconvenient positions, may readily cost two or three times as much peryc 
as the above: and in very hard material, as in cemented gravel and clay, four or live times as wucl 
for the dredging. The cost of towing, however, will remain as before, if wages are the same. 

The cost of dredgers, tugs, Ac, will vary of course with their capabilities, strength of construction 
style of finish, whether having accommodations for the men to live on board or not, Ac. When foi 
use in salt water, the bottoms of both dredgers aud scows should be coppered, to protect them from sea- 
worms; and if occasionally exposed to high waves, both should be extra strong. The most powerful 
machines on the St Lawrence cost about $45000 each ; and removed in 10 working hours on an average 
about 1800 cub yds in place, or 2520 in the scows. Good machines, capable, under similar circumstances 
of doing as much, may, however, be built for about $25000 to $40000. To remove this quantity to ti 
dist of *4 1 mile, would require two steam tugs, costing about $8000 to $10000 each : and 4 to ii scows 

(some to be loading while others are away,) bolding from 30 to 60 cub yds each ; and costing from $80Ci 
to $1500 each at the shop. Scows with two hoppers are best. Such a dredger would require at leasl| 

8 or 10 men, including captain, engineer, fireman, and cook. Each tug 4 or 5 men; and each scow 2 
men. The engineer should be a blacksmith ; or a blacksmith should be added. In certain cases a 1,1 
physician, clerk, assistant engineer, Ac, may be needed. 

Dredgers are often built on the principle of the Yankee Excavator, with but a single bucket or dip- ( 
per, of from 1 to 2 cub yds capacity. Hull about 25 by 60 ft. Draft 3 ft. Cylinder about 7 or 8 ins ,, 
diam ; 15 to 18 inch stroke; ordinary working pressure 50 to 80 lbs per sq inch, according to hardness . 
of material. Cost $8000 to $12000. Will raise as an average days' work (.10 hours) from 200 to 500 !l 
yds in place, or 280 to 700 in the scow, according to the depth, nature of the material. Ac. Require tt 

5 or 7 men in all aboard, including cook. Burn 3$ to 1 ton of coal daily. Tolerably large bowlders, 0] 
aud suuken logs, can be raised by the dipper.* 

When the material is hard and compacted, the buckets of dredgers should be armed with strong 
steel teeth projecting from their cutting edge. On arriving at such material, every alternate bucket 11 
is sometimes unshipped. By arranging the buckets so as to dredge a few feet in advance of the hull,: ti 
low tongues of dry land may be cut away ; the machine thus digging its own channel. The daily t) 
work in such cases will not average half as much as in wet soil. 


On small operations, dredgers worked by two or more liorses, 

instead of by steam, will answer very well in soft material: or even in moderately hard, by reduciug 
the size and number of the buckets. A two-horse machine will raise from 50 to 100 yards of ordinary 
mud in place, or 70 to 140 in the scow, per day, at from 12 to 15 ft depth. 

Soft material in small quantity, and at moderate depth, may he removed by the 
slow and expensive mode of the bag-scoop, or bag-spoon. 



This is simply a bag B, made of canvas or leather, and havine its 
mouth surrounded by an oval iron ring, the lower part of which is 
sharpened to form a cutting edge. It has a fixed handle h, and a 
swivel handle i. One man pushes the bag down into the mud by h, 
while another pulls it along by the rope g; and when filled, another 
raises it by the rope c, and empties it. If the bag is large, a wiud- 
lass may be used for raising it. The men may work from a scow or 
raft properly anchored. Or a long-handled metal spoon, shaped like 
a deeply-dished hoe, may be used by only one man ; or a larger spoon 
may be guided by a man, and dragged forward and backward by a 
horse walking in a circle on the scow, Ac, Ac. 


The weight of a cub y«l of wet dredged mud, pure sand, or gravel, averages 
about 13^3 tons; say 111 lbs per cub ft; muddy gravel, full tons; say 126 lbs per 
cub ft. Pure sand or gravel dredges easily; also beds of shells. Wet dredged clay 
will slide down a shoot inclined at from 5 to 1, to 3 to 1, according to its freedom 
from sand, Ac; but wet sand or gravel will not slide down even 3 to 1, without a free 
flow of water to aid it; otherwise it requires much pushing. 


* The writer has seen cases in which a circular saw for logs in deep water, would have been a 
very useful addition to a dredger. It should be worked by steam; and be adjustable to different 
depths. It would cost but about $500. 

The American Dredging Co, No. 234 Walnut St, Philada, make dredgers of 
many patterns ; aud contract for dredging on any scale. 































FOUNDATIONS, 


633 


it 

s 

J 

l 


FOUNDATIONS. 

A volume might be occupied by this important subject alone. We have space for 
only a few general hints ; leaving it to the student to determine how far they may 
be applicable in any given case. In ordinary cases, as in culverts, retaining walls, 
&c, if excavations, or wells, &c, in the vicinity, have not already proved that the soil 
is reliable to a considerable depth, it will usually be a sufficient precaution, after 
having dug and levelled off the foundation pits or trenches to a depth of 3 to 5 ft, to 
test it by an iron rod, or a pump-auger; or to sink holes, in a few spots, to the depth 
of 4 to 8 ft farther; (depending upon the weight of the intended structure;) to ascer¬ 
tain if the soil continues firm to that distance. If it does, there will rarely be any 
risk in proceeding at once with the masonry; because a stratum of firm soil, from 4 
to 8 ft thick, will be safe for almost any ordinary structure; even though it should 
be underlaid by a much softer stratum. If, however, the firm upper stratum is ex¬ 
posed to running water, as in the case of a bridge-pier in a river, care must be taken 
to preserve it from gradually washing away; or from becoming loosened and broken 
up by violent freshets; especially if they bring down heavy masses of ice, trees, and 
other floating matter. These are sometimes arrested by piers, and accumulate so as 
to form dams extending to the bottom of the stream ; thus creating an increase of 
velocity, and of scouring action, that is very dangerous to the stability both of the 
bottom and of the structure. When the testing has to be made to a considerable 
depth, it may be necessary to drive down a tube of either wrought or cast iron, to 
prevent the soil from falling into the unfinished hole. If necessary, this tube may 
be in short lengths, connected by screw joints, for convenience of driving: and the 
earth inside of it may be removed by a small scoop with, a long handle.f 

Borings in common soils or clay may be made 100 ft deep in a day or two by a 
common wood auger 1)4 ins diam, turned by two to four men with 3 ft levers. This will bring up 
samples. For this and other earth-boring tools see p 626. 

Iu starting the masonry, the largest stones should of course be placed at the bot¬ 
tom of the pit, so as to equalize the pressure as much as possible; and care should 
be taken to bed them solidly in the soil, so as to have no rocking tendency. The 
next few courses at least should be of large stones, so laid as to break joint thoroughly 
with those below. The trenches should be refilled with earth as soon as the masonry 
will permit; so as to exclude rain, which would injure the mortar, and soften the 
foundation. It is well to ram or tread the earth to some extent as it is being deposited. 

If the tests show that the soil (not exposed to running water) is too soft to support 
the masonry, then the pits should be made considerably wider and deeper: and after¬ 
ward be filled to their entire width, and to a depth of from 3 to 6 or more ft, (de¬ 
pending on the weight to be sustained,) with rammed or rolled layers of sand, gravel, 
or stone broken to turnpike size; or with concrete in which there is a good propor¬ 
tion of cement. On this deposit the masonry may be startl'd. The common practice 
in such cases, of laying planks or wooden platforms in the foundations, for building 

t Subterranean caverns in limestone regions are a frequent source of trouble, against which it is 
difficult to adopt precautions. 











634 


FOUNDATIONS. 


upon, is a very bad one. For if the planks are not constantly kept thoroughly wet 
they will decay in a few years; causing cracks and settlements in the masonry. 

Some portions of the circular brick aqueduct for supplying Boston with water, 
gave a great deal of trouble where its trenches passed through running quicksands, 
and other treacherous soils. Concrete was tried, but the wet quicksand mixed itself 
with it, and killed it. Wooden cradles, &c, also failed; and the difficulty was finally 
overcome by simply depositing in the trenches about two feet in depth of strong i 
gravel.* Sand or gravel, when prevented from spreading sideways , forms one of the j 
best of foundations. To prevent this spreading, the area to be built on may be sur¬ 
rounded by a wall; or by squared piles driven so close as to touch each other; or in 
less important cases, by short sheet piles only. But generally it is sufficient simply 
to give the trenches a good width; and to ram the sand or gravel (which are all the 
better if wet) in layers; taking care to compact it well agaiDst the sides of the trench 
also. Under heavy loads, some settlement will of course take place, as is the case 
in all foundations except rock. If very heavy, adopt piling, &c. See Grillage, p 641. 

When an unreliable soil overlies a firm one, but at such a depth j 

that the excavation of the trenches (which then must evidently be made wider, as well as deeper,) ? 
becomes loo troublesome, and expensive; especially when (as generally happens in that case) water j 
percolates rapidly into the trenches from the adjacent strata, we may resort to piles. See p 641. 

When making deep foundation pits in damp clay, we must remember that | 

this material, being soft, has, to a certain degree, a tendency to press in every direction, like water. 1 
This causes it to bulge inward at the sides; and upward at the bottom The excavations for tunnels, I 
or for vertical shafts, often close in all around, and become much contracted thereby before they can 
be lined ; therefore they should be dug larger than would otherwise be necessary. The bottoms of I 
canal and railroad excavations in moist clay are frequently pressed upward by the weight of the sides. I 

Dry clay rapidly absorbs moisture from the air, and swells, producing effects 

similar to the foregoing. Its expansion is attended by great pressure: so that retaining-walls backed 
with dry rammed clay will be in danger of bulging if the clay should become wet. It is a treacherous 

material to work in. For concrete foumlntioiiM. see ps 680 Ac. 

Am to the greatest load that may safely be trusted on an earth founda¬ 
tion, Rankine advises not to exceed 1 to 1.5 tons per sq ft. But experience proves that ou good com- I 
pact gravel, sand, or loam, at a depth beyond atmospheric influences, 2 to 3 tons are safe, or even 4 
to 6 tons if a few ins of settlement may be allowed, as is often the case in isolated structures without 
tremors. Years may elapse before this settlement ceases entirely. Pure clay, especially if damp, is 
more compressible, and should not be trusted with more than 1 to 2.5 tons, according to the case. All 

earth foundations must yield somewhat. Equality of pressure is a main 
point to aim at. Tremor increases settlements, and causes them to continue 

for a louger period, especially in weak soils, great care must be taken not to overload in such cases, I 
even if piled. Foundations in silty soils will probably settle, in years, at the rate of from 3 to 
12 ins per ton (up to 2 tons) per sq ft of quiet load, if not on piles. 

Fig 2 shows an easy mode of obtaining a foundation in certain cases. It is the 
“pierre perdue ” (lost stone) of the French; in English, “ random 
stone.” or rip-rap. 

It is merely a deposit of rough angular quarry stone thrown into the water: the largest ones being 
at the outside, to resist disturbance from freshets, ice, floating trees, &c. A part of the interior may 
be of small quarry chips, with some gravel, sand, clay, <fec. When the bottom is irregular rock, this 
process saves the expense of levelling it off to receive the masonry. For 2 or 3 feet below the surface 
of the water, the stones may generally be disposed by hand so as to lie close and firmly. Small spawls 
packed between the larger ones will make the work smoother, and less liable to be displaced by violence. 
Cramps or chains may at times be useful for connecting several of the large stones together for greater 

stability. l(ip-rap, liowcver, is apt to Mettle. 

If the bottom Im mo yielding aM to be liable to waMh nway in t 

freshets, it may, in addition, be protected, as in Fig 2, by a covering of the same kind 

ing all around the struc¬ 
ture. Or the main pile 
of stones may be extend¬ 
ed as per dotted line at d\ 
so that if the bottom 
should wash away, as per 
dotted line at o, the 
stones d will fall into 
the cavity, and thus pre¬ 
vent further damage. 
Sheet-piles, smay be 
driven as an additional precaution. For greater security, the bed of the river may 
be dredged or scooped under the entire space to be covered by the main deposit, as 
per dotted lines in Fig 3, to as great a depth as any scouring would be apt to reach; 

* Smeaton mentions a stone bridge built upon a natural bed of gravel only about 2 ft thick over- 
lying deep mud so soft that an iron bar 40 ft long sank to the head by its own weight. One of the 
piers, however, sank while the arches were being turned; and was restored by Smeaton. Although 
a wretched precedent for bridge buildiDg, this example illustrates the bearing power of a thick layer 
•>f well-compacted gravel. * 

























FOUNDATIONS. 


635 


et, this excavation also to he filled with stone. Such foundations are evidently best 
adapted to quiet water. The masonry should rest on a strong platform. 





Large deposit ,r of stone, as in these 
two figs, greatly increase the velocity, 
and the scouring action of the stream 
around them, especially in freshets; un¬ 
less the bottom on each side from the de¬ 
posit he dredged out to such an extent 
that the original area of tvater shall not 
he reduced. If the bottom is treacherous, 
this should he done before depositing the 
covering stones c, Fig 2. Judgment and 
experience are necessary in such matters, 
ns in all others connected with engineer¬ 
ing. Mere study will not guard against 
constant failures. Theory and practice 
must guide each other. 

Fig 3 is another simple method; and 
when it does not create too great an ob¬ 
struction to the navigation of the stream, or to the escape of its waters in time of high freshets, is a 
very effective one. Here the piles are first driven into the river bottom, for the support of the pier; 
then the deposit of stone is thrown in, for the support and protection of the piles; preventing them 
from bending under their loads; and shielding them from blows from floating bodies. The tops of 
the piles being cut off to a level, a strong platform of timber is laid on top of them, as a base for the 
masonry. The top of the platform should not he less than about 12 or 18 ins below ordinary low 
water, to prevent decay. Mitchell's iron screw pile; or hollow piles of cast iron, may be used instead 
of wooden ones. See pp 641 to 646. 

Figs 4 represent a convenient method of establishing a foundation in water, by 
means of a timber crib, A A, without a bottom. It should be built of 
squared timbers, notch¬ 
ed together at their 
crossings, as shown at 
Fig 5; each notch being 
% of the depth of the 
stick. By this means 
each timber is support¬ 
ed throughout its entire 
length by the one below 
it; and resists pulling 
in both directions. Bolts 
also are driven at the 
intersections; at least 
in the sides of the crib, 
to prevent one portion 
from being floated off 
from the other. The 
crib is thus divided into 
square or rectangular 
cells, from 2 to 4 or 5 ft 
on a side, according to 
the requirements of the 
case. The partitions 
between the cells are 
put together in the 
same manner as those 
at the sides of the cribs; 
and consequently, like 
the latter, form solid 
wooden walls. 


Fids Jjr 


The crib may be framed afloat, at any convenient spot; and when finished, may be towed to its 
final place, where it is carefully moored in position, and then sunk by throwing stone into a few 
cells provided with platforms, as at c c, for that purpose. These platforms should be placed a little 
above the lower edge of the cells, so as not to prevent the crib from settling slightly into the soil, and 
thus coming to a full bearing upon the bottom. After it has been sunk, all the cells are filled with 
rough stone. A stout top platform may be added or not, as the case may be ; also, a protection, 11, of 
random stone, to prevent undermining by the current. If the sides are exposed to abrasion from 
ice Ac they may be covered in whole or in part with plank, or plate iron ; and the angles strength¬ 
ened hv iron straps, Ac. In deep water, a foundation may be made partly of random stone, as in 
Figs 2 and 3; and on top of this may be sunk a crib, with its top about 2 ft under iow water, as a base 
for the masonry. This is much safer than random stone alone. 

Oil uneven rock bottom it may be necessary to scribe the bottom of the 

crib to fit the rock; or the crib may first be sunk by means of a loaded platform on its top, or by 
filling some of its cells, until its lowest timbers are within a short distance above the bottom. Being 
there kept in a horizontal position, small stones may be thrown into the cells, and allowed to find 
I their wav under the timbers of the crib, thus forming a level support for it. The cells may then be 
filled • and rip-rap deposited outside around the crib to prevent the small stones from being displaced. 













































































































636 


FOUNDATIONS, 


A crib with only an outside row of cells for sinking it may be 

built; and the interior chamber may be filled with concrete underwater. The masonry may theu 
rest on the concrete alone. If the crib rests upon a foundation of broken stone, the upper interstices 
of this stone should first be levelled off by small stone o- coarse gravel to receive the concrete of the 
inner chamber. 

Or a crib like Fig*. 4 may be sunk, and piles be driven in the cells, which 
mav afterward be filled with broken stone or concrete. The masonry may then rest ou the piles only, 
which in turn will be defended by the crib. If the bottom is liable to scour, place sheet-piles or 
rip-rap around the base of the crib. 

By all nicaiiN avoid a crib like e, Fig 5*4, much higher at one part 

than at another, if the superstructure s is to rest on the timber of the crib instead of on 
piles, or on concrete independent of the timber; for the high part of the crib will compress more under 
its load than the low part, and will thus cause the superstructure to lean or to crack. 

A crib either straight sided or circular, with only an outer row of cells for pud¬ 
dling may be used as a cofferdam (see cofferdams, p 637). The joints 

between the outer timbers should be well caulked ; and care be taken, by means of outside pile-planks, 
gravel, &c, to prevent water from euteriug beueath it. 

The caKt-iron Bridge acroHg fhe Schuylkill at C'hestnut St, 

Pliila, Mr. Strickland Kneass, Engineer, affords a striking example of crib 
foundation. The center pier stands ou a crib, an oblong octagon in plan ; 31 by 87 feet at base ; 24 
by 80 ft at top; and (with its platform) 29 ft high. Us timbers are of yellow pine, hewn 12 ins 
square : and framed as at Fig 5. The lower timbers were carefully cut or scribed to conform to the 
irregularities of the tolerably level rock upon which it rests. These were ascertained (after the 8 rt 
depth of gravel had been dredged off) in the usual manner of mooring above the site a large floating 
woodeu platform, composed of timbers corresponding in position with all those of the lower course 
of the intended crib, both longitudinal and transverse. Soundings were then taken close together 
along all these lines of timber. Most of the cells are about 3 by 4 ft on a side, in the clear. A few 
of them had platforms at the level of the second course from the bottom, for receiving stone for sink¬ 
ing the crib ; the others are open to the bottom. 


pie 

It 

Ot 

ic 

fi'i 

Cl 

i 

v 

it 

it 

it 

ti 

in 

k 

bt 

I! 

II 

li 

SI 


i 


t 

f 

] 


The crib was built in the water ; and was kept floating, during its construction, with its unfinished 
top continually just above water, by gradually loading it with more stone as new timbers were added. 
The stoue required for this purpose alone was 300 tons. When the crib w as towed into position, and 
moored, 150 tons more were added for sinking it. All the cells were afterward filled w ith rough dry 
stone, and coarse gravel screenings ; making a total of 1666 tons. A platform of 12 by 12 inch squared 
timber covered the whole; its top being 2ft ft below low water. The pier alone, which stands ou this 
crib, weighs 3255 tons; and during its construction it compressed the crib 64$ ius. The weight of 
superstructure resting ou the pier, may be roughly taken at 1000 tons more. 

Aii ordinary caisson is merely a,strong; scow, or a box with¬ 
out a lid; and with sides which may at pleasure be readily detached from its bottom. It is built on 
land, and then launched. The masonry may first be built in it, either in whole or in part, w hile 
afloat; and the whole being then towed into place, and moored, may be sunk to the bottom of the 
river, to rest upon a foundation previously prepared for it, either’by piling, if necessary ; or by 
merely levelling off the natural surface, &c’. The bottom of the caisson constitutes a strong timber 

platform, upon which the masonry rests; and 
is so arranged, that after it is sunk, the sides 
may be detached front it, and removed to be 
rebottomed for use at another pier, if needed. 
This detaching may be effected bv some such 
contrivance as that shown in Fig 6, where 
P P w is the bottom of the caisson, to w hich ! 
are firmly attached at intervals strong iron 
eyes t; which are taken hold of by hooks d. at 
the lower end of long holts E n, reaching to 
the top timbers S of the crib, where they are 
confined by screw nuts n. By loosening the 
nuts n, the hooks d can be detached from the 
eyes t; and the sides can then be removed 
from the bottom; there being no other connec¬ 
tion between the two. These hooks and eves 
are usually placed outside of the caisson; the 
screw nuts n being sustained by the projecting 
ends of cross pieces, as tt, Fig 9. The ini- ! 
proper position given them in our Fig was 
merely for convenience of illustrating the prin 

,, . 4 , , ciple. It will sometimes be necessary to have i 

one side detachable from the others, in order to float the caisson away clear from the finished pier , 
unless it be floated away before the masonry has been built so high as to render the precaution use¬ 
less. Fig 6 shows one of many ways of constructing a caisson ; with sides consisting of upright 
corner-posts, I; cap pieces S, on top ; and sills g at bottom, resting on the bottom platform PP w ■ 
intermediate uprights T, framed into the caps and sills; the whole being covered outside by one or 
two thicknesses of planking B, which, as well as the platform, should be well calked to prevent 
leaking. Tarpaulin also may be nailed outside to assist in this. The greatest trouble front leaking 
is where the sides join the platform. On top of the platform is firmly spiked a timber o o, extending 
all around it just inside of the inner lower edge of the sides or the caisson. Its use is to prevent 
the sides from being forced inward by the pressure of the water outside. The details of construction 
will of course vary with the requirements of the case. In deep caissons, inside cross-braces or struts 
from side to side, as at c c, Fig 7, will be required to prevent the sides from being forced inward bv 
the pressure of the water, as the vessel gradually sinks while the masonrv is being built within it 
As the masonry is carried up. the struts are removed; and short ones, extending from the sides of 
the caisson to the masonry, are inserted in their place. When the caisson is shallow, onlv the upper 
course of braces will be required, they also support a platform for the workmen and their materials 
In deep caissons, in order not to be in the way of the masons, the outer planking of the sides mar" 
in part, be gradually built up as the masonry progresses. It may sometimes be expedient to build 
the masonry hollow at first, with thin transverse walls inside to stiffen it if Decessary ; and to com- 
























FOUNDATIONS. 


637 


plete the interior after sinking the caisson. Indeed, masonry or brickwork, in cement, may thus be 
“ built hollow at first, resting on the platform; the masonry itself forming the sides of the caisson. 
rl Or the sides may cousist of a water-tight casing of iron, or wood, of the shape of the intended pier, 
e &c. This casing being couflned to the platform, becomes, in fact, a mould, in which the pier may be 
formed, and stink at the same time by filling it with hydraulic concrete. For 
h concrete foundation*, see p 680 &c. 

)r Oil rock bottom the under timbers of the platform may be cut to suit the irregularities 
as already stated under Cribs.” Or the bottom may be levelled up by first depositing large stones 
around the area upou which the caisson is to rest; aud then filling between these with smaller stones 
t aud gravel; testiug the depth by souudiug. Or a level bed of cement concrete may, with care, be 
5 deposited in the water. If there are deep narrow crevices in the rock, through which the concrete 
r may escape, they may be first covered with tarpaulin. Diving bells may often be used to advantage, 
in all such operations. But in the case of very irregular rock, it will often be better to resort to cof¬ 
fer-dams. The draft of a caisson (the depth of water which it draws) whether empty or loaded, can 
be found by page 236. Valves for the admission of water for sinking the caisson are 

3 usually introduced. If, after sinking, it should be necessary to again raise the whole, it is only 
, necessary to close the valves, and pump out the water. Guide "piles may be driven and braced along¬ 
side of the caisson, to insure its sinking vertically, and at the proper spot. Or it may be lowered by 
screws supported by strong temporary framework. 

, Assuming the uprights I, T, &c. Fig 6, to be sufficiently braced, as at cc, Fig 7, the following table 
will show the thickness of planking necessary for different distances apart of the uprights, (in the 
clear,) to insure a safety of six against the pressure of the water at different depths; and at the 
same time not to bend inward under said pressure, more than part of the distance to which 

' they stretch from upright to upright; or at the rate of J4 inch iu 10 ft stretch ; % inch in 5 ft, &c. 

I Such a table may be of use in other matters. 

Table of thickness of white pine plank required not to bend 
more than part of its clear horizontal stretch, under 
different heads of water. (Original.) 


Stretch 
iu Ft. 


40 

HEADS IN FEET. 

30 J 20 | 10 

5 




Thickness in Inches. 


3 


3hs 

3 

2% 


i* 

4 


4 'A 

4 


'1% 


6 


6 H 

6 

5H 


3 H 

8 


9 

8 

7 

5)4 

4 H 

10 


li M 

10 

8 % 

7 


12 


ny 2 

1 2 % 

10 \ 

»y 2 

6% 

15 


161* 

15 

13 

io hi 

8)4 

20 


22)4 

20 

17K 

14 

11 


C’ofTer«daniS are enclosures from which the water may be pumped out, so as 
to allow the work to be done in the open air. Their construction of course varies 
greatly. In still shallow water, a mere well-built bank of clay and gravel; or of 
hags partly filled with those materials when there is much current, will answer 
every purpose; or (depending on the depth) a single or double rowot sheet-piles; or of 
squared piles of larger dimensions, driven touching each other; their lower ends a 
few feet in the soil; and their upper ones a little above high water, and protected 
outside by heaps of gravelly soil or puddle, (as at P in Fig i ,) to pievent leaking. 
The sheet-piles may be of wood; or of cast iron, of a strong form. 

The snfliciency of a mere bank of well-packed earth in still 
wa ter is shown by the embankments or levees, thrown up in all countries, to pre¬ 
vent rivers from overflowing adjacent low lands. The general average ot (he levees 
along 700 miles of the Mississippi, is about 6 ft high; only 3 ft wide on top; side- 
slopes 114 to 1. In floods the river rises to within a foot or less ot their tops ; and 
frequently bursts through them,doing immense damage. They are entirely too slight.. 

The method of a single row of 12 by 12 inch squared piles, driven in contact with 
each other, (close piles,) and simply backed by an outer deposit of impervious soil, 
is very effective; and with the addition of interior cross-braces or struts, like cc tig 
7 to prevent crushing inward by the outside pressure of the water and puddle when 
pumped out, has been successfully employed in from 20 to 25 ft depth of water, in 
which there was not sufficient current to wash away the puddle. 1 he cross-braces 
are inserted successively, as the water is being pumped out; beginning, of course, 
with the unDer ones. The ends of these braces may abut on longitudinal timbers, 
bolted to the [tiles for the purpose. Another method is a strong? crib, com¬ 
posed of uprights framed into caps and sills; and covered outside with squared 
timbers or plank, laid touching each other, and well calked ; as in the caisson, b lg 
f, but without a bottom. Between t he opposite pairs ol uprights are strong interior 
struts, as c c, Fig 7, reaching from side to side, to prevent crushing mwaid. Ihe 
















638 


FOUNDATIONS 


upper series of these usually supports a platform for the workmen, windlasses, &c. 
The crib having been built on land, is launched, taken to its final place, and sunk by 
piling stones on a temporary platform resting on the cross-struts ; the bottom of the 
stream having been previously levelled off, if necessary, for its reception. 

To prevent leaking under the bottom of the crib, sheet-piles may be driven around it, their heads 
extending a few feet above its bottom; or a small deposit of outside puddle may be placed around it. 
as shown at the stone deposits 11, Fig 4. Or a broad flap of tarpaulin may be closely nailed arouud 
and a little above the lower edge of the crib; so arranged that it may be" spread out loosely on the 
river bottom, to a width of a few feet all around the outside of the crib: and the puddle may be placed 
upon it. Such a tarpaulin is also very useful in case the river bottom is somewhat irregular, and 
cannot be levelled off without too great expense; in which case the crib cannot come to a full bearing 
upon it; and consequently the water would leak or flow beneath freely. It is especially adapted to 
uneven rock ; where sheet-piles cannot be driven. An artificial stratum of impervious soil may, how¬ 
ever, be deposited on bare rock : in which case the sinking of the crib, and the subsequent operations 
will be the same as on a natural stratum. These expedients are evidently more or less applicable in 
other cases, where, to avoid repetition, they are not specially mentioned. 



S Plan ci( one end. s 


Fig 1 7 is another cril» coder-flam : in which the sides, instead of being 
planked longitudinally, as in the last instance, are sheathed with vertical sheet-piles 
s, driven after the cril» is sunk. It is much inferior to the last, owing to its greater 
liability to leak In one of this description, Fig 7, successfully used in lfi ft water, 
the dimensions of the crib were 34 ft by 8(i ft. Along each long side were 7 uprights t , t, 
l't ft long, 12 ins square, 12% ft apart. Into each opposite pair of these were notched, 
and held by dog-irons, 6 cross-braces c c, of 12 ins square. The distance between the 
two upper ones was 3 ft in the clear; gradually diminishing to 18 ins between the 
two lower ones, on account of the increased pressure of the water in descending On 
the outside of the uprights, and opposite the ends of the braces, were bolted longi- 


I 




tudinal timbers to support the outside pressure against the 3-inch sheet-piling xs. 
Other longitudinal pieces o o, confine the heads of the sheet-piles to the top of the 
crib after they are driven. The feet of the sheetpiles were cut to an angle, as at m; 
to make them draw close to each other at bottom in driving. 

The sheet-piles will drive in a far more regularand satisfactory manner, with the 
arrangement shown in Figs 8. Here o o are the uprights; c c are pairs of longitudinal 





























































FOUNDATIONS. 


639 


pieces, notched and bolted to the uprights, near both their tops and their feet; and 
at as many intermediate points as may be desired. The sheet-piles I, are inserted 


between these; and of course are guided during their descent much more perfectly 
than in Fig 7. The crib at top of p 636 may be used as a cofferdam. 


When the current is too strong to permit the use of outside puddle, P, Fig 7, th« 
principle of coffer-dam shown in Fig 9, is generally used ; in which both sides of the 
puddle are protected from washing away. The space to be enclosed by the dam is sur¬ 
rounded by two rows of firmly-driven main piles p p, on which the strength chiefly 
depends. They may be round. In deciding upon their number, it must be remem¬ 
bered that they may have to resist floating ice, or accidental blows from vessels, &c. 
M ith reference to this, extra fender -piles may be driven. A little below- the tops of 
the main piles are bolted two outside longitudinal pieces w w, called wales; and oppo¬ 
site to them two inner ones, as in the fig. The outer ones serve to support cross- 
timbers 11, which unite each pair of opposite piles, and steady them ; and prevent 
their spreading apart by the pressure of the puddle P. The inner ones act as guides 
for the sheet-piles s s, while being driven; after which the heads of the sheet-piles 
are spiked to them. In deep water these sheet-piles must be very stout, say 12 ins 
square ; to resist the pressure of the compacted puddle. 

A gangway m, is often laid on top of the cross-pieces t f, for the use of the 
workmen in wheeling materials, &c. The puddle P is deposited in the water in the 
space, or boxing, between the sheet-piles. It should be put in in layers, and com¬ 
pacted as well as can be done without causing the sheet-piles to bulge, and thus open 
their joints. The bottom of the puddle-ditch should be deepened, as in the fig, in 
case it consists, as it often does, of loose porous material which would allow water to 
leak in beneath it and the sheet-piles. This leaking under the dam is frequently a 
source of much trouble and expense. Water will find its way readily through almost 
any depth and distance of clean coarse gravelly and pebbly bottom, unmixed with 
earth. Sand is also troublesome; and if a stratum of either should present itself ex- 
• tending to a great depth, it w ill generally be expedient to resort to either simple 
cribs, F'ig4; or to caissons; with or without piles in either case, according to cir¬ 
cumstances. But if such open gravel, or any other permeable or shifting material, 
1 as soft mud, quicksand, &c, is present in a stratum but a few feet in thickness, and 
j underlaid by stiff clay, or other safe material, leaking may be prevented, or at least 
, much reduced, by driving the sheeting-piles 2 or 3 ft into this last; and by deepening 
the puddle-trench to the same extent. It may sometimes be better, and more con¬ 
venient, to dredge away the bad material entirely from all the space to be enclosed 
by the dam, and for a short distance beyond, before commencing the construction of 
the latter. If the dam, Fig 9, is (as it should be) well provided with cross-braces, 
like c c, Fig 7, extending across the enclosed area, the thickness or width o n of the 
puddle, need not be more than 4 or 5 feet for shallow- depths ; or than 5 to 10 ft for great 
ones: because its use is then merely to prevent leaking. But if there are no braces, 
it must be made wider, so as to resist upsetting bodily; and then,with good puddle, 
o » may, as a rule of thumb, be % of the vertical depth o l below high water; except 
when this gives less than 4 ft: in which case make it 4 ft; unless more should be 
required for the use of the workmen, for depositing materials, &c. Or if the excavation 
for the masonry is sunk deeper than the puddle, the dam must be wider; else it may 
be upset into the excavated pit. 


PLAN 


Tlie exeavaled soil may be 

raised in buckets by windlasses, or by hand, in 
successive stages. The pumps may be worked 
by hand, or by steam, as the case may require; 
as also the windlasses generally needed for 
lowering mortar, stone, &c. More or less leak¬ 
ing may always be anticipated, notwithstanding 
every precaution. 

Where a coffer-dam is exposed to a violent 
current, and great danger front ice, <fec, the ex¬ 
pensive mode shown in Figs 10 may become 
necessary. The two black rectangles c c, repre¬ 
sent two lines of rough cribs filled with stone, 
and sunk in position; one row being enclosed 
by the other; with a space several feet wide be¬ 
tween them. Sheet-piles p p are then driven 
around the opposite faces of the two rows of 

cribs; and the puddle is deposited within the boxing thus provided for it, as shown in the fig. 



Where the current is not strong enough to wash away gravel backing, we may, on rock especially, 
enclose the space to be built on, hv a single quadrangle of cribs sunk by stone; and after adopting 
precautions to prevent the gravel from being pressed in beneath the cribs, apply the backing.* 

Figs 10% show the plan, outside view, and transverse section, to a scale of 20 ft to 
an inch, of a coffer dam on rock, in 8 to 9 ft water, used successfully on the Schuylkill 
Navigation. 


* A pure clean coarse gravel is entirely unfit for such purposes, 
earth is essential for preventing leaks. 


A considerable proportion of 







































640 


FOUNDATIONS. 



Valuable hints for coffer-dams may be taken from what is said under “ Dams,” pp 
282 etc, where Fig 1 affords useful suggestions for coffer-dams also, on rock in shallow 
water. 

The mooring of larg-e caissons or cribs, preparatory to sinking 

them, is sometimes troublesome, especially in strong currents. It may be neces¬ 
sary to drive clumps of piles; or to temporarily sink rough cribs filled with stone, 
to which to attach the long guide-ropes by which the manoeuvring into position, Ac, 
is done. Frequently dams are left standing after the work is done ; if not in the way 
of navigation, or otherwise objectionable; inasmuch as the materials are rarely worth 
the expense of removal. But if removed, the piles should not be drawn out of the 
ground; but be cut off close to river bottom ; for if drawn, the water entering their 
holes may soften the soil under the masonry. It is often expedient to drive two 
rows of piles from the dam to the shore, for supporting a gangway for the workmen; 
or even for horses and carts; or for a railway for the easy delivery of large stones, Ac. 

folfiw-dains may be sunk tlirougii a soft to a firm soil, in' 
shape of a box of cribwork, either rectangular or circular, and without a bottom. 
This being strongly put together, and provided with proper temporary internal 
bracing, (to be gradually removed as the masonry is built up,) is floated into place; 
and after being loaded so as to rest on the soft bottom, is sunk by dredging out the 
soft material from inside. Additional loading will sometimes be required for over¬ 
coming t he friqtion of the soil against the outside; or it may even become necessary 
to dredge away some of the outer material also. On rock it may at times be 
expedient to drill holes in deep water, for receiving the ends of piles, or of iron rods, 
Ac. This may be done by means of long drill-rods, working in an iron tube or pipe 
sunk as a guide to the rod; with its lower end over the spot to be bored. Or a diving- 
bell may be used. Or a cy Under of staves 4 to 12 inches thick, long enough to 
reach above the surface, and having a broad tarpaulin fiapor apron around its lower 
edge, to be covered with gravel to prevent leaking: maybe sunk, and the water 
pumped out, to allow a workman to descend, and work in the open air. 


on rock. Uprights 6, about 1 ft square, and 10 ft apart from center to oenter 
along the aid" <"the dan.; and 10 ft in the clear, transversely of the dam, support two lines of hori¬ 
zontal stringers, t f; inside of which are the two lines of sheeting piles, s «, enclosing between them 
a width of 7 ft of gravel puddle. Two flat iron bars (t f, of the transverse section) tie together each 
pair of uprights 6 ft. These bars are % inch thick, by 2^ ins deep, aud 9 ft long. Their hooked ends 
fit into eye-bolts c, which pass through the uprights b ; outside or which they are fastened by keys, k, 
'see detail sketch.) Between the keys and ft, were washers. At the corners of the dam (see plan) 
were additional tie-bars, as shown. A small band or straw, as seen at y, wrapped around the tie- 
bars just inside of the sheet-piles : and kept in place by the puddle ; effectually prevented the leaking 
which generally proves so troublesome in such cases. The stout oblique braces, o o, were merely 
spiked to the outside faces of the uprights 6. They are not shown in the transverse section. This dam 
was built ou shore: in sections 30 to 40 ft long. These were floated into place, and weighted down, 
sheet piled and puddled with gravel. The dam had sluices by which water was admitted when 
necessary for preventing the outside head from exceeding 9 ft. The lengths of the uprights 6 b were 
first found by careful soundings. 


PLAN 


I*iles. When driven in close contact.as in Fig 11, for preventing leakage; for 
confining puddle in a coffer-dam : or for enclosing a piece of soft or sandy ground, to 
prevent its spreading when loaded; or if the outside soil should wash away from 


of pilOS delivered at wharf, Pbilada, 1886. Hemlock, 5 to 6 cts per foot lineal, Bay 
yellow pine, up to 35 ft, 6 cts, Southern yellow piue, 10 cts. 









































FOUNDATIONS. 


641 


around them, &c, they are called sheet-piles, 
they are wide; but fre- 


Generally these are thinner than 

t X 


quently they are square; 
and as large as bearing 
piles; and are then called 
dose piles. To make 
them drive tight to¬ 
gether at foot, they 
are cut obliquely as at 
f. Occasionally, when 
driven down to rock 
through soft soil, their 
feet are in addition cut 
to an edge, as at i, so as 
to become somewhat 
bruised when they reach 
the rock, and thus fit closer to its surface. Their heads are kept in line while driv- 
i n g> 5, y means of either one or two longitudinal pieces a and o, called wales or 
stringers. These wales are supported by gauge-piles, or guide-piles, previously driven 
in the required line of the work, and several ft apart, for this purpose. See Figs 8. 



A dog-iron d, of round iron, may also be used for keeping 
the edges of the piles close at top to those previously driven, both 
during and after the driving. Its sharp euds, cc, being driven into 
the tops of the wales ww, (shown in plan,) it holds the descending 
pile o firmly in place. Atw, d,p, Fig 11, are other modes occasionally 
used for keeping the piles in proper line. At^, the letters ss denote 
small pieces of iron well screwed to the piles, a, little above their feet, 
to act as guides; very rarely used. At m are shown wooden tongues 
tt, sometimes driven down between the piles after they themselves 
have been driven; to assist in preventing leaks. In some cases 
sheet piles are employed without being driven. A trench is first 
dug to their full depth for receiving them ; and the piles are simplv 
placed iu these, which are then refilled. Closer joints can be secured 



Fig 1.2 

in this manner than by driving. 


When piles are intended to sustain loads on their tops, whether driven all their 
length into the ground, or only partly so, as in Fig 3, they are called bearing? 
piles. They are generally round; from 9 to 18 ins diam at top; and should be 
straight, but the bark need not be removed. White pine, spruce, or even hem- 
> lock, answer very well in soft soils ; good yellow pine for firmer ones; and hard 
' oaks, elm, beach, &c, for the more compact ones. They are usually driven from about 
to 4 ft apart each way, from center to center, depending on (he character of the 
! soil, and the weight to he sustained. A i read-wheel is more economical than 
• the winch for raising the hammer, when this is done by men. Morin found that 
i the work performed by men working 8 hours per day. was 3900 foot-pounds per man, 
i per minute by the tread-wheel; and only 2600 by a winch. 


1 After piles have been driven, and their heads carefully sawed off to 

‘ a level, if not under water, the spaces between them are iu important cases filled up level with their 

tops with well rammed gravel, stone spawls, or concrete, iu order 
to impart some sustaining power to the soil between the piles. Two 
courses of stout timbers, (from 8 to 12 ins square, according to the 
weight to be carried) are then bolted or treeuailed to the tops of the 
piles' and to each other, as shown in the Fig, formiug what is called 
a grillage. On top of these is bolted a floor or plat¬ 
form of thick plank for the support of the masonry; or the timbers 
of the upper course of the grillage may be laid close-together to form the floor. The space below the 
floor should also, iu important cases, be well packed with gravel, spawls, or concrete. 
If under water, the piles are sawed off by a diver, or by a circular saw driven 
by the engine of the pile-driver, and the grillage is omitted. Instead of it the masonry or coucrete 
may be built iu the open air iu a caisson, which gradually sinks as it becomes filled ; or on a 

strong platform which is lowered upon the piles by screws as the work progresses. Or a strong 

caisson may first be sunk entirely under water, and then be filled with concrete, up to near 

low water; the caisson beiug allowed to remain. Or the caisson may form a cofferdam, to be first 
sunk, and then pumped out. If the ground is liable to wash away from around tlx; piles, as in the 
case of bridge piers, &c, defend it by sheet-piles, or rip rap, or both. 

The cost of a floatin'? steam pile driver, in Phiiada, scow 24 ft by 50 ft, 

lraft 18 ins, with oue engine for driving, and one (to save time) for getting another pile ready ; with 
me ton hammer, is about $6000; and foOO more will add a circular saw, &c, for sawing oft’ piles at any 
•eqd depth. Requires engineman, cook, and 4 or 5 others. Will burn about half a ton of coal per day. 
Driving 20 ft into gravel, and sawing off. will average from 15 to 20 piles per day of 10 hours. In 
uud about twice as many. On laud about half as many as in water. 



The gunpowder pile driver invented by Mr. Thomas Shaw, the well-known 
mechanical engineer of Phiiada, is a very meritorious machine. The hammer is worked by small 
jartridges of powder, placed one f>y one in a receptacle on top of the pile; and exploded by the ham¬ 
mer itself. It can rcadilv make 30 to 40 blows of 5 to 10 ft, per minute ; and, since the hammer does 
oot come into actual contact with the piles, it does not injure their heads at all; thus dispensing 
ivitil iron hoops kc, for preserving them. When only a slight blow is required, a smaller cartridge 

49 






































642 


FOUNDATIONS. 


is used. To drive a pile 20 ft into mud averages about one-third of a pound of 
powder; into gravel, 4 times as much. This machine does not assist in raising the 
pile, and placing it in position, as is done by ordinary steam pile drivers; the latter, 
however, average hut from 6 to 14 blows per minute. 

I*ilcs liave been driven by exploding small charges of dynamite 
laid upon their heads, which are protected by iron plates. 

Steam-hammer pile drivers, operating on the principle of that devised 
by Nasmyth about 1850, are economical in driving to great depths in difficult 
soils where there are say 200 or more piles in clusters or rows, so that the machine 
can readily be moved from pile to pile. 

The steam cylinder is upright, and is confined between the upper ends of two 
vertical and parallel I or channel beams about 6 to 12 ft long and 18 ins apart, 
the lower ends of which confine between them a hollow conical bonnet cast¬ 
ing 1 ,” which tits over the head of the pile. This casting is open at top, and through 
it the hammer, which is fastened to the foot of the piston-rod,* strikes the head or 
the pile. Each ot the vertical beams encloses one of the two upright guide-timbers, 
or “leaders,” of the pile driver, between which the driving apparatus, above tie- 
scribed, is free to slide up or down as a whole. 

When a pile has been placed in position, ready for driving, the bonnet casting is 
placed upon its head, thus bringing the weight of the beams, cylinder, hammer,and 
casting upon the pile. This weight rests upon the pile throughout the driving, the 
apparatus sliding down between the leaders as the pile descends. 

The steam is conveyed from the boiler to the cyl by a flexible pipe. When it is 
admitted to the cyl, the hammer is lifted about 30 or 40 ins, and upon its escape the 
hammer falls, striking the head of the pile. About 60 blows are delivered per min¬ 
ute. The hammer is provided with a trip-piece which automatically admits steam 
to the cylinder after each blow, and opens a valve for its escape at the end of the 
up-stroke. By altering the adjustment of this trip-piece, the length of stroke (and 
thus the force of the blows) can be increased or diminished. The admission and 
escape of steam, to and from the cyl, can also be controlled directly by the attendant. 
The number of blows per minute is increased or diminished by regulating the sup¬ 
ply of steam. 

In making the up-stroke, the steam, pressing against the lower cyl head, of course 
presses downward on the pile and aids its descent. 

Tbc chief advantage of these machines lies in the great rapidity 
with which the blows follow one another, allowing no time tor the disturbed earth, 
sand, &e, to recompact itself around the sides, and under the foot, of the pile. This 
enables the machines to do work which cannot be done with ordinary pile drivers. 
They have driven Norway pine piles 42 ft into sand. They are less liable than 
others to split and broom the pile, so that these may be of softer and cheaper wood. 
The bonnet casting keeps the head of the pile constantly in place, so that the piles 
do not “ dodge ” or get out of line. Their heads have, in some cases, been set on fire 
by the rapidly succeeding blows. 

These machines consume from 1 to 2 tons of coal in 10 hours, and 
require a crew of 5 men. They work with a boiler pressua-e of from 

50 to 75 lbs per sq inch. 

They are marie by R. J. & A. B. Cram, 80 Griswold St, Detroit, Mich, and 
by Vulcan Iron Works, 86 N. Clinton St, Chicago, Ill. 

The largest (No. 1) Vulcan machine has a hammer weighing' about 3640 
lbs; stroke. 36 ins; riiam of cyl. 12 ins; space between leaders, 19*4 ins. 
Cost (1886) including hose, fittings, &c, but exclusive of hoisting engines and boiler, 
about $1075 on cars at works. 

The Vulcan Works make a railroad car, fiirnishcri with a pi le-ri ri v in« 
machine, and 2 hor engines, with boiler, for raising the hammers. Either the 
Nasmyth, or the ordinary, hammer may be used. The car can be made self-pi 
pelling when desired. It is larger than an ordinary platform car, and is provided 
with lateral supports to enable it to drive piles to the right or left of the road bed. 
The leaders are hinged, so that they can be laid horizontally upon the car when not 
in use. 


tie 

,iit 

lime 

[riel 

ills’ 

.eep 


111 

}if! 
] till 
| to 
■it 


lull 

ill 


fills 

M 

»P 


«K 

El 


11 

ers' 


Oi 

Dd 


a 

ue 


Or 

)a-i 


eti 


toh 

cn 


lloi 

Al 


n?r 

r* 


;|f r 


Ai 

nd 


*In the Cram machine the hammer is fastened to the lower end of the cylinder, 
instead of to that of the piston-rod. The cylinder rises and falls with the hammer, 
and its weight is thus utilized in increasing the force of the blows. Steam is sup¬ 
plied to the cylinder through the piston-rod, which is made hollow for this purpose. 
The piston-rod is fixed in place between the vertical beams by means of a cross-head 
at its top. The piston is fastened to the foot of the piston-rod. 


m 
net 
I Dill 


: m 
n 








FOUNDATIONS, 


643 


id 

itl 

li 

sTi 

i( 

ell 


Rules for ttie Sustaining' Power of Piles. 

They differ very much. No rule can apply correctly to all conditions. The ground itself between 
the piles, in most cases, supports a part of the load ; although the whole of it is usually assigned to 
the piles. Again, in very clayey soils, there is greater liability to sink somewhat with the lapse of 
time, iu cousequence of the admission of water between the pile and the clay; thus diminishing the 
frictiou between them. The less firm the soil, the more will the piles lie affected by tremors ; which 
also tend in time to cause sinking. In some cases this sinking will not be that of the piles settling 
deeper into the earth around them ; but that of the entire compacted mass of piles and earth into 
which they were driven, settling down into the less dense mass below them. Piles are sometimes 
blamed for settlements which are really due to the crushiug (flatways) of the timbers which rest 
immediately upou their beads. See p 430, 


fi lii the fine London bridge across the Thames, each pile under some of the 
ij, piers sustains the very heavy load of 80 tons. They are driven hut 20 feet into the 
a stiff, blue London clay ; and are placed nearly 4 ft apart from center to center; which 
•in is too much for such piers and arches. At 3 ft apart scant, they would have had but 
ad 45 tons to sustain. They are 1 ft in diam at the middle of their length. Ugly set- 
iln tlements, some of them to the extent of about a ft, have occurred under these piers. 
Rlackfriars bridge, in the same vicinity, exhibits the same defect. By some 
this is ascribed in both cases to the gradual admission of water between the clay and 
the piles, perhaps by capillary action of the piles themselves; or perhaps hv direct 
leaking. It may, however, be owing in part to the crushing of the platforms on 
top of the piles; or to a bodily settlement of the entire mass of piled clay, into 
the uupiled clay beneath, under the immense load that rests upon it. This here 
i amounts to 5^ tons per sq foot of area covered by a pier; and is probably too much 
!f . to trust upon damp clay, when even the slightest sinking is prejudicial. 




Maj J. Sanders, U. S. Engs, experimented largely at Fort Delaware in river 
mud; and gave the following in the Jour. Franklin Inst, Nov 1851. For the safe 
load for a common wooden pile, driven until it sinks through only small and 
nearly equal distances, under successive blows, divide the height of the fall in ins, 
by the small sinking at each blow in ins. Mult the quot by the weight of the 
hammer, ram, or monkey, in tons or pounds, as the case may be. Divide the 
prod by 8, He does not state any specific coefficient of safety. 


on 

lid 

•ai 

1 


Example. At the Chestnut St Bridge, Philada, the greatest weight on any pile is 18 tons. 
Mr Kneass had the piles driven until they sauk %, or .75 of an inch under each blow from a 1200 ft 
hammer, falling 20 ft. Was he safe in doing so? Here we have the fall in ius= 20 X 12 = 240. And 
240 384000 

= 3201 and 320 X 1200 = 384000 B)s ; and -^-= 48000 lbs, =: 21.4 tons safe load by Maj San¬ 

ders’ rule. The soil was river mud. 


it 

t! 

M 

P 

II 


J 

fi 


! 

; 

»i 

ii 


•I 

fi 

•l 

i 


■ 

ii 

» 


Our own rule is as follows. Mult together the cube rt of the fall in ft; the wt of hammer in fts ; 
and the decimal .023. Divide the prod by the last sinking in ins. -f- 1. The quotient will be the 
extreme load that will be just at the point of causing more siuking. For the safe load take from 
oue twelfth to one half of this, according to circumstances. Or, as a formula, 

Cube rt of w Wt of hammer v 0 ,, 3 
Extreme load _ fall * n feet in pounds 

iu tons ~ Last sinking in inches -j- 1 


Example. The same as the foregoing at Chestnut St Bridge. Here the cube rt of 20 ft fall is 
2.714 ft. Hence we have 


Extreme load — 

iu tons 


2.714 X 1200 X .023 
/75 + 1 


74.9 

42.8 tons. 

1.7o 


Or say half of this, or 21.4 tons, the load for a safety of 2. Major Sanders* rule makes the safe 
load 21.4. The actual one is 18 tons. 

A safety of 2 is not enough for river mud. See “ Proper load for safety,” p 644. 

But although Major Sauders’ rule and our own agree very well in this instance if a safety of 2 be 
taken for each, they differ widely iu some bthers. Thus at Neullly Bridge, France, Perronet’s 
heaviest hammer weighed 2000 lbs, fall 5 ft, sinkage .25 of an iucb iu the last 16 blows; or say .016 
inch per blow. The piles sustain 47 tons each. Our rule gives 38.8 tons for a safety of 2; while San¬ 
ders' rule gives 515 tons safe load ! If, as we think probable, there was no actual sinking at the last 
blow, then our rule gives 39.3 tons for a safety of 2 ; while Sanders' gives infinity. 

At the Hull Docks, England, piles 10 ins square, driven 16 ft into alluvial mud. by a 1500 ft ham¬ 
mer, falling 24 ft, sank 2 ins per blow at the end of the driving. They sustain at least 20 tons each, 
or according to some statements 25 tons. Our rule gives 33.2 tons for the extreme load ; or 16.6 for a 
safety of only 2. Sanders gives for safety 12.06 tons. As before remarked, 2 Is not safety enough for 
mud.’ In mud, it is not primarilv the piles, but the piled soil that settles, bodily, for years. 

At the Uoyal Border Bridge, England, piles were very firmly driven from 30 to 40 ft in sand 
and gravel, iu some cases wet. Pine was first tried, but it split and broomed so badly under the hard 
driving, that American elm was substituted, with success. They were driven until they sank but .05 
inch per blow, under a 1700 ft monkey, falling 16 ft. They support 70 tons each. Our rule gives 47 
tous for a safety of 2 ; while Sanders gives 364 tons safe load I 

11 is the writer's opinion, however, that the piles did not aclually sink, as was (and always is, in 
such cases) taken for granted by the observers ; but that they were merely compressed or partially 
crushed by overdriving. Most of the piles were driven until they sank (7) only an inch under 150 
blows; but we doubt whether thev were any safer, or farther in the ground, than wheD they had re¬ 
ceived only one of them ; and consider suoh extreme precaution worse than useless. 

In gome experiments (187.3)-at Philada, a trial pile was driven 15 ft into soft river mud. bv a 
1600 1b hammer ; its last sinking being 18 ins under a fall of 36 ft. Ouly 5 hours after it was driven 
it was loaded with 6 5 tons ; which caused a siuking of but a very small fraction of an iuch. Our rule 










644 


FOUNDATIONS, 


By 


gives 6.4 tons as the extreme load. Under 9 tons it sank .75 of an inch ; and under 15 tons, 5 ft 
Maj Sanders' rule its safe load would be 2.14 tons. 

A U. 8. Govt trial pile, about 12 ins sq, driven 29 ft through layers of silt, sand, and clay, ham¬ 
mer 910 lbs, fall 5 ft, last sinking .875 of an inch, bore 26.6 tons ; but sank slow ly under 27.9 tons. 
Our rule gives 26 tons extreme load. 

French engineers consider a pile safe for a load of 25 tons, when it is driven to the refusal of 
1844 lbs, falling 4 ft; our rule gives 24.2 tons for safety 2. They estimate the refusal by its not sink¬ 
ing more than .4 of an inch under 80 blows. In many important bridges &c they drive until there is 
no sinking under an 800 lb hammer, falling 5 ft. Our rule here gives 81.5 tons extreme load ; or 15.7 
for safety 2. 

As to the proper load for safety, we think that not more than one-half the extreme load given 
by our rule should be taken for piles t/mroui/hly driven in firm soils: nor more than one-sixth when 
in tiver mud or marsh ; assuming, as we have hitherto done, that their feet do not rest upon rock. 

If liable to tremors, take only half these loads. 


5 


Piles may be made of any required size as regards either length or cross section, by bolt¬ 
ing and fishing together sidewise and leugtuwise, a number of squared timbers. 

Piles with blunt ends. At South Street Bridge, Phila, 1200 stout piles of Nova Scotia spruce 
with blunt ends were driven 15 to 35 ft, partly in strong gravel, by a common steam pile driver, at a 
total cost (piles and driving) of $7 to $8 each. At Wiiinfugton Harbor, Cal, Mr. C. B. Sears, 
U. S. Army. (Jour. Am. Soc. C. E., Dec 1876) found that in firm compact wet sand, after the first few 
blows the piles would not penetrate more than .5 to 1.5 ins at a blow, no matter how far the 2400 lb 
hammer fell. The unpointed ones of which there were many thousands, drove quite as readily to aver¬ 
age depths of 15 ft in this sand as the pointed ones, and with much less tendency to cant. As a higlt 
fall had no farther effect than to batter the heads he reduced it to 10 ft. which drove an average of 
about, .72 inch to a blow. To insure straight driving, the ends must be at right angles to the length. 
Instead of driving piles to moderate depths it mav at times tie better to mereh plant them butt 
down in holes bored by an auger like Pierce’s Well Borer. See p 626. This will atoid shaking 
adjaceut buildings. See “ In Mobile Bay," p 646. 

The ultimate friction of piles even with the bark on, and driven about 3 ft apart from C"i 
to cen probably never much exceeds about 1 ton per sq ft even when well driven into dense moist 
sand or loamy gravel; nor more than .5 to .75 of a ton in common soils and clays; or than .1 to .2 
of a ton in silt or wet river mud depending on the depth and density. 

The friction of east iron cylinders seems to be about .3 that of piles. 

There is a jjreat difference in the penetrability of different 

sands. Thus, in the Lary bridge, no special difficulty was found in driving piles 35 ft into deep wet 
sand ; while, in other wet localities, piles of very tough wood, well shod with iron, canuot be driven 
6 ft into sand, without being battered to pieces. The same difference has been found in the case of 
screw-piles. At the Brandywine light house these could not he forced more than 10 ft into the clean 
wet sand. Stiff wet clay (and clean gravels) also differ very much in this respect. Generally they 
are penetrable to any required depth with comparative ease; but we have seen stout hemlock piles 
battered to pieces itt driving 6 ft through wet gravel; and Mr. Rendel found that at Plymouth he 
‘could not by any force drive screw-piles more than about. 5 ft into the clay, which is not as stiff as 
the London clay,” on which the forementioned new London and Blackfriars bridges were founded; 
and into which even ordiuary wooden piles were driven 20 ft without special difficulty. 

A mixture of mud with the sand or gravel facilitates driving very much ; but before beginning an 
extensive system of piling, a few experimental ones should be driven, to remove doubt as to the 
trouble and expense that may be anticipated. Mere boring will often be but a poor substitute for this. 

As a general rule, a heavy hammer witlt a low fall, dri\es more pleasantly than a light one w ith a 
high fall. Where a hammer of % ton (1500 lbs) falling 25 ft, in a very strong ground, shattered the 
piles; one of 2 tons, (4500 ft>s.) with 7 ft fall, drove them satisfactorily. More blows can be made in 
the same time with a low fall; and this gi ves less time for t he soil to compact itself around the piles 
between the blows. At times a pile may resist the hammer after sinking some distance; but start 
again after a short rest; or it may refuse a heavy hammer, and start under a lighter one. It may 
drive slow ly at first, and more rapidly afterward, from causes that may be difficult to discover. The 
driving of one sometimes causes adjacent ones previously driven, to spring upward several feet. A 
pile is in the most favorable position when its foot rests upon rock, after its entire length has been 
driven through a firm soil, which affords perfect protection against its bending like an overloaded 
column; and at the same time creates great friction against its sides; thus assisting much in sus¬ 
taining the load, and thereby relieving the pressure upon the foot. A pile ntay rest upon rock, and 
yet be very weak ; for if driven through very soft soil, ail the pressure is borne by the sharp point; 
and the pile becomes merely a column in a worse condition than a pillar with one rounded end. See 
Fig 1, page 439 , Strength of Iron Pillars. In such soils^the piles need very little sharpening ; indeed, 
had better be driven without any ; or even butt end down. 

The driving of a pile in soft ground or mud will generally cause an adjacent one previously driven, 
to lean outwards unless means be taken tc prevent it. 

In piling an area of firm soil it is best to begin at its center and work outwards; otherwise the soil 
may become so consolidated that the central ones can scarcely be driven at ail. 

I'lasJ i<* reaction of th<‘ soil lias been known to cause entire piled areas 

to rise, together with the piles, before they were built upon. 

In very firm soil, especially if stony; or 
even in soft soil, if the piles are pointed, and 
are to be driven to rock; their feet should 
be protected by shoes of either wrought 
iron, as at a, s, and b, Figs 13; spiked to the 
pile by means of the iron straps w, forged 
to them; or of cast iron, as at c, where the 
ehoe is a solid inverted cone, the wide flat 
base of which affords a good hearing for the 
fiat bottom of the pile-point. The dotted 
line is a stout wrought-iron spike, well se¬ 
cured in the cone, which is cast around it; 
this holds the shoe to the pile. Regular 


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Fids l j 










































FOUNDATIONS. 


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wrought-iron shoes will generally weigh 18 to 30 lbs; but sheet iron may be used when the soil is but 
moderately compact; plate iron wheu more so ; and solid iron or steel points, from 2 to 4 ins square 
at t he butt, and 4 to 8 ins long, when very compact and stony. IIol<*s may Ik* 
drilled in rock for receiving the points of piles, and thus preventing "them 

from slipping ; by first driving down a tube, as a guide to the drill, after the earth is cleaned out of 
the tube. To preserve tlie lieads to some extent from splitting under tlie* 
blows of the hammer, they are usually surrounded by a hoop ft, Fig d; from U to 1 inch thick ; and 
1 '4 to 3 ins wide. These are, however, sometimes but imperfect aids ; for in hard driving the head 
will crush, split, and bulge out on all sides, frequently for many feet below the hoop: moreover, tne 
hoops often split open. The heads, therefore, often have to be sawed, or pared off several times 
before the pile is completely driven; and allowance must be made for this loss in ordering piles for 
any given work; especially in hard soil. Capt Turnbull, U S Top Eng, states that at the Potomac 
aqueduct, his pileheads were preserved from injury by the simple expedient of dishing them out to a 
depth of about an inch, and covering them by a loose plate of sheet iron ; as shown in section at e, 
Figs 13. A very slight degree of brooming or crushing of the head, materially diminishes the force 
of the ram. Piles may be driven through small loose rubble without much labor. Shaw's driver 
does not injure the heads. Piles which foot on sloping rock may slide when loaded. 

To <1 ri ve h pile head below water a wooden punch, or follower, as 

atp, Figs 13, may be used. The foot of this punch fits into the upper part of a casting / /. round or 
square, according to the shape of the pile ; and having a transverse partition o o. The lower part 
of the casting is fitted to the head of the pile t; and the hammer falls on top of the punch. When 
driving piles vertically in very soft soil, to support retaining-walls, or other structures exposed to 
horizontal or inclined forces, care must be taken that these forces do not push over the pile's them¬ 
selves ; for in such soils piles are adapted to resist vertical forces only, unless they be driven at an 
inclination corresponding to the oblique force. 

A broken pile may he drawn out, or at least be started, if not very 

firmly driven, by attaching scows to it at low water, depending on the rising tide to loosen it. Or a 
long timber may be used as a lever, with the head of an adjacent pile for its fulcrum. Or a crab 
worked by the engine of the pile driver. In very difficult cases the method devised by Mr J. Monroe, 
0 E, may be used. A 4 inch gas pipe 15 ft long, shod with a solid steel point, and having an outer 
shoulder for sustaining a circular punch, was thereby driven close to and '2 or 3 ft deeper than two 
piles driven 1'2 ft, in 37 ft water, and broken off by ice. Four pounds of powder were then deposited 
in the lower end of the pipe, and exploded, lifting the piles completely out of place. It will often be 
best to let a broken pile remain, and to drive another close to it. May be drawn by hydraulic press. 

Ice jullieres to piles with a force of about 30 to 40 lbs per sq inch, and in 

rising water may lift them out of place if not sufficiently driven. 

Icon piles ami cylinders. Cast iron in various shapes lias been much 
used in Europe for sheet piles: especially when intended to remain as a facing for the protection of 
concrete work, filled in behind and against them.* Cast iron cylinders, open at both ends, may be 
used as bearing piles; and may be cleaned out, and filled with concrete, if required. The frictiou in 
driving is greater than in solid piles, inasmuch as it takes place along both the inner and the outer 
surfaces. This may be diminished by gradually extracting the inside soil as they go down. Thev 
require much care, and a lighter hammer, or less fall than wondeu ones, to prevent breaking ; to 
which end a piece of wood should be interposed between the hammer and the pile; or the ram may he 
of wood. Hut it is better to use them in the shape of screw cylinders, which, 
moreover, gives them the advantage of a broad base, as in the following. 

Brunei's process. He experimented with an open cast-iron cylinder, 3 ft 
outer diam ; 1 % ins thick : in lengths of 10 ft, connected together by internal socket and joggle joints, 
secured by pins, and run with lead. It had a sharp-edged hoop or cutter at bottom : and a little 
above this, one turn of a screw, with a pitch of 7 ins, and projecting one foot all around the outside 
of the cylinder. By means of capstan bars and winches, he screwed this down through stiff'clay and 
sand, 58 feet to rock, on the bank of a river. In descending this distance the cylinder made 142 
revolutions ; sinking on an average about 5 ins at each. The time occupied in actually screwing was 
48J4 hours; or about 1 ft per hour. There were, however, many long intervals of rest for clean¬ 
ing away the soil in the inside. After resting, there was no great difficulty in restarting. The next 
fig will give an idea of the arrangement of the screw. 

The screw-pile of Alex. Mitchell, Belfast, consists usually of a rolled iron 
shaft A, Figs 14. from 3 to 8 ins diam; and having at its foot a cast-iron screw 
S S S. with a blade of from 18 ins to 5 ft diam. The screws used for light-houses, 
exposed to moderate seas, or heavy ice-fields, are ordinarily about 3 ft diam, have 
114 turns or threads, and weigh about 600 His. The round rolled shafts are from 
5 to 8 ins diam. They are screwed down from 10 to 20 ft into clay, sand, or coral, by 
about 30 to 40 men, pushing with 6 to 8 capstan bars, the ends of which describe a 
circle of about 30 to 40 ft diam. For this purpose a platform on piles lias frequently 
to he prepared. In quiet water, this may be supported on scows: or a raft well 
moored may he used when the driving is easy; or the deck of a large scow with a 
well-hole in the center for the pile to pass through. Roughly made temporary 
cribs, filled with stone and sunk, might support a platform in some positions. The 
platform must evidently he able to resist revolving horizontally under the great 
pushing force of the men at the capstan bars; and on this account it is difficult 
to drive screws to a sufficient depth, in clean compact sand, by means of a floating 
platform. The feet of the piles must he firmly secured to the screws, to prevent 

* Cast iron, intended to resist sea-water, should be close-grained, 
hard, white metal. In such, the small quantity of contained carbon is chemically combined with the 
metal; but In the darker or mottled irons it Is mechanically combined, and such iron soon becomes 
soft, (somewhat like plumbago,) when exposed to sea-water. Hard white iron has been proved to 
resist for at least 40 years without aiiy deteriotation ; whether constantly under water, or alternately 
wet and dry. Copper and bronze are but slightly and superficially affected by sea-water; but destruc¬ 
tive galvanic actiou takes place if diff metals are in contact. See p 218. 







FOUNDATIONS, 




their being lifted out of them by the upward force of waves against the super¬ 
structure. At y p, Figs 14, is shown a mode of splicing or uniting the different 
lengths or sections of a pile. The point of junction is at v\ rr is a stout iron ring 
Ibrged on to the lower pile p, 
about a foot or 18 ins below its 
top o. A strong cylindrical cast¬ 
ing n n, enclosing the ends of 
the sections, rests on this ring, 
and is pinned through the piles, 
as at tt. On this casting are 
also cast projections cec, for at¬ 
taching rods g g, and beams &c, 
necessary for bracing the struc¬ 
ture from pile to pile. The time 
actually required for driving a 
screw is from 2 to 10 hours, in 
favorable circumstances. 

At the Brandywine lighthouse, on 
a sand-bank of very pure sand, cov¬ 
ered fi or 8 ft at low water, and from 
11 to 13 ft at high, they could not be 
forced dowu, from a fixed platform, 
for more than 10 ft. At other places 20 ft in sand Is reached without much trouble, where the sand 
contains a good deal of mud, but its bearing power is then less. This (ultimate) ranges between 
about 1 and 6 tons per sq ft according to purity, depth, compactness, Ac, of the sand. In important 
cases the bearing power should be tested. 


Mitchell's piles have been screwed about 40 feet into a mixture of clay and sand, with screws 
4 ft diam. They pass through small broken stoue and coral rock without much difficulty ; and will 
push aside bowlders of moderate size. Ordinarily, clay or sand will present no great obstruction ; 
but occasionally either of them will do so. Perfectly pure cleau sand, as a general rule, gives most 
difficulty. At the Brandywiue shoal the driving was aided b.v a spur and pinion placed as low as the 
water permitted ; and the levers were worked by 30 men. The dauger of twisting off the shaft is 
the limit for screwing them. They are much used for the anchoring of chains for mooring buoys, Ac. 
Ou land, small screws, with short hollow shafts, make good durable supports for depot pillars, cranes, 
wooden telegraph poles, station signals in mariue surveying, &c, Ac. They can readily be unscrewed 
for removal. Horses or oxen may be used iu driving large screws. The Brandywine light-house 
stands on 9 screw-piles, which are surrounded by 30 others of 5 ins diam, as fenders. They have to 
resist not only moderate seas, but immense fields of floating ice, miles in extern. An unfinished 
structure was destroyed by ice, which at times injures the bracing of the standing one. 

'Text iMM'inus should b<» inside to ensure that the screws do not stop just 
above a very weak stratum which may endanger their bearing power. So with any piles. 


By means of a jet of water forcibly impelled through a tube l>y a force 
pump, the most obstinate sands (but not stiff clay or cemented gravel) will be loosened, and the sink¬ 
ing of screw-piles, or woodeu ones, or even the largest cylinders, be greatly facilitated. In a govern¬ 
ment pier at Cape llenlopen in very compact sand, in which 6 out of 7 
screws previously broke before reaching 10 ft, the use of the jet was found to remove more than 
three-fourths of the resistance.* The pile p to be sunk having first been placed in position as in Fig 
15, the lower open ends (I of a bent iron tube t s t of one and a 
quarter ins bore were stood upon the upper face of the screw disk, and 
there held firmly by 3 or 4 men while the pile was being screwed down 
by the capstan c, which was worked by a leading rope r. From the 
bend s of the pipe, a hose h, 2 ius diam. led to the force pump, the 
cylinder of which was 5 ins bore, and 9 ins stroke, aud worked about 
80 full strokes per minute, by a mule walking on a tread wheel on a 
floating platform/. There was now no trouble in screwing the piles to 
any required depth. Previous trials by playing the jet beneath the disk 
gave unsatisfactory results. 

In Mobile Bay several thousands of wooden piles, 
from 18 to 48 ins diam, were sunk from 10 to 20 ft into obstinate sand, 
at the average sinking rate of about 1 ft per second, entirely by means 
of jets. The jet was propelled by a city steam fire engine, on a steam¬ 
boat, through its own hose, with a one and a quarter inch nozzle. 

During the descent the nozzle n n was held loosely in its place near 
the foot of the pile, by two staples s s and by a string t reaching to the 
surface. The piles were suspended by their heads from shears, by the 
tackle of which their descent was regulated. The sand settled firmly 
around the piles iu a few minutes after they were sunk.t 



At Tpiissis Riv«*r. Alabama, lor iron cylinders 6 ft diam 

(enclosing piles, see p 651), in deep light shifting sand, the jet was forced by a small 
rotary pump of 200 to 300 revolutions per minute, through a canvas hose 3 ins diam, 
into a central conical cast iron vessel 10 ins diam. from which radiated 12 gas pipes 1 
inch diam, and about 30 ins long. At the outer end of each of these radii was an 
elbow to which was attached a long vertical pipe reaching down into the cyliuder, 
and made in 10 ft lengths with screw ends for prolonging them as the cylinder went 
down. This apparatus was raised and lowered by a light block and line; and by it 
alone each cylinder was sunk about 16 ft into the light sand in a few hours.! 



* Report Sec of War 1872. t John W. Glenn. C E, Van Nostrand, June 1874. 

} Gabriel Jordan, C E; Trans Am Soc C E, Feb 1874. 












































FOUNDATIONS. 


647 


At file T.evan Viaduct. Mr James Rrimleo, England, in a light 

it ' p andy marl of great depth, sunk hollow cast iron cylinders of 10 ins outer diaui, to a depth of ‘20 ft, 
f by means of a jet pipe 2 ins diam passing down inside of the cylinder, and through a hole in its base, 
6 | whion was a cast iron disk 30 ins diam, and 1 inch thick, strengthened by outside flanges. The con- 
I necting flanges of the cylinder sections are outside, thus impeding the descent, as did also the broad 
| bottom disk ; still 3 or 4 hours usually sufficed for the sinking of each, to 20 ft depth. Actual trial 
r showed that their safe sustaining power was about 5 tons per sq ft of bottom disk. 

At liOck lien viaduct each pier consists of two cylinders, open at both 
1 ends; of cast iron, 8 ft in diam; l^ins thick; in lengths of 6 ft, weighing 4 tons 
I each; and bolted together by inside flanges, with iron cement between them. The 
1 cylinders stand 8 ft apart in the clear; and are in 36 ft water. “ A strong staging 
I was erected; and 4 guide-piles driven for each cylinder. The several lengths being 
ift previously bolted together, these were lowered into their places. Each cylinder sank 
by its own weight one or two ft through the top mud, and then settled upon the sand 
| and gravel which form the substratum for a great depth. Into this last they were 
sunk about 8 or 9 ft farther, by excavating the inside earth under water, by means 
| of an inverted conical screw-pun. or dredger, of % inch plate iron. This was 
| 2 ft greatest diam, and 1 ft deep; and to its bottom was attached a screw about l ft 
| long, for assisting in screwing it down into the soil. Its sides had openings for the 
I entrance of the soil; and leather flaps, opening inward, to prevent its escape. From 
1 opposite sides of the pan, 3 rods of % inch diam projected upward 4 feet, and were 
| there forged together, and connected by an eye-and-bolt joint to a long rod or shaft, 
| at the upper end of which was a four-armed cross-handle, by which the pan was 

( screwed down by 4 men on the staging.” 

“ When a pan was full, a slide which passed over the joint at the bottom was lifted; and the pan 
was raised by a tackle. This pan raised about X cub ft at a time. A smaller one of only 1 ft diam, 

I and 1 ft deep, raising about % cub ft, was used when the material was very hard. By this means 
j the cyliuders were suuk at the rate of from 2 to 18 ins per day. The slow rate of 2 ins was caused 
by stones, some of them of 50 lbs. These were first loosened by a screw-pick, which was a bar of 
iron 3 ft long, with circular arms 12 ins long projecting from the sides. After being loosened by this, 
the stones were raised by the pan. The expense of all this apparatus was very trifling; and the ex¬ 
cavation was done easily and cheaply. After the excavation was finished, aud the cylinder suuk, 
before pumping out the water, concrete (gravel 2, hydraulic cement 1 measure) was filled in to toe 
depth of 12 feet, by means of a large pau with a movable bottom; and about 12 days were left it. to 
harden. The water was then pumped out, aud the masonry built in open air. In some of the cylui 
ders, however, the water rose so fast, notwithstanding the 12 ft of concrete, that the pumps could not 
keep them clear; and 6 ft more of concrete had to be added in those. Piually random-stone, or rough 
dry rubble, was thrown in around the outsides of the cylinders, to preserve them from blows and 
undermining.” * The masonry extends 20 ft above the cylinders, and above water. 

The vacuum and plenum processes. We can barely allude to 
the general principles of these two modes of sinking large hollow iron cylinders. In 
the vacuum process of Dr. Lawrence Holker Potts, of London, the cylinder 

c, Fig 16, while being sunk, is closed air-tight at top, by a 
trap door, opening upward. A flexible pipe p, of India- 
rubber, long enough to adapt itself to the sinking of the 
cylinder, and provided with a stopcock s, leads from the 
cylinder to a vessel v; which may be placed on a raft, or a 
scow, or on land, as may suit circumstances. The cylinder 
being first stood up in position, as in the fig, the water is 
pumped out, and the interior soil removed if the cylinder 
has sunk some distance by its own weight. The cock 
s is then closed, and the air is drawn out from the vessel » 
by an air pump. The cock is then opened, and most of the air in the cylinder rushes 
into the void vessel v ; thus leaving the cylinder comparatively empty, and therefore 
less capable of resisting the downward pressure of the external air upon its top. 
This pressure, as is well known, amounts to nearly 15 lbs on every sq inch ; or nearly 
1 ton per sq ft of area of the top. Consequently the cylinder is forced downward in 
the bed of the river, by this amount of pressure, in addition to its own weight. At 
the same time, the pressure of the air upon the surface of the water is transmitted 
through the water to the soil around the open foot of the cylinder; so that if this 
soil be soft or semi-fluid, it will be pressed up into the nearly void cylinder, in which 
is no downward pressure to resist it. The descent varies from a few' inches, to 4 or 5 
ft each time. The process is then repeated, by admitting air again into the cylin¬ 
der, opening the trap-door, removing the water and soil, as before, &c. Additional 
lengths of cylinder may be bolted on, by means of interior flanges. 

It is atlupfed only to soft soi Is, and to wet sandy ones ; but is not sufficient¬ 
ly powerful in very compact ones; nor does it answer where obstructions from bowlders, logs, &c, occur; 

*Hollow Iron Piles either cast or wrought with solid pointed feet, to be driven by the hammer 
falling inside of them and striking against the top of the solid foot, are a recent device of great use in 
many cases. They are made in sections of which enough can be gradually united to reach any 
required depth. They avoid the danger of bending which attends striking the top. The iron feet are 
swelled outwardly a little to diminish earth-friction against the pile above them. 



TiylO 


















648 


FOUNDATIONS, 



the removal of which requires men to enter the cylinder to its foot; which they cannot do in the rarefied 
air. The pipe p should be of sufficient diam to allow the air to leave the cylinder rapidly, so that the 

outer pressure may act upon the top as suddeuly as possible. 

At the Goodwiu Sands light house, England, hollow cylinders ft in diam, were sunk o4 It into 
saud by this process, iu about <i hours; where a steel bar could be driven only 8 It by a sledge-ham¬ 
mer. Others, 12 ms in diam, have been sunk 10 ft into sand within less than au hour. In this last 
instance the air-pump had two barrels, 4)$ ins diam, 10 inch stroke, w orked by 4 men. The pipep 
was of lead, and only % inch diam. 

The plenum process, invented by Mr JTriger, 
of France, consists in forcing air into the cylinder 
C C, Fig 17, to such an extent as to force out the 
water, compelling it to escape beneath the open foot, 
into the surrounding water. The interior of the cylin¬ 
der being thus left dry to the bottom, men pass down it 
to loosen and remove the soil at and below its base. When 
this is done, they leave; the compressed air is allowed to 
escape; and the cylinder, being no longer sustained by 
the upward pressure of the compressed air beneath its 
top, sinks into the cavity, or the loosened material at its 
foot. Fig 17 shows the simple arrangement by which 
workmen are enabled to enter or leave the cylinder, 
without allowing the compressed air to escape; as well 
as the general principle of the entire process. 

L I, is a separate small chamber, the air-lock, which is 
removed when a new length of pipe is to be added; aud afterward 
replaced aud firmly bolted on. This chamber has a small air tight 
door d, by which it can be entered from Without; and auother, o, 
opening into the cylinder. The flaps, t, h, of botli doors, open in¬ 
ward, or toward the cylinder. This chamber also has two stopcocks; one. a, in its floor, communi¬ 
cating with the cylinder; and one e. above, communicating with the open air. At s is a bent, tube, 
also with acock, which passes air-tight through the side and the bottom of the air lock. Through 
it the compressed air is forced into the cylinder, by an air force pump or condenser: and through it 
the same air is allowed to escape at a later period. A siphon is shown at nnn. A drum w is used 
fog hoisting theexcavated material from the bottom, to the air-lock : its axle i i passes air-tight through 
stuffing-boxes iu the sides of the lock ; the hoisting being done by men outside. This is the general 
arrangement employed by Mr W. J. McAlpine, CE. of New York, at Harlem bridge; and from his 
description of it, ours has been condensed. The cylinders were there 6 ft diam, ins thick, and in 
lengths of 9 ft, bolted together through inside flanges /, as the sinking went on. The air-lock is 6 ft 
diam, by nearly <> ft high ; with sides of boiler iron ; and top and bottom of cast iron. 

Now suppose the cylinder C C to be let down, aud steadied in position, as in the fig; and the air¬ 
lock L I. to be adjusted on top of it. The next process is to force in air through the curved tube s ; 
the flap t of the lower door o. and the cock a, being previously closed. As the compressed air accu¬ 
mulates in the cylinder, it forces out the water; which escapes partly beneath the bottom of the c.i 1- 
inder, aud partly by rising through the siphon tin, and flowing out at </■ The door o being already 
closed, and that at d open, the air in the air-lock is iu the same condition as that outside; so that 
workmen can euter it readily. Having done so, they close the door d. and the cock e; and open the 
cock a, through which condensed air from the cylinder rushes upward, soon filling the air-lock. 
When this is done, the flap t is opened, and the men descend through the door oby a ladder, or by a 
bucket lowered by the drum w, to the bottom. Here they loosen and excavate the material as deep 
as they can ; and, filling it into a bucket or hag, they signal to those outside, who raise it to the air¬ 
lock. When done, they ascend to the uir-lock, close the door o. and the cock a; and open the cock e, 
through which the condensed air in the lock soon escapes, leaving the internal air the same as that 
outside. The door d is then opened, the buckets of earth are removed, and the men go out. Finally 
the cock at s is opened, the condensed air in the cylinder escapes through it to the outside air. and 
the cylinder sinks by its own weight into the cavity and loosened soil prepared for it at its base, and 
which is now forced up into the cylinder by the rush of the returning water. The process is then 
repeated. The sinking will often vary from 0 to 10 or more feet at one operation. Until depths of 
40 or 50 ft, most tneu can endure the pressure of the coudensed air ; but as the depth increases this 
becomes more difficult, and positively dangerous to life. Cast-iron cylinders 15 ft diam ; aud great 
caissons, Fig 18, have been thus sunk ; but ut times at great expense aud trouble. 

The cylinder should he g-uided in its descent by a strong frame, which 
may be supported by piles. Otherwise it will be apt to tilt, aud thus give great trouble to settle it 
upon its exact place. Have been sunk iu deep water by divers underminiug inside. 


The plenum process as applied at the South St bridge, Philada, 

by Mr. John W. Murphy, contracting engineer, differs materially from that described above ; and 
moreover deserves notice on account of the great simplicity and efficacy of his plant. This consisted 
partly of two canal boats, decked, each 100 ft long, by 17J^ ft wide, and 8 ft depth of hold. They 
were anchored parallel to each other, 15 ft apart. Supported by the boats, aud over the space between 
them, was a strong four-legged shears about 50 ft high; at the top of which was attached tackle for 
handling the cast iron cylinders. In the hold of one of the boats was a ISiirleigli 
Compressor having two pistons of 10 ins diam, aud 9 ins stroke; together with 
ifs boiler. On thedeck of the same boat stood a vertical air-tank or regulator, 

22 ft long, by 2 ft diam, made of quarter inch boiler iron. This served to maintain a supply of com¬ 
pressed air in the submerged cylinder iu case of an accidental stopping of the compressor; which 
otherwise would probably be fatal to the laborers in the cylinder. The condensed air flowed from 
this air-tank to the uir-lock of the cylinder through a hose 4 ins diam, made of gum elastic and can¬ 
vas, and so long, and so placed, as to extend itself as the cylinder went down, thus maintaining the 
communication at all limes. Entirely across both boats, and across the interval between them, ex¬ 
tended two heavy wooden damps, each 3 ft wide by 18 ins high; each composed 




















FOUNDATIONS, 


649 


of three pieces of 12 X 18 Inch timber strongly bolted together. At the centers of these clamps the 
two inner vertical sides which faced each other were hollowed out to the depth of a foot by concavi¬ 
ties corresponding to the curve of the cylinders. The distance apart of the clamps was regulated by 
two stroug iron rods, having screws and nuts at their ends for that purpose. Thus when a section 
of a cylinder was hoisted by means of the shears into its position over the space between the two 
boats, the two concavities of the clamps wore brought into contact with it. and the nuts being then 
screwed up, the cylinder was firmly held in place bv the clamps. The shears could then be used to 
raise another section of the cylinder to its place upon the first one, that the two might be bolted to¬ 
gether. By repeating this process the height of the cylinder would soou become too great to allow 
the shears to place another section upon it; in which case the nuts of the screws were slightly 
loosened, and the cylinder was allowed to slip down slowly into the water until its top was hut a 
little above the surface. The screws were then again tightened, and the cylinder again held fast 
until other sections were added and bolted to it. When there was danger that the upward pressure 
of the condensed air might lift a cylinder, the clamps were raised by the shears clear of the boats; 
then tightened to the cylinder, aud a platform of plauks laid upon them, and loaded with stone. 

The air-lock was so arranged as not to require to be removed when a new sec¬ 
tion was to be bolted on. This was effected as follows. Sections of the cylinder were bolted together 
in the manner just described, until its foot rested on the bottom, with its top a few feet above high 
water. A heavy cast iron diaphragm inches thick, to form the floor of the 
air-lock, was then placed on top. Then was added another 10 ft high section of the cylinder, to form 
the chamber of the air-lock. These were bolted together ; and then another diaphragm was added 
at top to form the roof of the air-lock. These diaphragms were furnished with openings, and with 
doors and valves corresponding with those shown in Fig 17. and remained permanently in the 

cylinders when the work was finished. If the depth of soil to be passed through before reaching 
rock is so great as to require other sections of cylinder to be bolted on above the top of the air-lock, 
this may be done to any extent, inasmuch as it is immaterial whether the air-lock is under water or 

not. To keep the cylinder both air- and water-tight the faces of 

the flanges before being bolted together were smeared with a mixture of red and white lead and cote 
ton fiber. 

At the South St brid ii'e the cylinders were 4, 6, and 8 ft diam ; in lengths 
or sectious 10 ft long They were all \ % inch thick. Inside linnges 2% ins wide. Ij4 thick, with bolt¬ 
holes 1J4 inch diam. by 5 ins apart from center to center. The bottom edge has no flange. A 10 ft 
section of an 8 ft evlinder weighs 11600 lbs; of a 6 ft one, 10800; of a 4 ft one, 6800. An 8 ft dia¬ 
phragm. 2800 R)s ; 6 ft, 1600; 4 ft, 783. The rock under the soil was quite uneven in places ; but was 
levelled off as the cylinders went down. These were then bolted to it by cast iron hrackets. 
The work went on. day and nijfht. summer and winter: with no inter¬ 
ruption from the tides, floods, or floating ice; and the thirteen columns were sunk, filled with con¬ 
crete, aud completed in 11 months ; much of which was consumed in levelling off the rock, and bolt¬ 
ing the brackets. The want of guides caused much tilting, trouble, and delay. 

Rise and fall of tide about. 7 ft. Greatest depth of soil, gravel, &c, passed "through. 30 ft: least. 6 
ft. Depth of water about 25 ft. The work was under charge of John Anderson, a very skillful and 
energetic superintendent of such matters. The entire neat cost of the cylin¬ 
ders in place, and filled with hydraulic concrete, was probably not far from $120 per ft of total length 
for the 8 ft ones ; $S0 for the 6 ft ; and $50 for the 4 ft diaius. There were three gangs of workmen ; 
and each gang worked 4 hours at a time. See a full and very instructive description with engrav. 
ings, by I). M. Stauffer. Superintending Engineer for the city, in the Journal of the Franklin Inst, 
Nov., 1872. From it the above few items are taken. Mr. Anderson’s firm (Anderson & Barr, Tribune 
Building, N. Y.) have since successfully sunk a number of such cylinders, including (1884-5) four of 
wrougbt-iron, 8 ft diam, 66 ft long, at an angle of 45° with the hor, intended as struts to preveut the 
movement of one of the abut piers of Chestnut St bridge, Phila. 

Cast iron cylinders have cracked through, around their entire circum¬ 
ference, in many parts of the U. S. in very cold weather; owing to the diff of contraction of the 
iron, and of the concrete filling. Ignorant use of them may be attended by great danger. 


The shaded part of Fig 18 shows a transverse section of the caisson of yellow- 
pine timber and cement, for the lirooklyn tower of East River (N Y) 

suspension bridge, of 1600 ft clear span. It is 168 ft long at bottom, and 102 ft wide. 
A longitudinal section resembles the transverse one, except in being longer, and in 
showing more shafts .T. Of these there are 6, arranged in pairs, for expedition and as 
a precaution against accident. Namely, two water-shafts J, each 7 ft by 6^ ft across, 
for removing by buckets and hoisting apparatus, the material excavated beneath the 

caisson; together with such 
water as may accumulate at 
o o; two air-shafts of 21 ins 
diam, through which air is 
forced from above, to expel 
the water from the chamber 
C S S I) below the caisson, so 
as to allow the laborers to 
work there at undermining; 
the expelled water escaping 
under the foot C D of the cais¬ 
son, into the river; and two 
supply shafts of 42 ins diam, 
for admitting laborers, tools, 
&c. The several shafts of course have air-chambers on top, on the same principle as 
Fig 17, to prevent the escape of the compressed air in s s. 


















FOUNDATION'S 


650 


The shafts are of J4 inch boiier iron. The foot C D nine timbers high, is continuous, extending 
entirely around the caisson ; its bottom is shod with cast iron ; its four corners are strengthened by 
wooden knees 20 ft long. 

From the bottom, up to the line N, N, 14 ft. the caisson is built of horizontal layers of timbers one 
foot square: the layers crossing each other at right angles; and the timbers of each layer touching 
each other well forced and bolted together; and all the joints tilled with pitch. To aid in preventing 
leakage, the nuts and heads of the screws have India-rubber washers; also all outside seams, as well 
as all the seams of the layer of timbers N, N, are thoroughly calked: and a layer of tin, enclosed 
between two layers of felt, is placed outside of each outer joint; and over the entire top of the layer 
next below N, N. 

When the caisson was built up to N. N. on land, it was launched, floated into position, and anchored ; 
after which were added for sinking it. fifteen courses of timbers one ft square; and laid one ft apart 
iu the clear; with the intervals filled with concrete. The top course A B is of solid timber, to serve 
as a floor for supporting machinery, &c. It was sunk some feet below the very bottom of the 
river, iu order to avoid the teredo. 

Cribs are sunk outside of the caisson, to form temporary wharves for boats carrying away excavated 
material ; and for vessels bringing stone, &c. 

When the caisson was suuk, and the water forced out from the chamber or space CSS D. workmen 
began to excavate uniformly the enclosed area of river bottom, so as to allow the cai«son to descend 
slowly until it reached a firm substratum. The space C S S D, as well as the shafts, was then filled up 
solid with concrete masonry. A coffer-dam was built on top of the caisson ; and in it the regular 1 
masonry of the tower was started. The total height of this tower including the caisson, is about 300 
ft. For full details see report. 1873, of W. A. Koebliug the chief engineer. 

Hollow oylintlorfl, or oilier (onus of brickwork or msi- 

sonry, with a strong curb or open ring of timber or iron beueath them, may be 
gradually sunk by undermining and excavating from the inside; aud form very stable foundations. 
Under water this may be done by properly shaped scoops, with or without the aid of the diving-bell, , 
according to the depth, &c. On land it will often be the most economical and satisfactory mode, 
especially in firm soils. The desceut may be assisted by loading them, if, as sometimes happeus, the t 
friction of their sides agaiust the earth outside prevents their sinking by their own weight. A brick 
cylinder, 46 ft outer diam, walls 3 ft thick, has been suuk 40 ft iu dry sand aud gravel, without auy 
difficulty. It was built 18 ft high, (on a wooden curb 21 ins thick,) aud weighed 300 tons before the 
sinking was begun. The interior earth was excavated slowly, so that the sinking was about 1 ft per 
day ; the walls being built up as it sauk. Tunnel shafts are at times so sunk. 

On the Rhine for a coal shaft, a brick cylinder 25^ feet diam was first thus 
sunk by its own weight 76 ft through saud and gravel; then an interior one. 15 ft diam, was suuk in 
the same way to the depth of 256 ft below the surface; of which depth all the 180 ft below the first 
cylinder was a running quicksaud. At 256 ft friction rendered the cylinder immovable. The quick 
sand was removed by boriug; no pumping was done ; but the water was permitted to keep the cyl full. 

The eutire foundation for a large pier of masonry has been sunk in this manner, in a single mass ; 
a sufficient number of vertical openings being left in it for the workmen to descend, or for tools to be 
inserted for undermining. This is generally a very slow aud tedious operation, especially under 
water. It may often be expedited by diving-bells or by diviug-dresses. It will generally be better to 
make the mass wider at bottom than above it, so as to diminish friction agaiust the outside earth. 

On land, water may at times be used for softeuiug the bottom earth. By keepiug the interior of such 
hollow masonry dry, it may even be built downward from the surface; by undermining ouly a por¬ 
tion of its circumference at a time, filling said portion with masonry, aud then removing and filling 
the other portion ; and so on iu successive stages of 2 or 3 ft downward at a time. This mode may be 
adopted also when friction has stopped the sinking of a mass by Us own weight wheu undermined. 

The sand pump as used at the St Louis bridge will often be of service iu rais¬ 
ing saud from cylinders while being sunk in water. With a pump pipe of 3.5 ins bore, aud a water 
jet under a pressure of 150 lbs per sq inch, 20 cub yds of sand per hour were raised 125 feet. A jet of 
air has also been successfully used in the same way, as at the East River, N Y, suspension bridge, ke. I 

Fascines. On marshy or wet quicksand bottoms, foundations may be laid by ] 
first depositing large areas of layers of fascines, or stout twigs and small branches, 
strongly tied together in bundles from b to 12 ft long, and from 6 ins to 2 ft in diam. 

The layers or strata of bundles should cross each other. A kiud of floating raft or large mattress j 
is first made of these, and theu suuk to the bottom by being loaded with earth, gravel, stones, <tec. 

In this manner the abutments aud piers of the great suspension bridge at Kieff. in Russia, with spans | 
of 4 40 ft, were founded in 1852, on a shifting quicksand. There the fascine mattresses extend 100 ft 
beyond the bases of the masonry which rests upon them. 

Fascines may be used iu the same way for sustaining railway embankments. &c, over marshy 
ground, but they will settle considerably. 

Kami -piles. We have already alluded to the use of sand well rammed in layers 
into trenches or foundation pits; but it may also be used in soft soils, in the shape 
of piles. A short, stout wooden pile is first driven 5 to 10 feet or more, according to 
the case. It is then drawn out, and the hole is filled with wet sand well rammed. 
The pile is then again driven in another place, and the process repeated. The inter¬ 
vals may be from 1 to 3 ft in the clear. Platforms may be used on these piles as on 
wooden ones. If the sand is not put in wet, it will be in danger of afterward sink¬ 
ing from rain or spring water. In this case, as with fascines, it is well to test the 
foundation by means of trial loads. Some settlement must inevitably take place 
until all the parts come to a full bearing: but it will be comparatively trifling. The 
same occurs in every large work to some extent; as in a roof or arch of great span, 
whether of wood, iron, or masonry ; so also with all tall piers, walls, &c, <fcc. Sandy 
foundations under water should be surrounded by stout well-driven sheet-piling, to 
prevent the enclosed sand from running out in case the outer sand is washed away ; 
and should also be defended by a deposit of random-stone. See Sand-piles, p 626. 







651 


ROCK-DRILLING. 


On bad bottoms under water, small artificial islands of good soil have 

Deeu deposited; and the masonry founded upon them. Caual locks and other structures may at 
VT be advantageously rounded in this way in marshy soils. If necessary, a depth of several feet 
of the bud soil may be dredged out before the firmer soil is deposited; and the latter may be weighted 
by a trial load to test its stability. 


The mode of laying a foundation under water, by building the masonry upon a timber platform 
HlK> vo water, upheld by strong screws, and lowered into the water as the work 
is finished in the open air. a course or two at a time, has of late been much employed with entire 
success, in large bridge-piers in deep water. It however is not new. It was suggested more than 
100 years ago by Belidor. 

Piles are driven 6 to 10 ft apart around the space to be occupied by the pier; having their tops con- 
uecteii by heavy timber cap-pieces. These last uphold the screws, which work through them. The 
whole is braced against lateral motion. 


A clump of piles well drivex ; and then enclosed by an iron cylinder sunk to a 
firm bearing, and filled with concrete, is an excellent foundation. The piles may 
extend to the top of the cylinder, and thus he enclosed in the concrete. Such an 
arrangement has been patented by S. B. Cushing, C. K , Providence, R. I. The cyl¬ 
inder and concrete serve to protect the piles from sea-worms, and from decay above 
low water; and are not intended to support the load above them. 

Cost of a (living* outfit, with two dresses, air-pump and tubes, about $750, 
or $GiiO with cheaper pump. Alfred Hale & Co, 30 School St, Boston, Mass. 

Two men can work the air-pump to 50 ft depth. 


STONEWORK. 


Where work is done on a large scale, blasting can sometimes be done at from 10 
to *20 per cent less cost per cubic yard by means of machine drills and 
dynamite, than by liand drills and gunpowder. Ordinarily, how¬ 
ever, tile cost is about the same, and the advantage of the newer 
methods consists rather in economy of time, convenience,and having the work more 
entirely under control. In ordinary railroad work in average hard rock, and when 
common labor costs $1 per day of ten hours, the cost per cubic yard, for loosening, 
will ordinarily range between 30 and 60 cts, including tools, drilling, powder, &c; 
average 45 cts. 

IIol es for l>lastin<>', drilled by hand, are generally from 2*4 to 4 ft 

deep; and from 1J4 to 2 ins diam. Churn-drilling; is much more expeditious 
and economical than that by jumping, mentioned below. The churn-drill is merely 
a round iron bar, usually about 1*4 his diam, and 6 to 8 ft long; with a steel cutting 
edge, or bit, (weighing about a lb, and a little wider than the diam of the bar,) welded 
to its lower end. A man lifts it a few inches; or rather catches it as it rebounds, 
turns it partially around; and lets it fall again. By this means he drills from 5 to 
15 feet of hole, nearly 2 ins diam, in a day of 10 working hours, depending on the 
character of the rock. From 7 to 8 ft of holes 1% ins diam, is about a fair day's 
work in hard gneiss, granite, or compact siliceous limestone; 5 to7 ft in tough c<»m- 
pact hornblende; 3 to 5 in solid quartz; 8 to 9 in ordinary marble or limestone; 9 to 
10 in sandstone; which, however, may vary within all these limits. When the hole 
is more than about 4 ft deep, two men are put to the drill. Artesian, and oil wells, 
in rock, are bored on the principle of the churn-drill. See also diamond drill, p 052. 

The jumper, as now used, is much shorter than the churn-drill. One man (the holder) sitting 
down, lilts it slightly, and turns it partly around, during the intervals between the blows from about 
8 to 12 lb hammers, wielded by two other laborers, the strikers. It can be used for holes of smaller 
diameters than can be made by the churn-drill; because the holder can more readily keep the cutting 
cud at the exact spot required to be drilled. It is also better in conglomerate rock ; the bard siliceous 
pebbles of which deflect the churn-drill from its vertical direction, so that the hole becomes crooked, 
and the tool becomes bouud in it. The coal conglomerates are by no means hard to drill with a 
jumper. The jumper was formerly used for large deep holes also, before the superiority of the churn- 
drill became established. 

Either tool requires resharpening at about each 6 to 18 inches depth of hole; and the wear of the 
steel edge requires a new one to be put on every 2 to 4 days. With iron jumpers, the top also be¬ 
comes battered away rapidly. As the hole becomes deeper, loDger drills are frequently used than at 
the beginning. The smaller the diameter of the hole, the greater depth can be drilled in a given 
time; and the depth will he greater in proportion than the decrease of diam. Under similar circum¬ 
stances, three laborers with a jumper will about average as much depth as one with a churn-drill. 

The hand-drill, in which the same man uses both the hammer and the short drill, ischiefly used 
for shallow holes of small diam. With it a fair workman will drill about as many feet of hole from 
<i to 12 ins deep, and about % inch diam, as one with a churn-drill can do in holes about 3 ft deep, and 
2 ins diam. in the same time. (July the jumper or the hand-drill can be used for boring holes which 
are horizontal, or much inclined. 







652 


MACHINE ROCK-DRILLS. 


MACHINE ROCK-DRILLS. 


Art. 1. machine Rock-drills bore much more rapidly than hand drills; 
and more economically, provided the work is so great as to justify the preliminary 
outfit. They drill in any direction, and can often be used in boring holes so located 
that they could not be bored by hand. They are worked either by steam directly ; 
or by air, compressed by steam or water power into a tank called a “receiver,” and 
thence led to the drills through iron pipes. The air is best for tunnels and shafts, 
because, after leaving the drills, it aids ventilation. 

Art. 2. Such <1 rills are of I wo kinds; rotating- drill* and 
B>orciission drills. In the former, the drill-rod is a long tube, revolving about 
its axis. The end of this tube, hardened so as to form an annular cutting-edge, is 
kept in contact with the rock, and, by its rotation, cuts in it a cylindrical hole, gen¬ 
erally with a solid core in the center. The core occupies the core-barrel. Art 8. 
The drill-rod is fed forward, or into the hole, as the drilling proceeds. The debris 
is removed from the hole by a constant stream of water, which is led to the bottom 
of the hole through the hollow drill-rod, and which carries the debris up through 
the narrow space between the outside of the driil-rod and the sides of tin* hole. 

In percussion drill*, the drill-rod is solid, and its action is that of the 
churn drill, p 651. 

Art. 3. In the Rraudt (European) rotary drill, the cutting-edge at the 
end of the tubular drill-rod is armed with hardened steel teeth. It is pressed against 
the rock under enormous hydraulic pressure, and makes but from 5 to 8 revolutions 
per minute. 

Art. 4. The Diamond drill is the only form of rotary rock-drill exten¬ 
sively used in America. In it, the boring-rod consists of a number of tubes jointed 
rigidly together at their ends by hollow interior sleeves. 

Art. 5. The boring-bat. Fig 1, is called a ’‘core-bit.” Its cutting-edge 
has imbedded in it a number of diamond* as shown. These are so arranged as 
to project slightly from both its inner and outer edges. Annular spaces are thus 
left between core and core-barrel, and between the latter and the walls of the hole. 
These spaces permit the ingress and egress of the water used in removing the debris 
from the hole, and, at the same time, prevent the core from binding in the barrel, or 
the latter in the hole. 


ITig. 1 



Art. <>. Just above the “core-bit,” the 44 COre-1 inter,” Fig 2, is screwed to 
the barrel. This is a tube about 8 ins long and of the same outer diam as the 
barrel. Inside it is slightly coned, with the base of the cone upward, and fur¬ 
nished with a loose split-ring, K, toothed inside, and similarly coned. While the 
drilling is going on, this ring encircles the core closely, and remains loose from the 
outer cylinder; but when the drilling is stopped, and the drill rod begins to be 
raised, the ring is caught and raised by the outer cylinder: and, by reason of its 
beveled shape, is pressed hard against the core of rock, which is pulled apart close 
to its foot by the power which lifts the drill-rod. 

Art. 7. This power is supplied by a rope-drum, fastened to the top of the 
frame which supports the drill and worked by the same engine which rotates the 
drill-rod. The rope from the drum passes up to a pulley at the top of a derrick, 
and thence down to the upper end of the drill-rod. The consideral>le*height of the 
derrick enables from 40 to 50 feet of the drill-rod to be removed in one piece. 

Art. H. Above the “core-lifter ” is the 44 core-barrel.” This is a wrought- 

iron tube from S to 16 ft long. It j 8 spirally 

grooved outside, to permit the ascent of the water and debris from the hole; and is 
sometimes set with diamonds on its outer surface, t<> prevent wear. The hit, lifter 
and barrel are of uniform outer diam, a little less than the diam of the hole The 
outer diam of the drill-rod varies from about 1% ins for 2-inch barrel to 5)4 ins for 12- 
inch barrel. /z 















MACHINE ROCK-DRILLS. 


653 


^ hore R is not desired to preserve the core intact, a “boring-. 

Svr'mV H ♦ ?’ "' H * V b r ,,Se 'l i,isread of tl,e “core-bit.” Fig 1. This is a solid bit 
im 0 ] . / !t 1 f. pel ' lorHted Wlth holes which allow the water to pass out from 
ieieuce * l0d '’ aild 18 armeU Wltl ‘ dii ‘i»ouds, some of which project beyond its circum- 

ne^mfnof?* drill ' ro . d revolves at a speed of from 200 to 400 revolutions 
i 1 « r, ? e en «’ ,,,e * b >’ which it is rotated, consists usually of two cvlin- 

deis, eithei fixed or oscillating, operated by steam or compressed air, and working 

n noSi rf/ 68 to , eac1 ' 1 ,,tll cr By means of cranks they turn a shaft, which cone 
m uni cates its motion, through bevel gearing, to the drill-rod. The latter is fe«l 
now u, as the hole progresses, either by other bevel gearing driven bv the 
same engine ; or by being attached to a cross-head which connects the piston rods of 
2 the piston rods being.parallel with the drill rod. 

.vri. 11 . ibe diamond drill bores perfectly circular boles, in straight 
lines and in any direct ion, to great depths; from 300 to U.uo feet 
being not uncommon. This, with the fact that it beings up unbroken 
cores, from 8 to 16 ft long, which show the precise nature and stratification of i lie 
rock penetrated, renders it very valuable in test-boring, prospecting of mines, Ac. 
they are also furnished of sufficient size to bore holes from 6 to 15 ins diam, for 
artesian wells. The roundness of the holes bored enables the use of casing of 
neatly as great diam as that of the hole; and their straightness is advantageous in 
case a pump has to be used. 

Art. 12. In soft, rock a bit may drill through 200 ft or more without, resetting. 
On the other hand, in very hard rocks, similar drills will wear out in 10 ft or less. 
In 1883-4. a diamond drill by the Am’n Diamond Rock Boring Co, weighing com¬ 
plete about 1400 tbs. and costing about $2800, bored, in 1428 hours of actual boring, 
53 holes of 2 ins diam, and aggregating 9141 lineal ft. Average length of hide 172.5 
ft. Average rate^ 6.4 lin ft per hour; greatest, 12.8. Average total cost, 
about 96 cts per lin ft. The rock was principally limestone, with some quartz and 
sandstone. The holes were bored at angles varying from 0° to 45° with the vertical. 

As a rough average we may say that in ordinary rocks, as granite, lime¬ 
stone, and hard sandstone, these drills will bore deep holes, 2 to 3 ins diam, at from 
1 lo 2 ft per hour, and at a cost of from $1 to 82 per ft. 

Art. 13. These drills are made of many widely different sizes, and with 
different mountings, depending upon the nature of the work to be done. 

Tbey are made by the Penna Diamond Drill Co, Pottsville, Pa; American 
Diamond Rock-Boring Co, office 15 Cortlandt St, New York ; M. C. Bullock Mfg Co, 
Chicago, III, and others. These companies usually contract to do the drilling them¬ 
selves. They also sell the machines, generally under restrictions as to the location 
ami extent of the territory in which they are to lie used. The prices depend, 
to a great extent, upon the nature of these restrictions. Roughly, the card prices 
for some of the leading sizes, are, in 1886, as follows : 


Diam 
of hole. 

Greatest 
length of 
hole. 

Weight 

of 

machine. 

Card price, 
1886. 

ins. 

ft. 

tbs. 

$ 

IK 

250 

400 

1500 

2 

500 

1000 

2000 

2 

2000 

3500 

4000 

4 

2000 

6200 

6000 

- 1— - 


Art. 14. In percussion drilling- machines, the drill-bar is driven 
forcibly against the rock by the pressure of steam or of compressed 
air, acting upon a piston, P, Fig 4, moving in a cylinder, C C, Figs 4 and 5; and 
makes about 300 strokes per minute. The rotation of the drill-bar is accomplished 
automatically, as explained in Art 27. 

Art. 15. The cylinder, C C, is free to slide longitudinally in the fixed 
frame or shell, S S, Fig 5, to which it is attached, and which, in turn, is fixed to the 
tripod or other stand (see Arts 18 and 19) upon which the machine is supported. 

Art. 1(1. The drill-rod, R, corresponding to the churn drill, p 651, is 
fastened, by an appropriate chuck, K, to the end of the piston-rod, 0. The drilling 
is begun with a short drill-rod, and with the cylinder as far from the hole as the 
length of the shell, S, will permit. As the bit penetrates the rock, the cylinder is 
fed forward,* either automatically or by hand (see Art 28), as far as the length of 


* Bv forward, or downward, we mean totuard, the hole which is being drilled. By back¬ 
ward, or upw T ard,/ro//t the hole. 















654 


MACHINE ROCK-DRILLS. 


the shell permits. The drilling is then stopped, by shutting off the steam * and the 
cylinder is run back, by reversing the motion of the feeding apparatus. The short 
drill-bar is then removed, and, if the drilling is to be continued, a longer one is sub¬ 
stituted in its place, and the process repeated. 

Art. 17. Inasmuch as the act of drilling wears the edges of the bit, thus reduc¬ 
ing its diam somewhat, the hole will of course be tapering, or of 
slightly less diam at bottom than at top. The second bit must therefore be of 
slightly less diam than the first; say from ^ to % inch less; the third must be less ' 
than the second, and so on. On the other liami, in long holes, the drill-bar will 
seldom he in a perfectly straight line, so that the bit, instead of striking always in 
the same spot, will describe a circle, and thus enlarge the hole. 

Art. 1H. The shell, S, in which the cylinder slides, is provided with an arrange¬ 
ment by which it may he clamped, either to a tripod, as in Fig 5, or to a long 
liar or column, along which it may slide. The column, if hor, may rest upon 
two pairs of legs; or it may be braced, in any position, against the opposite sides of 
a narrow cut, or against the floor and ceiling of a tunnel-heading, &c, in which case 
one of its ends is provided with a screw which is run out so as to cause tlie two ends 
of the col to press firmly against tlie opposite rock walls; or rather against wooden 
blocks which are always placed between each end of the col and the rock. In any 
case, the supports of the drill are so jointed that it can bore in any direction. 

Art. If). Frequently the drill is clamped to a short arm, which, in 
turn, is clamped to the column, and projects at right angles from it. The arm may 
be slid lengthwise of the column, and may be revolved around it, and thus may he 
placed in any desired position, and there clamped. This gives the drill a greater 
range of motion, and enables it to bore holes over a greater space tliau would other¬ 
wise he possible without moving the column. 

Art. 20. In tunnels, one or more drills may be mounted upon a drill-car¬ 
riage, travelling upon a railroad track running longitudinally of the tunnel. 
Upon this track the carriage is moved up to the work, or run back from it when a 
blast is to be fired. The gauge of the track may be made wide enough to admit of 
a second track, of narrower gauge, running underneath the dr.11-carriage. Upon 
said narrower track the cars are run which carry away the debris. Drill-carriages 
are less commonly used in this country than in Europe. 

Art. 21. The pressure used in the cylinders of percussion drills is 
usually from about tit) to 70 lbs per sq inch. In an hour, one will drill 
a hole from 1 to 2 ins diam, and from •'! to 10 ft deep, depending ou the character of 
the rock and the size of the machine at from 10 to 25 Cts per tin ft witli labor at 
$1 per day. A bit requires sharpening at about every 2 to 4 ft depth of 
hole. One blacksmith and helper can sharpen drills for 5 or 6 machines. 

Art. 22. The bits are of many different shapes, varying with 
the nature of the work to he done. For uniform hard rock, the bit has two cutting- 
edges, forming across with equal arms at right angles to each other. For seamy 
rock, the arms of the cross are equal, but form two acute and two obtuse angles with 
each other, as in the letter X. For soft rock, the cutting-edge sometimes has the 
shape of the letter Z. 

Art,. 2S. Each drill requires one man to operate it. Two or three men 

are required fur moving the heavier sizes from place to place. One man can attend 
to a small air-compressor aud its boiler. 

Art. 24. Figs 4 and 5 represent the “ Eclipse” percussion drill of the Inger- 
soll Rock-I)i ill Co, No 10 Park Place, New York. Fig 5 shows the drill, mounted 
(its is most frequently the case) upon a tripod. Fig 4 is a longitudinal section through 
the cylinder, valve-chest, and piston. 

Art. 2«>. Ihe cylinder, C, is provided at each end with a rubber cushion, 

N, for deadening the blows of the piston, which, in all percussion drills, is liable, at 
times, to stiike either cylinder-head. The side of each cushion nearest the piston is 
protected by a thin iron plate. The cushions hove to be renewed from time to time. 

Art. 26. The valve, V, is shaped somewhat like a spool. The bolt, B, 
passes loosely through its center and guides it. Steam is admitted from the boiler 
to the steam-chest, and occupies all of the space between the two end flanges of the 
valve, except it. It drives the valve alternately from one end of the valve-chest to 
the other, and back, according as one end or the other is relieved from opposing 
pressure by being put into communication with the exhaust, E, by way of the pas¬ 
sages, D I)' aud F F'. D and b' communicate with the ends of the steam-chest 
through passages not shown; while F and F' communicate, through similar pas¬ 
sages, with the exhaust, E. The piston has an annular channel, L L', encircling it 
Whatever the position ot the piston, one of the passages.!) or D', is always, by means 
ot this channel, in communication with its corresponding passage, F or F / , leading 

* To avoid repetitions, we will use the woid steam to signify either steam or compressed air, 
whichever happens to be used. r 







MACHINE ROCK-DRILLS. 


655 


to the exhaust. Thus, one or the other end of the valve-chest is always in com¬ 
munication with the open air; and to that end the valve is driven by the pres of 
the steam surrounding it, admitting steam to the cyl, C, from the other end. 

Art. 27. Tile rotation of the piston, and, with it, that of the drill- 
bar, is effected thus: The spirally-grooved, cylindrical steel bar, A, called a ride¬ 
bar, passes through and works in, the rifle-nut, 11, which is firmly fixed in 
the end of the piston, and has spiral grooves corresponding with those on the rifle- 
bar. Said bar is fixed, at its upper end, to the ratchet-wheel, J, the pawls of which 
are so arranged that, on the clown stroke of the piston, the rifle-nut, H, acting upon 
the grooves on the rifle-bar, causes it, and, with it, the ratchet-wheel, to revolve 

about their common axis. The weight and mo¬ 
mentum of the piston, &c, are such that it thus 
readily turns the ratchet-wheel without itself 
turning. Thus the bit is prevented from rotating 
while delivering its blow. But, on the up stroke, 
the tendency of the rifle-nut is to turn the rifle- 
bar and ratchet-wheel in the opposite, direction ; 
and as this is prevented by the pawls, the rifle- 
bar remains stationary , while the piston, piston- 
rod, and drill are made to revolve about their 
common axis. 

Art. 28. The feed-screw, M, is col¬ 
lared, at its upper end, to the fixed frame, Q. It 
is thus prevented from moving longitudinally 
when revolved by means ot the crank fixed to its 
top. Its lower end works in a nut, T, fixed to the 
cylinder, which last is thus moved longitudinally 
backward or forward as the crank is turned. 




Large drills are frequently furnished with an automatic feeding arrange¬ 
ment in addition to the hand-crank. In this arrangement, when the cylinder 
requires feeding forward, and when, consequently, the piston is running nearly to 


































































656 


MACHINE ROCK-DRILLS. 


the forward limit of its stroke, the piston presses against a cam projecting into 
cvl near the forward end, and presenting an inclined plane to it. The motion 
tiiis cam, by means of an exterior axle, running alongside ot the cyl and lurnisl i 
at its top with a dog, turns a ratchet-wheel fixed to the feed-screw. W hen desit 
the automatic feed may be thrown out of gear, and the feed moved by hand. 

Art. 29. The tripod leg's consist of wrought-iron tubes, W W. These : 
screwed at their upper ends into sockets, X X. At their lower ends, they rece 
the pointed and tapering steel bars, Y Y, about 2 or 3 ft long. The legs may 
lengthened or shortened by turning tlie set-screws, Z Z, thus regulating the distal 
to which the bars, Y Y, can enter the legs. The clamps, l> b, have L-shaped hoi -* 
of inch to 1 inch round iron forged to them. On these hooks the weight 
d d. are hung, which hold the machine down against tli6 upward reaction ot 
blows. ' 

Art. 30. The following table gives the principal dimensions of tin 1 < 
drills, with the diams and lengths of holes to which each is adapted. 

for prices, apply to the Co at the above address. We give the card prices 1 
1886, These may be taken as giving, approximately, the present range of prices 
percussion drills of any first-class make. Size II is used for submarine work, hea 
tunneling, and deep rock cutting. G and F for tunneling,street grading, quarrvi 
and sewer work. B, D, and 0 for general mining purposes. Bis adapted only 
very light work. In asking for estimates on drills and compressors, give the fulK 
possible description (accompanied by a sketch) of the work to be done, stating 
present and proposed extent. State whether the work is on the surface or undi 
ground. State how far the steam or compressed air will be carried. Give depth 
holes to be drilled, nature of rouk, Ac. Percussion drills are sold without restricti 
as to the purpose or extent to which they are to be used. 


LIST OF INGERSOLL “ECLIPSE” PERCESSIOX ROCK 

dkileim; machines. 


Inner diam of cylinder.ins. 

Length of full stroke. “ 

“ feed. “ 

“ machine*. “ 

Wt of machine, unmounted_lbs. 

“ tripod, without the wfs. “ > 
“ three wts for tripod legs. “ > 
“ column, arm and clamp. “ ) 

Diam of hole drilled.ins. 

Maximum depth of vertt hole.ft. 

Prices. 

Machine, unmount'd,ivitho'tdrills. ) 
Bet of drills for above depth of hole. > 
Tripod, with weights. ) 


Column, 8 ft long, with arm & clamp.. 


Letter designating the size of the machine. 


A 

R 

c 

D 

E 

F 

O 

F 

m 


2 H 

3 

34 

354 

*4 

i 

3 

4 

5 

6 

6 

6^ 

7 

1 

12 

20 

24 

21 

24 

26 

34 

31 

86 

34 

36 

40 

42 

53 

60 

6( 

80 

155 

11*5 

230 

250 

345 

605 

67( 

c 

125 

125 

125 

125 

150 

275 

27c 

l i 

250 

250 

250 

250 

350 

400 

401 

( 

200 

280 

280 

2*0 

420 

420 

42C 

bi to % 

H to 14 

1 to 2 

1 to 2 

l to 2 

1 4 to 24 

2 to 4 

3 t c 

X 

4 

8 

10 

12 

16 

30 

40 


$ 

$ 

$ 

$ 

$ 

$ 

8 

( 

230 

255 

280 

300 

350 

375 

425 

X ) 

8 

18 

21 

31 

to 

139 

400 

l 

45 

45 

45 

45 

45 

55 

55 


Column 44 in 1 ! diam. 

Column 6 ins diam. 

X 

80 

| 80 

| 80 

| 80^ 

^110 

j 110 

j 110 


* From top of hand'e of feed-crank to lower end of piston at the end of the down stroke, 
t For greatest advisable length of lior holes, deduct one-fourth flout these depths. 

J Machine A is mounted on a small frame. Price, so mounted, $150. Hole 18 ins deep. Drills $ 


Art. 31. Tlie drills of different makers differ chiefly in th 

methods by which the valve is operated. In some this is done, as in the itigerso 
“ Eclipse” drill, Art 26, by the pres of steam. In others, the valve is moved b 
a lever or tappet, which projects into the cylinder so as to come into contat 
with, and lie moved by, the piston at each stroke. As these strokes are made wit 
great force some 300 or more times per minute, such valve-gear is necessarily subjec 
to great wear. 

Art. 32. In the “ kittle Giant Drill,” made by the Rand Drill Co 

office 23 Park Place, New York, the valve, V, Fig 6. is slid backward and forwari 
in the same direction in which the piston is moving, by the tappet, T, which i 
pivoted at p. The inclined lower corners of this tappet ride up as they come, altci 
nately, in contact with the shoulders, ss, of the piston. 

Art. 33. The same Co have recently, 188-i, brought out two lie 
drills, the 44 Economizer ” and the “Slugger”; in each of which t J 
vaJve, as in the Ingersoll “ Eclipse” drill, is moved by steam, but upon a quite < 






































Fig.S 

the downward stroke. The piston-rod, o, is made lighter than in 

itlier <1 ri I is. This gives a greater surface under the piston for the pressure of 
/•» steam on the up stroke, and, consequently, greater lifting power. This is use- 
• when the drill sticks in the hole. 

4 he tripod legs are of bar iron. Their length is adjustable. 

43 


MACHINE ROCK-DRILLS. 657 


• rent principle. In these two drills, there is no steam cushion for the piston 
strike against on the down stroke, the force of which is thus more completely 
upended upon the rock. The cushion behind or above the piston, on the return 
roke, is formed by exhaust steam. Both of these drills cut off steam before 
61 « completion of either stroke, thus using the steam expansively. On the down 
i oke, the “Economizer” cuts off earlier than the “ Slugger.” Hence its name. 
1 both machines the point of cut- 


:, c -f is fixed when the machine is 
made. 

Art. 34. In the improved 
t Burleigh drill, the valve, V, 
Fig 7, is moved by two tappets, 
l' T', which are alternately struck 




100 

lit! 


lie 

oy the ends ol the piston, P. Bur¬ 
nt leigli Bock-Drill Co, mfrs, 
itchburg, M;iss; office 115 Liberty 
„New York. 

Art. 35. In the “Dynamic 1 
.^k-drill, invented by Prof Be 
Olson Wood, and developed 
,id manufactured by the Cray- 
•ion iV Bouton MIg Co, 15 

Cortlandt St, New York, the valve is attached to a valve-piston, V, Fig 8 , which 
js moved backward and forward by steam, which is r.dmitted so as to act alternately 
upon its two ends. The admission of 
his steam is controlled by a small 
nxiliary valve, a. A hub on the 
tack of the auxiliary valve fits in the 
piral groove shown on the plug, n. 
his plug is constantly pressed down- 
J 1 yard (as the Fig stands) by steam 
l pressing upon its upper shoulder, but 
H) it is lifted at each forward stroke by 
~ the conical surface of the piston, P, 
pressing against its foot. It thus 
moves constantly up and down, carry¬ 
ing the valve, a, with it. By turning 
the plug, n, by means of the adjusting- 
stem, s, the hub of the valve is made to occupy a higher or lower point in the spiral 
groove, and thus the stroke of the piston may be varied, or may be confined to any 
part of the cylinder. 

In this drill, unlike the Ingersoll, Art 27, the )>isloii rotates while making 



















































































































658 MACHINE ROCK-DRILLS. 


Art. 36. The hand rock-drilling' machine of the Pierce Well 
Excavator Co, New York, and Long Island City, N Y, is a percussion drill. It is 
worked by a crank which turns a disc about 2 ft in diaui. The disc has a semi-circular 
slot, iu which works the arm which raises the drill-rod. This arm, in rising,compresses 
a coil-spring, which, on the down stroke, drives the drill against the rock. An iron 
ball, weighing 30 lbs or more, is furnished with each machine. This ball may be 
screwed to the top of the drill-rod, for giving greater force to the blows of the drill. 
The ball may be used without the spring, by disengaging the latter. 

The drill makes about 40 strokes of 10 or 12 ins per minute; and bores holes from 
% to 2% ins diam. It can be arranged to drill to depths of 30 ft and over. For 
sharpening the bits, it has an emery wheel attached, which is turned by the crank. 
The latter, at such times, is thrown out of gear with the disc. 

The drill is mounted on a rectangular two-legged frame, about 5 ft high 
by 2 ft wide, made of iron tubes. To the top of this frame a third leg is attached, 
by adjusting which the angle of the drill-rod with the vert may be changed. Like 
other percussion drills worked by hand-power, this one ceases to work to advantage 
when said angle exceeds about 45°. These drills cost, ready for work, in 1886, 
$225 each. They weig h from 200 to 400 lbs. They are moved from place to place 
like a wheelbarrow, the disc serving as a wheel. 

Art. 37. Channeling consists iu making long, deep, and narrow cuts iu 
the rock. Iu this way large blocks can be gotten out without blasting and the con¬ 
sequent danger of fracture. This is ordinarily done by boring a row of holes about 
an inch apart in the clear, and then breaking down the intermediate spaces by 
means of a blunt tool, called a breach. This is called broach channeling. 
For this purpose a steam drilling machine is mounted upon a hor bar resting upon 
two pairs of legs. The hor bar is placed over the intended row of holes, and the 
drill is slid along upon it from one hole to the next. In using the broach , the rotat¬ 
ing apparatus is thrown out of gear, so that the edge of the broach maintains its 
position in line with the row of holes. 

Art. 38. The Saunders patent channeling machine, of tho 

Ingersoll Co, consists of a rock-drilling machine, having, in place of the usual drill¬ 
ing-bit, a gang of tools consisting of a number of chisels, clamped together side by 
side, and thus forming a cutting tool about 7 ins long by % inch wide. This tool 
has as many cutting-edges (each as long as the tool is wide) us there are chisels. 
The machine is supported upon a carriage, moving on a track parallel with the 
channel to be cut. The tool is of course not rotated; but the rifle-bar, A, Fig 4, is 
employed to move the carriage along the track about an inch after each blow. Tho 
carriage remains stationary while a blow is being struck. Under favorable circum¬ 
stances this machine has cut from SO t<> 100 sq ft of channel per day of ten 
hours. Its weight, including carriag;e, is about 5000 lbs. Cost, 1886, about $2300. 

A valve is provided, by which, if desired, the steam may be shut olF trom 
the piston on the down stroke, so that said stroke may be made with only the weight 
of the piston, rod, and drill. 

Art. 3!). The Ingersoll Co have a special appliance, designed by Mr. W. L. 
Saunders, C E. for drilling and blasting rocks under w ater, even 
when they are covered by a considerable depth of mud. 

Art. 40. Air compressors for rock-drills, as made and used in this coun¬ 
try, are mostly hor, direct-acting engines. That is, the axes of the steam- and air- 
cylinders are hor; and the piston-rod passes directly from the steam-cylinder into 
the air-cylinder. A fly-wheel is attached, by a crank and connecting-rod, to the 
piston-rod. Sometimes the steam-engine is separate from the compressor, and the 
power is conveyed to the latter by belts or gearing; or water-power may be used in 
the same way. The air is forced into a receiver, which is generally a plate-iron 
cylinder, 3 or 4 ft in diam, and 5 to 12 ft long. 

If the air- or pumping-cylinder of the compressor is so arranged as to take in air 
on one stroke only, and force it out into the receiver upon the return stroke, it is 

siligic-ucting.” It, at each stroke, it both takes iu and forces out air, it is 
“double-acting.” If the compressor has only one air-cylinder, it is “sill- 
glt 1 .” II it has two, and thus practically consists of two single compressors, it is 
“duplex.” 

The valves may be either “ poppet” valves, held in place by springs, and 
operated by the pressure of the air itself; or slide valves, operated by eccentrics 
and rods, as in steam-engines. 

Tho compression of the air develops heat. This is removed either by causing 
cold water to circulate through the air-piston, and through jackets surrounding the 
air-cylinder; or by injecting it into the air-cylinder iu the form of spray. Or both 
methods may be used together. 

Art. 41. Compressors are furnished by the Ingersoll Rock-Drill 

Co, 10 Park Place, New York; Rand Drill Co, 23 Park Place, New York; Burleigh 


Cl 


Hi 

iDia 

(Lei. 

jxi 


|C» 

Afl 


Oh 


It 


let 

*1 

*r 



» 

Ml 


- 


1N( 






MACHINE ROCK-DRILLS. 659 


Rock-Drill Co, 115 Liberty St, New York ; Graydon & Denton Mfg Co, 15 Cortlandt 
St, New York; Clayton Steam Pump Works, office, 45 Dey St, New York, and others. 

The following partial list of Clayton compressors, compiled from data 
riven by the makers, shows the dimensions and performance of each. 
vVe give also a list of their receivers. Por prices, apply to the Co as above. 
Ye give the card prices of 1886 as an approximation. 

CLAYTON DOIJBEE-ACTING AIR-COMPRESSORS. Partial List. 


Number, designating the size 
of the machine. 


Duplex Direct-acting* Compressors. 


Diani of steam-cylinders.ins. 

“ air “ .I...ins. 

Length of stroke.ins. 

Number of revolutions per minute. 

Cub ft of free air compressed per minute.Actual. 

Approximate wt of compressor. lbs. 

Approx number of rock-drillsf with 3-incli cyls sup¬ 
plied with air at 60 lbs per sq inch. 

Card price, 1886.*.$ 


Single Direct-acting* Compressors. 


Diarn of steam-cylinder.ins. 

“ air “ .bis. 

Length of stroke.bis. 


Number of revolutions per minute 


Cub ft of free air compressed per minute.Actual. 

Approx wt of compressor......... 

Approx number of rock-drillsf with 3-incli cyls sup¬ 
plied with air at 60 lbs per sq inch.. 

Card price, 1886.*...* 


1 


4 

7 

8 

10 

14 

18 

8 

10 

14 

18 

12 

13 

15 

24 

f 120 

100 

100 

80 

< to 

to 

to 

to 

(140 

130 

120 

90 

136 

210 

438 

900 

3000 

7000 

15000 

25000 

2 

4 

8 

18 

1320 

1540 

3025 

6050 

8 

10 

14 

18 

8 

10 

14 

18 

• 12 

13 

15 

24 

(120 

100 

100 

80 

\ to 

to 

to 

to 

(140 

130 

120 

90 

68 

105 

219 

450 

1650 

3850 

8250 

13750 

1 

2 

4 

9 

725 

850 

1650 

3300 


* The price of a compressor alone, to be worked by a separate steam-engine or water-power, is of 
course less than that of the above compressor and engine combined. 

1 Some makes of rock-drills require much more air than others. 


Air-Receivers; vertical or horizontal. 


Diara, 

Ins. 


33 

30 

36 

40 


L’gth, 

Ft. 

Approx 
wt, lbs. 

Card Price, 
i886. 

Diam, 

Ins. 

L’gth, 

Ft. 

Approx 
wt, tbs. 

5 

700 

$100 

40 

8 

1675 

7 

890 

120 

40 

10 

1900 

8 

1560 

150 

40 

11 

2000 

6 

1600 

160 

40 

12 

2100 


Card Price, 
1886. 


$168 

200 

212 

225 


The above prices include brass-face pressure-gauge, glass water-gauge, safety-valve, 
blow-off valve, try-cocks, flanges, and connections to automatic feed on compressor. 


























































660 


EXPLOSIVES. 


GUNPOWDER. 

The explosive force of powder is about 40000 lbs, or 18 tons, per square 
inch. Its weight averages about the same as that of water, or 62}^ lbs per 
cubic foot; hence, 1 lb = about 28 cubic inches. In ordinary quarrying, a cubic 
yard of solid rock in place, for about 1.9 cubic yards piled up after being quar¬ 
ried,) requires from ^ to ^ In very refractory rock, lying badly for quarry¬ 
ing, a solid yard may require from 1 to 2 lbs. In some of the most successful tt 
great blasts for stone for the Holyhead Breakwater, England, (where several » 
thousands of lbs of powder were usually exploded by electricity at a single si 
blast,) from 2 to 4 cubic yards solid were loosened per lb ; but in many instances 
not more than 1 to yards. Tunnels and shafts require 2 to 6 lbs per solid If 
yard; usually 3 to 5 lbs. Soft, partially decomposed rock frequently requires e 
more than harder ones. Usually sold in kegs of 25 lbs.* ii 

Weight of powder in one foot depth of hole. 


Diameter of hole In inches. 

1 I IX ! I 2 | 'VA I 3 | 3% 1 4 | 4>$ I 5 I 

Weight of powder in pounds and ounces avoir. 

0 '5 | 0"8 | 0"11 | t"4 | 2 | 2"13 | 3"14 | 5”0 | 6"6 | 7"14 | 

* Price, 1886, in Atlantic cities, about $2.50 per 25 ft keg. 




9''8 |11 "5 


l* 


I 


o 







MODERN EXPLOSIVES. 


661 


MODERN EXPLOSIVES. 


'• Art. 1. Most of the explosives, which, of late years, have been taking 
'j ;he place of gunpowder (p 660), consist of a powdered substance, part ly saturated 
' with nitro-glycerine,a fluid produced by mixing glycerine with nitric and sulphuric 
icids. 

Art. 2. Pure liitro-Kiycerine, at 60° Fall, has a sp grav of ljB. It is odor- 
ess, nearly or quite colorless, and has a sweetish, burning taste. It is poisonous, 
jven in very small quantities. Handling it is apt to cause headaches. It is insoluble 
in water. At about 306° Fah it takes fire, arid, if unconfined, burns harmlessly, 
unless it is in such quantity that a part of it, before coming in contact with air, be¬ 
comes heated to the exploding point, which is about 380° Fah. 

N-G, and the powders containing it, are always exploded by means of 
/sharp percussion. See Arts 36, &c. After N-G is made, great care is required to 
wash it completely from the surplus acids remaining' in it from the 
iprocess of manufacture. Their presence, either in the liquid N-G, or in the powders 
Containing it, renders the N-G liable to spontaneous decomposition, which, by rais¬ 
ing the temperature, increases the danger of explosion. 

Art. 3. N-G freezes at about 45° Fah. It is then very difficult of ex¬ 
plosion, and must be thawed gradually, as by leaving it fora sufficient length 
of time in a comfortably warm room, or by placing the vessel containing it in a sec¬ 
ond vessel containing hot water, not over 100° Fah; but never by exposing it to 
intense heat, as in placing it before a fire, or setting it on a stove or boiler. Extra 
strong caps are made for exploding N-G and its powders when frozen. 

Art. 4. N-G, owing to its incompressibility, is liable to explosion 
through accidental percussion. This, and its liability to leak¬ 
age, render it inconvenient to transport and handle. Hence it is rarely used in 
the liquid state in ordinary quarrying and other blasting. In the oil regions of 
Penna.it is largely used in oil wells, in order to increase the flow. For this 
purpose it is confined in cylindrical tin casings, from 1 to 5 inches diam, called tor¬ 
pedo-shells. These are suspended from, and lowered into the well by means of, a 
cord or wire wound on a reel; and are destroyed when the charge is exploded. 
They are about 1 inch less in diam than the well, and contain usually from one to 
twenty quarts = 3 lbs, 5*/^ oz to 66 lbs, 6J^ oz of N-G. They are pointed at their 
lower ends, in order to facilitate their passage through the oil or water which may 
be in the well. When a greater charge than about 66% lbs is required, two or more 
of these shells are placed in the well, one on top of another, the conical point on 
the lower end of each one fitting into the top of the one next below. In this case, 
the N-G is fired by means of a cap or series of caps placed in the top of the charge 
before it is lowered. When the charge is in place, the caps are exploded by elec- 
| tricity led to them by conducting wires, as in Art 37, or (as in^the method more 
icommonly practised) by letting a weight fall on them. 

When a well has been repeatedly torpedoed, and a cavity has thus been formed in 
it so large that the space surrounding a torpedo would interfere too greatly with 
the effect of the explosion of the N-G on the walls of the well, the latter is placed 
directly in the well, by lowering a tin cylinder, filled with it, and provided with an 
automatic arrangement which allows the N-G to escape when at the bottom of the 
well. The N-G is then tired by a torpedo suspended on a line, and having caps 
placed in its top. These caps are exploded by a leaden or iron weight sliding down 
the line, or by electricity. When the rock is seamy, the N-G is confined in short 
cylindrical tin shells, lowered into the cavity, and fired by a torpedo. N-G and tor¬ 
pedoes of N-G, and of “ Atlas ” and “ Hercules ” powders, are furnished by The Tor¬ 
pedo Co of Delaware, office Warren, Pa. Price of N-G, 1886, about 75 cts per lb. 
It is also used for increasing the flow of springs of water. It of course cannot be used 
in hor or upward holes, such as often occur in tunneling, &c. 

Art. 5. N-G explodes so suddenly that very little tamping: is re¬ 
quired. Moist sand or earth, or even water, is sufficient. This, with the fact 
that N-G is unaffected by immersion in water, and is heavier than water, render it 
particularly suitable’for sub-aqueous work, or for holes containing 
water, provided the rock has no seams which would permit the N-G to escape. If 
tlie rock is seamy, the N-G must be confined in a water-tight casing. Such 
casings, however, necessarily leave some spaces between the rock and the explosive, 
and these diminish considerably the effect of the latter. 

Art. 6 . The great explosive force of N-G is due partly to the very 
large volume of gas into which a small quantity of it is converted by explosion, and 








662 


MODERN EXPLOSIVES. 


partly to the suddenness with which this conversion takes place, the gases being 
liberated almost instantaneously * while with gunpowder their liberation requires j 
a longer time. The suddenness of the explosion increases its effect, not only by 
applying all of its force practically at one instant, but also by greatly heating (he 
gases produced, and thus still further increasing their volume. 

Art. 7. Tim liquid condition of N-G is useful in causing it to fill tile drill¬ 
hole'completely, so that there are no vacant spaces in it to waste the force 
of the explosion. On the other hand, the liquid form is a disadvantage, because, j 
when thus used without a containing vessel in seamy rock, portions of the N-G leak 
away and remain unexploded and unsuspected, and may cause accidental explosion 
at a future time. 

Art. 8. is stored in tin cans or earthenware jars. If 

properly washed from acid it does not injure tin. For transportation, these cans or 
jars are packed in boxes with sawdust, or in padded boxes, and loaded in wagons. 
The R R companies do not receive it. 

Art. 9. When N-G and its compounds are completely exploded, tile gnses 
{fiven out are not troublesome, but those resulting from incomjdete explosion, 
such as generally takes place, or from combustion, are very offensive. 

Art. 10. For convenience, we apply the name “ dynamite ” to any explo¬ 
sive which contains nitro-glyceiine mixed with a granular absorbent; ‘‘true 
dynamite ” to those ill which the absorbent of the N-G is “ Kieselguhr,”f or 
some other inert powder which takes no part in the explosion; and “false 
dynamite to those in which the absorbent itself contains explosive substances 
other than N-G. 

Art. 11. The absorbent, by its granular and compressible condition, 
acts as a cushion to the N-G, and protects it from percussion, and from 
the consequent danger of accidental explosion. 

N-G undergoes no change in composition by being absorbed; and it then freezes, 
burns, explodes, Ac, under the same conditions as to pressure, temperature, Ac, as 
when in the liquid form. The cushioning effect of the absorbent merely renders it 
more difficult to bring abont sufficient percussive pressure to cause explosion. The 
absorption of the N-G in dyn enables the latter to be used in hor holes, or in holes 
drilled upward. 

Art. 12. N-G and dyn explode much more readily when rigidly 
confined, as by a metallic vessel, or by the walls of a hole drilled in rock, than 
when confined by a yielding substance, as wood. Therefore the fact that dyn, not 
being liquid, can be packed in wooden boxes, renders it safer than N-G which lias to 
be kept in stone or metal vessels. 

Art. 13. True dynamites must contain at least about 50 per 
cent of N-G. Otherwise the latter will be too completely cushioned by the absorbent, 
and the powder will lie too difficult to explode. False dynamites, on (lie contrary, 
may contain as small a percentage of N-G as may be desired; some containing as 
little as 15 percent. The added explosive substances in the false dynamites generally 
contain large quantities of oxygen, which are liberated upon explosion, and aid in 
effecting the complete combustion of any noxious gases arising from the N-G. 

Art. 14. Dynamites which contain large percentages of 
N-<i explode (like the liquid N-G. Art 6) with great suddenness, tending to shatter 
the rock in their vicinity into small fragments. They are most useful in very hard 
rock. In such rock, No 1 dynamite, or that containing 75 per cent of N-G, is 
roughly estimated to have about 6 times the force of an equal wt 
of gunpowder. 

For soft or decomposed rocks, sand, and earth, the lower grades 

of dynamite, or those containing a smaller percentage of N-G, are more suitable. 
They explode with less suddenness, and their tendency is rather to upheave large 
masses of rock, Ac, than to splinter small masses of it. They thus more nearly re¬ 
semble gunpowder in their action. | 

Judgment must be exercised as to the grade and quantify of explOMivc 
to be used in any given case. Where it is not objectionable to break the rock into 
small pieces, or where it is desired to do so for convenience of removal, the higher, i 
shattering grades are useful. Where it is desired to got the rock out in large masses, 
as in quarrying, the lower grades are preferable. 

For very difficult work in hard rock, and for submarine blasting, the highest 
grades, containing 70 to 75 per cent of N-G, are used. A small charge of these does 
the same execution as a larger charge of lower grade, and of course does not require 

1 

—---—--- 

* Such sudden liberation of gas is called “ detouation.” 

t Kiesclguhr is an earthy, silicious limestone, composed of the fossil remains of small shells. 
Each shell acts as a minute receptacle for nitro-glycerinc. Kieselgubr is found in Hanover, Germany, 
and in New Jersey. 






MODERN EXPLOSIVES. 


663 


In submarine work their sharp explosion is not 


For general railroad work, ordinary tunneling, mining of ores, &c, the a 
i'e grade. containing 4u per cent of N-G, is used ; for quarry in;;, ; 
it; for blasting stumps, trees, piles, &c, 30 per cent; for sand 


aver- 
35 per 

and 


the drilling of so large a hole, 
leadened by the water 
For 

:en 

ear til. 15 per cent 

Art. 15. Dynamite, like N-G, can be readily exploded under 

water, provided it is so immersed as not to be scattered; but long exposure 
to water is injurious to it. In the higher grades, the water, by its greater 
ifflnity for the absorbent, drives out the N-G. lu the lower grades it is apt to wash 
away the salts used as additional explosives. 

Art. 16. In dyns containing a large percentage of N-G, the latter is liable 
to exude ill liquid form, or to “leak,” especially in warm weather, and then to 
explode through accidental percussion. The same danger exists, even though the 
percentage of N-G be small, if the absorbent has but small absorbing power, and is, 
consequently, easily saturated. 

Art. 17. True dyn resembles moist brown sugar. Its properties are 
generally those of the N-G contained in it. Thus, it takes fire at about 350° F, and 
burns freely. It freezes at 45° F, and is then difficult to explode. It is not exploded 
by friction, or by ordinary percussion, but requires, for general purposes, a strong 
cap, or exploder, containing fulminating powder, see Arts 36, 38, &c. It may, how¬ 
ever, be exploded by a priming of gunpowder, tightly tamped, and fired by an ordi¬ 
nary safety-fuse. 

Art. 18. The charge should fill the cross section of the 

hole as completely as possible. If water is not standing in the hole, the cartridge 
should be cut open before insertion, so that the powder may escape from it and fill the 
j hole; or the powder may be simply emptied from the cartridge into the hole. 

’’ Art. 19. For blasting' ice in place, holes are cut in it, and a number of dyn 
cartridges (one of which must contain an exploding cap) are tied together and low- 
a i ered from 1 to 5 ft into the water. They are fired as soon as possible after immer- 
6 siou, to avoid the danger of freezing. Electrical exploders (Arts 37, &c,) are best 
for sub-aqueous work. 

Art. 20. Dyn is useful for breaking up pieces of metal, such as old 
cannon, condemned machinery, “salamanders” (masses of hardened slag) in blast 
(furnaces, &c. In cannon, the dyn is of course exploded in the bore. In other pieces, 
small holes are generally drilled to receive it; but plates, even of considerable thick¬ 
ness, may be broken by merely exploding dyn upon their surface. 

Art. 21. For blasting trees or stumps, one or more cartridges are 
fired in a hole bored in the trunk or roots, or under the latter. This shatters both 
i trunk and roots. A tree may be felled neatly by boring a number of 
small radial holes into it, at equal short dists in a hor line around its circumf, and, 
j by means of an electric battery (Arts 37, &c), exploding simultaneously a small 
{ charge of dyn in each. Or a single long cartridge may be tied around the trunk of 
i a small tree, and fired. 

Art. 22. Piles may be blasted in the same way as trees; or a hole may 
i be bored for the cartridge in the axis of the pile; or the cartridge may be simply 
, tied to the side of the pile at any desired lit. 

Art. 23. The higher grades of dyn, like N-G, reqnire but little tamp¬ 
ing. Use a wooden tamping-bar, never a metullic one, for any explosive. If 
a charge of dyn “ hangs fire,” it is danger-ous to attempt to remove it. Remove 
the tamping, all but a few ins in depth, on top of which insert another cartridge, 
containing an exploder, and try again. See electiical exploders, Arts 37, <fce. Dyn, 
like N-G, if frozen, must be thawed gradually, by leaving it in a warm room, far 
from the fire; or by placing it in a metallic vessel, which is then placed in another 
| vessel containing hot water. The water should not be hotter than can be borne by 
, the hand. Otherwise the N-G is liable to separate from the absorbent. The N-G in 
| dyn may freeze without cementing together the particles of the absorbent; in 
which case the powder of course is still soft to the touch. An overcharge of 
N-G, or of dyn, is liable to be burned, and thus wasted, giving off offensive gases. 

Art. 24. Dyn is sold in cylindrical, paper-covered cart¬ 
ridges. from % to 2 ins in diam, and 6 to 8 ins long, or longer. They are fur¬ 
nished to order of any required size, and are packed in boxes containing 25 lbs or 50 
j lbs each. The layers of cartridges are separated by sawdust. 

Art. 25. Some of the It It companies decline to carry dyn or N-G in any 
shape. Others carry dyn under certain restrictions, based upon State laws; pro¬ 
viding that it must be dry (i e, that no N-G shall be exuding from it); that boxes 
and cars containing it shail be plainly marked with some cautionary words, as “ex¬ 
plosive,” “dangerous,” &c; that the cartridges shall be so packed in the boxes, and 
the boxes so loaded in the cars, that both shall lie upon their sides, and the boxes 







664 


MODERN EXPLOSIVES. 


be in no danger of falling to the floor; that caps, Ac, shall not he loaded in the same 
car with dyn, &c, &c. 

Art. 2t>. A great many varieties of <lyn are made. They differ 
(generally hut slightly) in the composition of the absorbent, and in the method of 
manufacture. Each maker usually makes a number of grades, containing different j 
percentages of N-G, Ac, and gives to his powders some fanciful name. 

Art. 27. The following table of explosives made by the Repauuo 
Chemical Co, at Thompson’s Point, N J, ottice Wilmington, Del. and known as “ At¬ 
las ” powders, gives the percentage of N-G in each, and the approx card 
price for 1886. It gives a general idea of the range of American dyns. 


Brand. 

Percentage 
of N-G., 

Card price, 
1886, 

cts per fl>. 

Brand. 

Percentage 
of N-G. 

Card price, 

1886, 

cts per tt>. 

A 

75 

36 

D + 

33 

21 

B + 

60 

32 




B 

50 

28 

E + 

27 

19 

Cf 

45 

26 

E 

20 

16 

c 

40 

24 





Tlie absorbents contain: in “A” brand, 18 per cent wood pulp and 7 
per cent carbonate of magnesia; in “C” brand (the average grade), 46 per cent 
nitrate of soda (soda saltpetre), 11 per cent wood p»iIp, and 8 per cent carbonate of 
magnesia; in “E” brand, 62 per cent nitrate of soda, 16 per cent wood pulp, Ac, and 
2 per cent carbonate of magnesia. 

Art. 28. “Miner’s Friend” powder, made by the Hecla Powder Co, 
office 289 Broadway, New York, contains nitrate of soda, wood pulp, resin, and car¬ 
bonate of magnesia. It freezes at 42°, and is then, like other dyn, difficult to ex- j 
plode. When used under water, the cartridges should not be broken, because the 
powder is injured by direct contact with water. Their “ Hecla ” powder is a 
lower grade. It is in granulated form, like ordinary blasting powder, but is said to 
be much stronger. It is intended as a substitute for it. 

Art. 2i>. “ Giant ” powder is made by Atlantic Dynamite Co, office 245 
Broadway, New York. No 1 is dyn proper, containing 75 per cent N-G, and 25 per 
cent Kieselguhr obtained near their works in New Jersey. Their lowest grade, 
branded “ M,” contains 20 per cent N-G. The name “giant powder” was originally 
applied to dynamite in general. 

Art. 30. Other brands are “ Hercules” powder, Hercules Powder Co, 40 
Prospect St, Cleveland, 0; and “ .1 udson R R I» powder.” Atlantic Dyn Co., 
New York, a substitute for ordinary blasting powder. It is put up in water-proof 
paper bags, of 6J4, 12}/), and 25 lbs each, and these are packed in wooden boxes hold¬ 
ing 50 lbs each. The same Co furnish also “ Jndson F F I' dynamite,” a 
higher grade, in cartridges of the usual shape, packed in 50-lb boxes. 

Art. 31. “ Raekaroek ” cartridges, furnished by Rendrock Powder Co, 23 
Park Place, New York, are said to contain no N-G, and to be entirely inexplosive 
until immersed, for a few seconds, in an inexplosive liquid furnished by the same 
Co. They are then allowed to stand for 15 mins, after which they may be used at 
any time. They are fired in the same way as dyn, and can be used under water. 
The infra claim that they “approximate N-G in strength, and are stronger than 
dyn.” 

Art. 32. Tl»e following' explosives are made and used in 
Europe, but have not yet been regularly imported into the U S. 

Fompressed gun-cotton, made at Stowmarket, Eng, is cotton dipped in 
a mixture of nitric and sulphuric acids, then reduced to a fine pulp, and made into 
discs 1 to 2 ins thick, and % to 2 ins diam, or larger. It is generally used wet, for 
the sake of greater safety. It then requires extra strong caps or primers. Roughly 
speaking, it is about as strong as dyn No 1, but is less shattering in its effect. Being 
lighter than dyn, it requires larger holes; and, owing to its rigidity, is less easily 
inserted, and does not fit the hole so completely. When dry, it is very inflammable, 
but, if not confined, it burns harmlessly. It contains no liquid, to freeze or to exude; 
and is safe to handle. 

Art. 33. Tonite consists of finely divided gun-cotton mixed with nitrate of 
baryta. It is made by the Cotton Powder Co, Limited, at Faversham, Eng. It is 
compressed into candle-shaped cartridges having, at one end, a recess tor the recep¬ 
tion of an exploder containing fulminate of mercury. The cartridges weigh about 
the same as dyn. They are generally made waterproof. 














MODERN EXPLOSIVES. 


665 


Art. 34. Forcite, Li tliofracteur, and Dunlin are foreign makes of 
nitro-glycerine explosives. In Dualin the absorbent is sawdust. It has greater 
bulk than dyn for a given wt, and requires larger holes. 

Art. 3i>. Explosive gelatine is made by the Nobels Explosives Co, Lint 
(office Glasgow, Scotland), at their several works in England. It is a transparent, 
pale yellow, elastic substance, and is composed of 90 per cent N-G and 10 per cent 
gun-cotton. It is less sensitive than dyn to percussion, friction, or pressure, and is 
not affected by water. Its specific gravity is 1.6. It burns in the open air. For 
complete detonation a special primer is required. The addition of a small propor¬ 
tion of camphor renders it still less sensitive, and increases its explosive force. The 
camphor evaporates to some extent. 

In some experiments on the power of different explosives to increase the contents 
of a small cavity in a leaden block, explosive gelatine caused an increase 50 per cent 
greater than that caused by dyu No 1. In hard rock the diff would probably have 
been greater. The increase was 10 per cent less than that caused by N-G. 

Art. 36. Tlie cap or exploder, used with ordinary safety fuse for ex¬ 
ploding N-G and dyn, is a hollow copper cylinder, about % inch diam, and an inch 
(or two in length. It contains from 15 to 20 per cent, or more, of fulminate of mer¬ 
cury, mixed with other ingredients into a cement, which fills the closed end of the 
cap. The oap is called “single-force,” “triple-force,” &c, according to the quantity 
of explosive it contains. 

The end of the fuse, cut off square, is inserted into the open end of this cap, far 
enough to touch the fulminating mixture in it. In doing this, care must be taken 
not to roughly scratch the latter. The neck of the cap is then pinched, near its 
open end, so as to hold the fuse securely. The cap, with the fuse thus attached, is 
then inserted into the charge of N-G or dyn, care being taken not to let the fuse 
come into contact with the explosive, which would then be burned and wasted. If 
a dyn cartridge is used, the fuse, with cap, is first inserted into it. The neck of the 
cartridge is then tied around the fuse with a string, and the cartridge is then ready 


f to be placed in the hole and fired. 

Art. 37. The Siemens mag-neto-electric blasting- appa¬ 
ratus, now in general use, consists of a wooden box about as large as a transit- 
box. Outside it has two metallic binding-posts with screws, for attaching the two 
wires leading to the exploder. From the top of the box projects a handle at the 
end of a vert bar. This bar, which is about as long as the box is high, is made so 
as to slide up and down in it, and is toothed, and gears with a small pinion inside 
the box. When a blast is to be fired, the bar is drawn up, by means of the handle, 
as far as it will come. It is then pressed quickly down to the bottom of the box. 
In its descent it puts iuto operation, by means of the pinion, a magneto-electric 
machine inside the box. This generates a current of electricity, which increases in 
force with the downward motion of tho bar, but which is confined to a short circuit 
of wire within the. box , until the foot of the bar strikes a spring near the bottom of 
if the box, breaking the short circuit and forcing the electricity to travel through the 
two longer “leading wires,” which lead it from the two binding-posts on the outside 
j of the box to the cap or exploder placed in the charge. 

Art. 38. TIie cap used with this machine is similar to that used with safety 
3 fuse (Art 36), except that its mouth is closed with a cork of sulphur cement, through 
which pass the two wires leading from the electric machine. The ends of these 
wires project into the fulminating mixture in the cap. They are inch apart, but 
are connected by a platinum wire, which is so fine as to be heated to redness by the 
current from the battery. Its heat ignites the fulminate and thus explodes the cap. 

These exploders, called platinum caps, or (improperly) platinum fuses, cost. 
1886, about 3 to 10 cts each, depending upon the length (from 1 to 16 ft) of the two 
cotton-covered wires attached to them. With gutta-percha-covered wires, 20 to 40 
cts. The outer end of each of these short wires is connected with the electrical ma¬ 
chine by a cotton-covered 44 leading wire ” costing 1 cent per ft. 

Art. 39. Wliere a number of holes are to be fired simul¬ 
taneously (thus increasing their effect), each hole has a platinum cap inserted 
into its charge, and one of the short wires attached to each cap is joined to one of 
those of the next cap, so that at each end of the series of caps there is one free end 
of a short wire. Each of these two ends is fastened to the end of one of the leading 
wires, placing the whole series “in one circuit.” Where the holes are too far apart 
for the caps to be thus joined by the short wires attached to them, the ends of the 
latter are connected by cotton-covered 44 connecting- wires,” costing, 1886, 
30 cts per pound. 







666 


MODERN EXPLOSIVES. 


Art. 40. The magneto-electrical machine weighs about. 16 lbs, at) 
costs, 1886, siae No 3, $25. It can fire about 12 caps at once. A larger size. No • 
costs $50; and a still larger ijne, said to be capable of firing over 50 holes at one< 
$ 100 . 

Frictional electric blasting 1 machines, costing about $75 each, ar 
now (1884) nearly obsolete. 

Caps for ordinary fuse and for electrical firing, fuses, wires, electrical machine 
&c, are made by Latlin & Hand Powder Co, 29 Murray St, New York, and are sold b 
most of the makers of, and dealers in, explosives, rock-drilling machines, <ftc. 

Art. 41. Simultaneous firing of a number of holes can be conveniently accon 
plished only by electricity. Electric blasting apparatus is specially useful fr 
blasting under water, where ordinary fuses are apt, especially at great depths, t 
become saturaied and useless. 

If an electrical machine fails to fire a charge, it is known that the charge cannr 
explode until the attempt is repeated. Therefore no time need be lost, and no risk 
run, on account of “ hanging fire.” 

' 


i i • .... 

il 1 . I • * 

■ 1 -.tl. u i boo « h » •UtiAK; 


. i > 


• : t 7 !.' 




si ii now «|i 






X* * J 






In : till'll (I c«r» *+ i» i 




“ l 








COST OF STONEWORK. 


667 


an 


w 


Cost of quarrying: stone. After the preliminary expenses of purchasing 
trie site of a good quarry; cleaning off the surface earth and disintegrated top rock • 
and providing the necessary tools, trucks, cranes, Ac; the total neat expenses for 
1 fitting out the rough stone for masonry, per cub yard, ready for delivery, may be 
roughly approximated thus: Stones of such sizes as two men can readily lift meas¬ 
ured m piles, will cost about as much as from to the daily wages of a quarry 
laborer. Large stones, ranging from % to 1 cub yd each, got out by blasting, from 
tijj l to daily wages per cub yd. Large stones, ranging from 1 to 1^ cub yds each, in 
which most of the work must be done by wedges, in order that the individual stones 
®hall come out, in tolerably regular shape, and conform to stipulated dimensions; 
fill fmm 2 to 4 daily wages per cub yard. The smaller prices are low for sandstone, 
;,t(J while the higher ones are high for granite. Under ordinary circumstances, about 
\/.\ cu ^ yds of good sandstone can be quarried at the same cost as 1 of granite; or, 
noi in other words, calling the cost of granite 1, that of sandstone will be so that 
ski the means of the foregoing limits may be regarded as rather full prices for sandstone; 
rather scant ones for granite: and about fair for limestone or marble. 


< ost of dressing; stone. In the first place, a liberal allowance should be 
made for waste. Kven when the stone wedges out handsomely on all sides from 
"he quarry, in large blocks of nearly the required shape and size, from to % of 
tne rough block will generally not more than cover waste when well dressed. In 
moderate-sized blocks, (say averaging about % a cub yard each,) and got out by 
blasting, from K to 34 will not be too much for stone of medium character as to 
straight splitting. About the last allowance should also be made for well-scabbled 
rubble. The smaller the stones, the greater must be the allowance for waste in 
dressing. In large operations, it becomes expedient to have the stones dressed, its 
far as possible, at the quarry; in order to diminish the cost of transportation, which, 
when the distance is great, constitutes an important item—especially when by land, 
j and on common roads. 


A. Stonecutter will first take out of wind; and then fairly patent-hammer dress, about 8 
to 10 sq ft of plain face in hard granite, in a day of 8 working hours; or twice as much of such infe¬ 
rior dressing as is usually bestowed on the beds and joints; aud generally on the faces also of bridge 
masonry, &c, when a very fine finish is not required. In good sandstone, or marble, he can do about 
J4 more than in granite. Of finest hammer finish, granite , 4 to 5 sq ft. 

Cost of masonry. Every item composing the total cost is liable to much 
variation; therefore, we can merely give an example to show the general principle 
upon which an approximate estimate may be made; assuming the woges of a 
laborer to be $2.00 per day of 8 working hours; and $3.50 for a mason. Tlic 
monopoly of quarries affects prices very much.* 

Com! of ashlar facing; masonry. Average size of the stones, say 5 ft 
long, 2 ft wide, and 1.4 thick; or two such stones to a cub yd. Then, supposing the 
stone to be granite or gneiss, the cost per cub yd of masonry at such wages 

will be, Gettiug out the stone from the quarry by blasting, allowing % for waste in 


dressing; 1% cub yds, at $8.00 per yard... $4.00 

Dressing 14 sq ft of face at 35 cts. 4.90 

“ 52 “ beds aud joints, at 18 cts. 9.36 


Neat cost of the dressed stone at the quarry. 18.26 

Hauling, say 1 mile; loading aud unloading... 1.20 

Mortar, sav.....:.. ..40 

Laying, including scaffold, hoisting machinery, superintendence, &c. 2.00 


Neat cost....... 21.86 

Profit to contractor, say 15 per ct.. 3.28 


Total cost. 25.14 

Dressing will cost more if the faces are to be rounded, or moulded. If the stones are smaller than 
we have assumed, there will be more sq ft per cub yd to be dressed, &c. 

If in the foregoing case, the stoues be perfectly well dressed on all sides, including the back, the 
cost per cub yd would be increased about $10; and if some of the sides be curved, as in arch stones, 
say $12 or $14; and if the blocks be carefully wedged out to given dimensions, $16 or $18; thus 
making the neat cost of the dressed stone af the quarry say $28, $31, or $35 per cub yd. 


# The blocks of granite for Bunker 11111 monument averaging 2 cub yds each, were 
quarried by wedging, and delivered at the site of the monument, at a neat actual cost of $5.40 
per cub vd ; by the Monument Association ; from a quarry opened by themselves for the purpose. The 
Association received no profit; their services being voluntary. The average contract offers for the 
same were $21.30! The actual cost of getting out the rough blocks at the quarry was $2.70. Load¬ 
ing upon trucks at quarry, about 15 cts. Transportation 8 miles by railway and common road, $2.55. 
Total, $5.40. In 1825 to 1845 ; common unskilled labor averaging $1 per day. 

Ill 1S8<», liranitc blocks about a cub yd each, with dressed beds and joints, 
but with only a 2 inch draft around the showing-face, (which is left rough,)>re del’d on the wharf at 
Philada, from Port Deposit, Md, by McCleuahau At Bros, at $16 each. 






















668 


COST OF STONEWORK. 


The item of laying will be much increased if the stone has to be raised to great heights; or if it h 
to he much handled; as when carried in scows, to be deposited in water-piers, Ac. Almost eve 
large work presents certaiu modifying peculiarities, which must be left to the judgment of the cn 
neer and contractor. The percentage of contractors’ profit will usually be less on large works th 
on small ones. 


Cost of ashlar facing 1 masonry. If the stone be samlston 

with good natural beds, the getting out may be put at $3.00 per cubic yard. Face dressing at 26 ( 
per sq ft: say $3.64 per cubic yd. Beds and joints 13 cts per sq ft; say $6.76 per cub yd. The ne 
cost, laid, $17.00. 

And the total cost of large well scabbled range 
sandstone masonry in mortar, may be taken at about $10 per cub y 


Cost of large scabbled granite rubble, such as is generally used ; 

backing for the foregoing ashlar ; stones averaging about % cub yd each : 

Cost per 

Labor at $1 per day. cub yd of 

masonry. 

Getting out the stone from the quarry by blasting, allowing for waste in 

scabbling; 1^ cub yds at $3.00. $3.43 

Hauling 1 mile, loading and unloading . 1.20 

Mortar; (2cub ft, or 1.6 struck bushels quicklime, either in lump or ground ; 

and 10 cub ft, or 8 struck bushels of sand, or gravel; and mixing). 1.50 

Scabbling; laying, including scaffold, hoisting machinery, Ac. 2.50 


Neat cost. 8.63 

Profit to contractor, say 15 per ct. 1.30 


Total cost. 9.93 ( , ( i 

Common nibble of small stones, the average size being such as tw 
men can handle, costs, to get it out of the quarry, about 80 cts per yard of pile 
or to allow lor waste, say $1.00. Hauling 1 mile, $1.00. It can he roughly scabblei 
and laid, for $1.20 more; mortar as foregoing, $1.50. Total neat cost, $1.70; or, wit 
15 per ct profit, $5.40, at the above wages for labor. 


la 


With Smaller stones, such as one man can handle, we may say, stone 70 cts; hauling $1 
laviug and scaffold, tools Ac, $1: mortar $1.50 Making the neat cost $4.20; or with 15 per ct profit. $4.8c 
Neat scabbled irregular range-work costs from $2 to $3 more per yd than rubble: according to tbecharac 
ter of the stoue Ac. The lay ingof thin walls costs more than that of thick ones, such as abutments Ac. 

The cost of plain H inch thiek ashlar facings for dwellings ic ii 

Philada, in 1886, is about as follows per square foot showing, put up, including everything. Sand 
stoue, $1.50 to $2.25. Pennsylvania marble, $2.50. New England marble, $2.75 to $3.25. Granite 
$2.25 to $2.75. If 6 ins thick, deduct one-eighth part. First. Class artificial StOII< 
could be made and put up at one-third the price. See p 681. North Kiver bine StOI14 
flags, 3 ins thick, for footwalks, put down, including gravel Ac. 70 cts per sq foot. Height! ' 
street pavement, with gravel, complete, $3.50 per sq yard in Eastern cities. 11 

When dressed ashlar facing is backed by rubble, the expense per cub yard of th< J 
entire mass will of course vary according to the proportions of the two. Thus, il ,| 
ashlar at $12 per yd, is backed by an equal thickness of rubble at $5, the mean cos t> 
will be ($12 + $5) 2 = $8.50; or if the rubble is twice as thick as the ashlar thci * 

($12 -f $5 + $5) -> 3 = $7.33, &c. Such compound walls are weak ant jj 
apt to separate in time, as also walls of cut stone backed by concrete, or by brick' 
from unequal settlement of the two parts. , 

At times the contractor must be allowed extra in opening new quarries; in formin; 1 
short roads to his work ; in digging foundations ; or for pumping or otherwise draining them, who 
springs are unexpectedly met with ; for the centers for arches, Ac; unless these items are expressly 
included in the contract per cub yd. * | 

» • ' i- • I 

For quantity of masonry in walls of wells, see p 158. 

Approximate cost of buildings per cubic loot, at Philada price 

in 1873 ; including every cub ft of space from roof to cellar tioor. Plain brick dwellings, such as most 
of those iu Philada, 12 to 15 cts. Better class, highly fiuished throughout, 15 to 18 cts. First class, | 
with cut stoue fronts, 20 to 30 cts. Plain brick churches, public schools, court-houses, theaters, Ac 
12 to 16 cts. Ornate Gothic churches with much cut stoue facing, 30 to 45 cts, exclusive of spires 
Barge plain brick or rubble K R shops, depots, station-houses, Ac. 9 to 12 cts; or with ornarneutal 
finish and best materials, 15 to 20 cts. First class city stores, marble fronts, high stories, fire-proof, 
(so called,) throughout to roof; best materials and workmanship, 18 to 25 cts. 

Small buildings cost more per cub ft than large ones of the same 

finish. Also isolated or corner buildings, cost more thau those which have two party-walls. 

In Philada dwelling* of brick, the carpentry and lumber usually cost 
each about one-fourth of the entire building, memorial Hall of the Centen¬ 
nial Buildings cost 68 cts per cub ft. exclusive of iron dome. 


*In Philadelphia, in 1886, cellar and other walls of rough rubble, $3 to $4 per perch of 22 cub ft of 
wall. Outside walls with a facing of broken range rock-work of sandstone, (as common in Gothic 
churches,) $5 to $7 per 22 cub ft, including everything. 



















669 


MORTAR, BRICKS, ETC. 


MOETAE, BEICKS, &c. 

i 


Art. 1. Mortar. The proportion of 1 measure of quicklime, either in ir- 
igular lumps, or ground.* and 5 measures of sand, is about the average used for 
jpmmon mortar, by good builders in our principal Atlantic cities; and if both 
laterials are good, and well mixed (or tempered) with clean water, the mortar is 
lertainly as good as can be desired for such ordinary purposes as require no addi- 
< on of hydraulic cement. The bulk of the mixed mortar will usually exceed that 
f the dry loose sand alone about 34 part. 

i Quantity required. 20 cub ft, or 16 struck bushels of sand, and 4 cub ft, or 

.2 struck bushels of quicklime, the measutes slightly shaken in both cases, will make abt 22 J$ cub ft of 
lortar; sufficient to lay 1000 bricks of the ordinary average size of 814 by 4 by 2 ins, with the coarse 
lortar joints usual in interior house-walls, varying say from % to 34 inch. With such joints, 1000 
uch bricks make 2 cubic yards of massive work. Nearly one-third of the mass is mortar. For 
utside or showing joints, where a whiter and neater looking mortar is required, house-builders in- 
rease the proportion of lime to 1 in 4, or 1 in 3. For mortar of fine screened gravel, for cellar-walls 
f stone rubble, or coarse brickwork. 1 measure of lime to 6 or 8 of gravel, is usual ; and the mortar 
s good. In average rough massive rubble, as in the foregoing brickwork, about one-third the mass is 
iortar: consequently a cubic yard will require about as much as 500 such bricks; or 10 cubic feet, (8 
,rUck bushels) of sand; and 2 cub ft, or 1.6 bushels of quicklime. Superior, well-scabbled rubble, 

irefully laid, will contain but about -j of its bulk of mortar; or 53^ cub ft sand, and 1.1 cub ft lime, 
jr cub yard. 

! For public engineering works, especially in massive ones, or where exposed to dampness, an addi- 

iion should be made in either of the foregoing mortars, of a quantity of good hyd 
ement, equal to about 34 °f the lime; or still better, 34 of the lime should be 
Quitted, and an equal measure of cement be substituted for it. If exposed to water while 
uite new. use little or no lime outside. 

With bricks of 8% by 4 by 2 ins, the following are the quantities of mor- 
!;ar ami of bricks for a cubic yard of massive work. 


Thickness 

Proportion of Mortar 

No. of Bricks 

No. of Bricks 

of Joints. 

in the whole mass. 

per cub yard. 

per cub loot. 

A inch . 


. 638 . 

. 23.63 


574 . 21.26 


3 <« 

<< 3 

. 522 . 

. 19.33 

8 




1 It 

<1 1 

. 475 . 


2 




5 << 

11 4 

. 433 . 



i In estimating for bricks in massive work, allow 2 or 3 per ct for waste; 

I nd in common buildings, 5 per ct. or more. Much of the waste is incurred in cutting bricks to fit 
Ingles, &c. In Philadelphia a barrel of lump lime is allowed for 1000 bricks; or for 2 perches (25 cub 
t each) of rough cellar-wall rubble. Somewhat less mortar per 1000 is contained in thin walls, than 
! u massive engineering structures ; because the former have proportionally more outside face, which 
oes not require to be covered with mortar; but thin walls involve more waste while building; so that 
,oth require about the same quantity of materials to be provided. Careful experiments show that 
aortar becomes harder, and more adhesive to brick or stone, if the proportion of lime is increased, 
lence, on our public works the proportion of one measure of quicklime to 3 of sand, is usually spec- 
fied, but probably never used. , „ 

Lime usually sold In lump, by the barrel.* of about 230 tbs net, 
>r 250 fts gross. A heaped bushel of lump lime averages about 75 lbs. Ground quicklime, 
oose, averages about 70 lbs per struck bushel; and 3 bushels loose just fill a common flour barrel; but 
rum 3.5 to 3.75 bushels, or 245 to 280 lbs can readily be compacted into a barrel. 

General remarks on mortar and lime. On too great a pro- 

rortion of our public works, the common lime mortar may be seen to be rotten and useless, where it 
las been exposed to moisture; which will be carried by the capillary action of earth to several feet 
ibove the natural surface: or as far below the artificial surface of embankments deposited behind 
ibutments, retaining-walls, &c. The same will frequently be seen in the soffits of arches under em¬ 
bankments. Common lime mortar, thus exposed to constant moisture, will never harden properly. 
Even when very old and hard, it absorbs water freely. Cement also does so, but hardens. 

Krickdust, or burnt clay, improves common mortar; and makes it hydraulic, 
in localities where sand cannot be obtained, burnt clay, ground, may be substituted; and will gen- 
jrallv give a better mortar. . . 

Protection of quicklime from moisture, even that of the air, is 
absolutely essential, otherwise it undergoes the process of air-slacking, or 


* Price of quicklime in lump in Philadelphia, about 25 cts per bushel. Bar 
sand $1.40 per cubic yard. J. Bex Allen, 131'J Washington Ave. 
























670 


MORTAR, BRICKS, ETC 


spontaneous slacking, by which it becomes reduced to powder as when slacked by water as usua 
but without heating, and with but little swelliug. As this air slacking requires from a few months t 
a year or more, depending on quality and exposure, it gives the lime time to absorb sufficient carboni 

acid from the air to injure or destroy its efficacy. But quicklime will keel 


good for a Iontime if first ground, and then well packed in air-tigh 


barrels. The grinding also breaks down refractory particles found in all limes, and which injure th 
mortar by not slacking until it has been made and used. For the same reason it is better that lint 
should not be made into mortar as soon as it is slacked, but be allowed to remain slacked for a day o 
two (or even several) protected from rain, sun, and dust. 




Lime slacked in great bulk may char or even set fire to wood. 

Lime paste and mortar will keep for years, and improve, if wel 

buried in the earth. Also for months if merely covered in heaps under shelter, with a thick layer o 
sand. The paste shrinks and cracks in drying ; but the sand in mortar prevents this. 

As approximate averages varying much according to the character ant ! 

degree of burning of the limestone; aud to the fiueuess or coarseness of the sand, oue measure o',I; 
good quicklime, either in lump, or ground ; if wet with about !4 a measure of water, will withjn les 
than an hour, slack to about 2 measures of dry powder. And if to this powder there be added abou 
% more measures of water, and .'1 measures of dry sand, and the whole thoroughly mixed, the restil 
will be about 3*^ measures of mortar. Or the same slacked dry powder, with about 1 measure ol 
water, and 5 measures of sand, will make about o% measures of mortar. In both cases the bulk ol 
the mortar will be about % part greater than that of the drv sand alone. If % of a measure of water 
be used for slacking, the result, instead of a dry powder, will be about measures of stiff paste ; oil 
with I whole measure of water for slacking, the result will be about measures of thin paste, ofi 
about the proper consistence for mixing with the sand. Very pure, fat limes, slack quickly, and ttiakt j 
about from 2 to 3 measures of powder; while poor, meagre ones, require more time, and swell less 
Slow slacking, and small swelling, in case the lime has been properly burut, are not in general baril 
properties ; but on the contrary, usually indicate that it is to some extent hydraulic. Iu this case ill 
makes a better mortar; especially for works exposed to moisture, or to the weather. Very pure liuie>, 
are the worst of all for such exposures; or are bad tveather-limes; aud iu importaut works, should] 
never be used without cement. 


.4 


ku 

It 

V 


« 


Shell lime appears to be about the same as that from the purest limestones;I! 

but that from chalk is still more inferior, and will not bear more than about 1 measures of sand ; f 
its mortar never becomes very hard. Madrepores (commonly called coral) appear to furnish a lime 8 
intermediate between those of chalk and limestone. They require to be but moderately burnt. 

The average weight of common hardened mortar is about 105 to 115 IbsjH 

per cub ft. 

Grout is merely common mortar made so thin as to flow almost like cream. 

It is intended to fill interstices left in the mortar-joints of rough masonry ; but unless it contains a i 1 
large amouut of cement, it is probably entirely worthless ; since the great quantity of water injures 
the properties of lime; aud moreover, its ingredients separate from each other; the sand settling be 
low the lime. Besides this, it will never harden thoroughly in the interior of thick masses of ma 
sonry; indeed, the same may probably be said of any common lime mortar. Iu such positions, it has } 
been found to be perfectly soft, after the lapse of many years. 

Both the sand and the water for lime mortar, should he free from clity an«l 
salt. The clay may be removed by thorough washing; hut it is extremely dif-j 

ficult to get rid of the salt from seashore sand, even by repeated washings. Enough will generally I 
remain to keep the work damp, and to produce efflorescences of nitre ou the surface; whether with 
lime, or with cement mortar. Slacking by salt water gives less paste thau fresh. 

Mortar should not be mixed upon the surface of clayey ground ; but a rough board, brick, or stone 
platform should be interposed. Pit sand sifted from decomposed gneiss, and other allied rocks, is ex¬ 
cellent for mortar; its sharp angles making with the lime a more coherent mass than the rounded 
grains of river or sea sand. Mortar should be applied wetter in hot than iu cold weather; especially ' 
in brickwork; otherwise the water is too much absorbed by the masonry, and the mortar is thereby 
injured. 

The tenacity, or cohesive strength, that is, the resistance to a pull 

of good common lime mortar of the usual proportions of lime and sand, and 6 months old, is about 
from 15 to 30 lbs per sq inch ; or .06 to 1.9 tons per sq ft. With less sand, or with greater-age, it will 
i>e stronger. 

The crushing strengt h of good common mortar 6 months old is from 150 

to 300 lbs per sq inch, or 9.7 to 19.3 tons per sq toot. . 

The sliding resistance, or that which common mortar opposes to any 

force tending to make one course of masonry slide upon another, is stated by Roudelet, to be but 5 lbs 
per sq inch ; or about oue third of a ton per sq ft, in mortar 6 months old. 

Transverse strength of good common mortar 6 months old. A bar 1 

inch square and 12 ins clear span, breaks with a center load of 4 to 8 Sis. 

The lime in mortar decays wood rapidly, especially in close, 

damp situations. Still the soaking of timber for a week or two iu a solution of quicklime iu water 
appears to act as a preservative. Iron, so completely embedded in mortar as to exclude air aud 
moisture, has been fouDd perfect after 1400 years; but if the mortar admits moisture the iron decays. 
So, probably, with other metals. 


The adhesion to common bricks, or to rough rubble at any 

age will average about ^ of the cohesive strength at the same age; or say 12 to 24 lbs per sq inch, or 
.75 to 1.5 ton per sq ft at 6 months old. If care lie tnken to exclude dust entirely, by dipping each 
brick into water before laying it, or by sprinkling the stone by a hose, &e, the adhesion will be in¬ 
creased. On the other hand, much dust may almost prevent auy adhesion at all. The precaution of 
wetting is especially necessary in very hot weather, to prevent the warm bricks or stone from kill* 
Injj the mortar by th»> rapid absorption and evaporation of its water. The adhesion to very 
smooth hard pressed bricks, or to smoothly dressed or sawed stoue is considerably less. 












MORTAR, BRICKS, ETC. 


671 


ai 

to 

lie 


he 

lie 

or 


of 

)i 

of 

!S3 

tit 

lit 

of 

of 


Art. 2. Bricks, size, wt, Arc.* * A common size in our eastern cities is 

about 8.25 X 4 X 2 ins; which is equal to 66 cub ins; or 26.2 bricks to a cub ft; or 707 bricks to a cub 
yard. For the number required with mortar, see table, p 669. 

In ordering a large number a minimum limit of dimension should be specified in order to prevent 
fraud. A brick X inch less each way than the above, contains but 52.5 cub ins ; thus requiring full 
25 per cent more bricks to do the same work ; in addition to 25 per ct more cost for laying, which is 
generally paid for by the 1000. 

The weight of a $?oo<l common brick of 8.25 X 4 X 2 ins, will aver¬ 
age about 4.5 lbs; or 118 lbs per cub ft = 3186 lbs or 1.42 tons per cub yard ; or 2.01 tons per 1000 . 
A g'ood pressed brick of the same size will average about 5 tbs, = 181 tbs 
per cub ft = 3537 lbs or 1.58 tons per cub yd ; or 2.23 tons per 1000. 
Immersed in water, either of them will in a few minutes absorb from 

X to % lb of water; the last being about y of the weight of a hand moulded one; or of its own 
bulk. Since the weight of hardened mortar averages but little less than that of good common brick, 
we may for ordinary calculations assume the weight of such brickwork at 1.4 tons per cub yard : 13 
tons per perch of 25 cub ft; or 116 lbs per cub ft; or for machine-moulded, at 1.56 tons per cub yd; 
1.44 tons per perch ; or 129 lbs per cub ft. 

Allowing for the usual waste in cutting bricks to fit corners, jambs, &c, the average number of 
8)4 X 4 X 2, required per sq foot of wall is as follows: 


Thickness of Wall. No. of Bricks. 

8)4 ius, or 1 brick. 14 

12% “ or IX “ . 21 

17 “ or 2 “ 28 

21 X “ or 2% “ 35 

25%“ or 3 •* ... 42 

Laying per day, a bricklayer, with a laborer to keep him supplied with materials, will in 
common house walls, lay on an average about 1506 bricks per day of 10 working hours. In the neater 
outer faces of back buildings, from 1000 to 1200; in good ordinary street fronts, 800 to 1000; or of the 
very finest lower story faces used in street fronts, from 150 to 300, depending on the number of angles, 
&c. In plain massive engineering work, he should average about 2000 per day, or 4 cub yds; uud 
in large arches, about 1500, or 3 cub yds.t 

Since bricks shrink about yj part of each dimension in drying and burning, the moulds should be 
about yy part larger every way than the burnt brick is intended to be. 

Good well-burnt bricks will ring when two are struck together. 

At the brick-yards about Philadelphia, a brick-moulder’s work is 2333 bricks per day; or 14000 per 
week. He is assisted by two boys, one of whom supplies the prepared clay, moulding sand, and 
water; while the other carries away the bricks as they are moulded. A fourth person arranges them 
in rows for drying. About % of a cord, or 96 cub ft of wood, is allowed per 1000 for burning. Where 
coal is used, the kilns are fired up with anthracite, and the finishing is done with bituminous. One 
ton of coal, in all, makes 4500 bricks. 


■ Paving? with bl*iclt. Tn our cities this is done over a 6-inch layer of gravel, which 
should be free from clay, and well consolidated. With bricks of 8%X4X2 ius, with joints of from X 
‘ to % inch wide, a sq yard requires, flatwise, as is usual in streets, 38 bricks; edgewise, 73; endwise, 
r 149. An average workman, with a laborer to supply the bricks and gravel, will in 10 hours pave about 
i 2000 bricks; or 53 sq yds flat, 27 edgewise, 13 endwise. When done, sand is brushed into the joints. 


Art. 3. The crushing' strength of briehs or course varies greatly. A 
rather soft one will crush under from 450 to 600 ibs per sq inch ; or about 30 to 40 tons per sq it; while 
U a first-rate machine-pressed one will require about 200 to 400 tons per square foot. This last is 
j about the crushing limit of the best sandstone; % as much as the best marbles or limestones ; and X 
as much as the best granites, or roofing slates. But masses of brickwork crush under much smaller 
loads than single bricks. In some English experiments, small cubical masses only 9 inches on each 
, edge, laid in cement, crushed under 27 to 40 tons per sq ft. Others, with piers 9 ins square, and 2 ft 
I 3 Ins high, in cement, only two days after being built, required 44 to 62 tons per sq ft to crush them. 
i Another, of pressed brick, in best Portland cement, is said to have withstood 202 tons per sq ft; and 
I with common lime mortar only X as much. See page 436. 

It must, however, be remembered, that cracking and splitting usually commence under about one- 

| half the crushing loads. To be safe, the load should not exceed X or jyj- of the crushing one; and 
so with stone. Moreover, these experiments were made upon low masses; but the strength decreases 
with the proportion of the height to the thickness. 

The pressure at the base of a brick shot-tower in Baltimore, 246 feet high, is estimated at 6X tons 


Tlie Peerless Bricli Co, office No. 1003 Walnut St, Philada. make superb smooth 
(not shining) bricks of various shapes and colors (as white, black, gray, buff, brown, red, Ac) for 
ornamental architectural purposes. Their standard size is 8% X 4% X 2% — 82 cub ins, or X larger 
than the above 8X X 4 X 2 ins. The color extends throughout the body of the brick. With a few 
of these judiciously distributed among common bricks, beautiful architectural effects may be pro¬ 
duced, both indoors and out, at far less cost than iu stone. For prices and illustrated catalogue, 

address as above. . . . , 

The same Co will if a sufficient order is given, furnish voussoir bricks for specified radii, but or 
the quality and finish of ordinary good hard brick, at from 25 to 50 per ct. advance on prevailing 
market rates of common plain ones. . . . , - 0 

* Prices in Philada in 1886. Bricks alone; Salmon, or soft, $o per 1000. Hard brick, $8. 
Back stretchers (generally used for the facings of back buildings, Ac), $11. Paving biick, $11. 

Pressed, (for lower stories of first-class fronts,) $22. , , . , 

i Bricklaying; including mortar; averaging an entire dwelling. $7 per 1000. Best pressed bricks 
in firs t-cl ass fronts, $15. “ Tuck ” fronts, laid with steel wire, special rates. 















672 


MORTAR, BRICKS, ETC, 


per sq ft; and in a brink chimney at Glasgow, Scotland, 4(58 feet high, at 9 tons. Professor Rankii: 
calculates that iu heavy gales this is increased to 15 tons, on the leeward side. The walls of both ai 
of course much thicker at bottom than at top. With walls 100 feet high, of uniform thickness, tl 
pressure at base would be 5.4 tons per sq ft. 

With our present imperfect knowledge on this subject, it cannot be considered safe to expose eve 
first-class pressed brickwork, in cement, to more thau 12 or 15 tons per sq ft; or good haud-mouldei ! t 
to more than two-thirds as much. 

Tensile strength of brick, 40 to 400 lbs per sq inch ; or 2.6 to 26 tons per sq ft ; 
Tlie English roil of brickwork is 306 cub feet, or 11^ cub yards; am 

requires about 4500 bricks of the English staudard size; with about 75 cub ft of mortar. The Englisl 1 
hundred of lime, is a cub yd. 

frozen mortlir. There is risk in using common mortar in cold weather, if the col< j 
should continue long euough to allow the frozen mortar to set well, the work may remain safe, but i I j 
a warm day should occur between the freezing and the setting of the mortar, the sun shiuing on ou, j « 
side of the wall may melt the mortar on that side, while that on the other side may remain frozei 1 1 
hard. In that case, the wall will be apt to fall; or if it does not, it will at least always be weak ; fo jj; 
mortar that has partially set while frozen, if then melted, will never regain its strength. By th, ' 
writer’s own trials hydraulic cements seemed not to be iujured by freezing. 

Experiments for rendering; brick masonry impervious tc 
water. Abstractor a paper read before the American Society of Civil Engineers, May 4 , 1870 J 
by William L. Dearborn, Civil Engiueer, member of the Society. 

The face walls of the Back Bays of the Gate-houses of the new Croton reservoir, located nortt 
of Eighty sixth Street, in Central' Park, were built of the best quality of hard-burnt brick; laid in 
mortar composed of hydraulic cement of New York, and sand mixed in the proportion of one measurt 
of cement to two of sand. The space between the walls is 4 ft: and was filled with concrete. The fact 
walls were laid up with great care, and every precaution was taken to have the joiuts well Ailed and 
insure good work. They are 12 ins thick, and 40 ft high ; aud the Bays when full generally have 36 fl 
of water in them. 

When the reservoir was first filled, and the water was let into the Gate-houses, it was found to filter 
through these walls to a considerable amount. As soon as this was discovered, the water was drawn 
out of the Bays, with the inteution of attempting to remedy or preveut this infiltration. After care , 
fully considering several modes of accomplishing the object desired. I came to the conclusion to try j 
“ Si lvester’s Process for Repelling Moisture from External Walls." 

The process consists in using two washes or solutions for covering the surface of brick walls ; one ' 
composed of Castile soap aud water; and one of alum and water. The proportions are: three-quar¬ 
ters of a pound of soap to one gallon of water; and half a pound of alum to four gallons of water 
both substances to be perfectly dissolved in the water before being used. 

The walls should be perfectly clean and dry ; and the temperature of the air should not be below 
50 degrees Fahrenheit, when the compositions are applied. 

The first, or soap wash, should be laid ou when at boiling beat, with a flat brush, taking care not 
to form a froth on the brickwork. This wash should remaiu tweutv-four hours ; so as to oecorue dry 
and hard before the second or alum wash is applied; which should be done iu the same manner as 
the first. The temperature of this wash when applied may be60° or 70°; and it should also remain 
twenty-four hours before a second coat of the soap wash is put on ; and these coats are to be repeated 
alternately until the walls are made .mpervious to water. 

The alum and soap thus combined form an insoluble compound, filling the pores of the masonry, 
and entirely preventing the water from penetrating the walls. 

Before applying these compositions to the walls of the Bays, some experiments were made to test 
the absorption of water by bricks under pressure after being covered with these washes, in order to 
determine how many coats the wall would require to render them impervious to water. 

To do this, a strong wooden box was made, put together with screws, large enough to hold 2 bricks; 
and on the top was inserted an iuch pipe forty feet long. 

In this box were placed two bricks after being made perfectly dry, and then covered with a coat of 
each of the washes, as before directed, and weighed. They were theu subjected to the pressure of a 
column of water 40 feet high ; and, after remaining a sufficient length of time, they were taken out 
and weighed again, to ascertain the amount of water they had absorbed. 

The bricks were then dried, aud again coated with the washes and weighed, and subjected to press¬ 
ure as before; and this operation was repented until the bricks were found not to absorb any water. 
Four coatings rendered the bricks impenetrable under the pressure of 40 ft head. 

The mean weight of the bricks (dry) before being coated, was 3% lbs; the mean absorption was 
one-half pound of water. An hydrometer was used in testing the solutions. 

As this experiment was made "in the fall and winter. (1863,) after the temporary roofs were put od 
to the Gate-house, artificial heat had to be resorted to, to dry the walls and keep the air at a proper 1 
temperature. The cost was 10.06 cts per sq ft. As soon as the last cont had become hard, the water 
was let into the Bavs, and the walls were found to be perfectly impervious to water, and they still 
remain so in 1870, after about 6)< years. 

Brick arch (pootwav op High Ruidqk). The brick arch of the footway of High Bridge is the 
arc of a circle 29 ft 6 iu radius ; and is 12 in thick ; the width on top is 17 ft; aud the length covered 
was 1381 ft. 

The first two courses of the brick of the arch are composed of the best hard-burnt brick, laid edge¬ 
wise in mortar composed of one part, by measure, of hydraulic cement of New York, and two parts 
of sand. The top of these bricks, and the inside of the granite coping against which the two top 
courses of brick rest was, when they were perfectly dry, covered with a coat of asphalt oue-half an 
iuch thick, laid on when the asphalt was heated to a temperature of from 360° to 518° Fahrenheit. 

On top of this was laid a course of hrick flatwise, dipped in asphalt, and laid when the asphalt was 
hot; and the joints were run full of hot asphalt. 

On top of this a course of pressed brick was laid flatwise in hydraulic cement mortar, forming the 
paving aud floor of the bridge. This asphalt was the Trinidad variety; and was mixed with 10 per 
cent, by measure, of coal tar; and 25 per cent of sand. A few experiments for testing the strength 
of this asphalt, when used to cement bricks together, were made, and two of them are given below. 

Six bricks, pressed together flatwise, with asphalt joiuts, were, after lying six months, broken. 
The distance between the supports was 12 ins ; breaking weight. 900 lbs ; area of single joint, 28}£ sq 
ins. The asphalt adhered so strongly to the brick as to tear away the surface iu many places. 








CEMENT, CONCRETE, ETC. 


673 


it* 

till 

iw 


ft. 

nd 

in 

AM 

iii 

ttl 

for 

He 

to 


Two bricks pressed together end to end. cemented with asphalt, were, after lying 6 months, broken. 

The distance between the supports was Ft) ins; area of joint, 8>$ sq ins; breaking weight, 150 lbs. 

The area of the bridge covered with asphalted brick, was 23005 sq ft. There was used 94200 lbs of 
asphalt, 33 barrels of coal tar, 10 cub yds of suud, 93800 bricks. 

The time occupied was 109 days of masons, and 148 days of laborers. Two masons and two labor¬ 
ers will melt aud spread, of the. first coat, 1050 sq ft per day. The total cost of this coat was 5.25 
cents per sq ft, exclusive of duty on asphalt. There were three grooves, 2 ins wide by 4 ins deep, 
made eutirely across the brick arch, and immediately under the first coat of asphalt, dividiug the 
arch into four equal parts. These grooves were tilled with elastic paint cement. 

This arrangement was intended to guard against the evil effects of the contraction of the arch in 
wiuter; as it was expected to yield slightly at these points, aud at no other point; and thou the 
elastic cement would prevent any leakage there. 

The entire experiment has proved a very successful one, and the arch has remained perfectly tight. 

In proposing the above plan for workiug the asphalt with the brickwork, the object was to avoid 
depending on a large continued surface of asphalt, as is usual iu coveriug arches, which very fre- 
queutly cracks from the greater contraction of the asphalt than that of the uiasoury with which it is 
iu contact; the extent of the asphalt on this work being only about one-quarter of au inch to each 
brick. This is deemed to be an essential element iu the success of the impervious coveriug.” 

A cheap and effective process for preventing the percolation of water through the arches of aque¬ 
ducts, and even of bridges, is a great desideratum. Many expensive trials with resinous compounds 
have proved failures. Hydraulic cement appears to merely diminish the evil. Much of the trouble 
is probably due to cracks produced by changes of temperature. 


The white efflorescence so common on walls, especially on those of brick, 
■th is due to the presence of soluble salts in tint bricks and mortar. These are dissolved, 
“ and carried to the face of the wall, by rain and other moisture. (Sulphate of magne- 
,[( sia (Epsom Salt) appears to he the most frequent cause of the disfiguration. In many 
nd places mortar lime is made from dolomite, or magnesian limestone, which often con- 
id tains 30 per cent or more of magnesia; which also occurs frequently in brick clay. 
Coal generally contains sulphur, most frequently in combination with iron, forming 
the well-known “ iron pyrites ”, The combustion of the coal, as in burning the lime- 
re .;stone or clay, in manufactures, in cooking etc, converts the sulphur into sulphurous 
■7 acid gas, which, when in contact with magnesia and air, as in the lime or brick kiln, 
or in the finished wall or chimney, becomes sulphuric acid and unites with the mag- 
“jnpsia, forming the soluble sulphate. We are not aware of any remedy that Mill pre- 
,, vent its appearance under such circumstances; but the formation of the sulphate may 
be prevented by the use of limestone and brick-clay free from magnesia. See also p 
« 678. 


Art. 4. Hydraulic cements.* Certain limestones, when burnt, will not slack 
with water; but when the burnt stoue is finetv ground, and made into a paste, it possesses the pro¬ 
perty of hardening under water; aud is therefore called hydraulic cement. So long as the propor¬ 
tion of those ingredients which impart hydraulicity, is so small that the burnt stoue will slack ; but 
still make a paste or a mortar, which will harden under water, it is called hydraulic lime. This does 
jnot harden so promptly, or to so great a degree, as the cements. Hydraulic limes slack more slowly, 
and swell less, in proportion to their hydraulicity ; some requiring mauy hours. Artificial hydraulic 
times and cemeuts, of excellent quality, may be made by mixing lime and clay thoroughly together; 
then moulding the mixture into blocks like bricks; which are first dried, then burnt, and finely 
a;rouud. The celebrated artificial English Portland cement, is made bv grinding together in water 
jhalk and clay. The fine particles are floated away to other vessels, and allowed to settle as a paste; 

which is then collected, moulded, dried, burnt, and ground. Natural Port¬ 
land is that made from limestone, or other material of very rare occurrence, 

which combines naturally that proportion of lime and clay which gives the above artificial Portlands 
their pre-eminence. This alone constitutes its difference from our common natural hyd cemeuts. 

The weight of the Koseudale, and oilier American cements, 

together with some foreign ones, will be found on page 382. Saylor's Port¬ 
land weighs about 120 tt>s per struck bushel. 

The writer found by 10 years' trial that if, after setting, dampness is absolutely excluded, Cements 
preserve Iron, lead, zinc, copper, and brass; aud that Plaster of Purls preserves all except 
iron, which it rusts somewhat unless galvanized. Lime-mortar probably preserves all of them, 
if kept free from damp. 

Protection from moist ure, even that of the air, is very essential for the 
nreservation of cemeuts, as well as of quicklime. On this account the barrels are generally lined 
with stout paper. With this precaution, aided by keeping the barrels stored in a dry place, raised 
ibove the ground, the cement, although it may require more time to set, will uot otherwise very 

* Prices of hyd cements in I’hilada, 1886, by the large importing firm of 
Samuel H. French ’& Co, corner of York Avenue and ■ Gallon'hill St. English 
Portland cement, $2.00 to $3.25 per barrel of about 400 11.s gross; according to 
quality and quantity. German Portland, $2.40 to $3.00 pc* barrel of about 
400 lbs gross. Saylor's American Portland, $2.35 ^$460 per barrel «»t 
400 lbs gross. Rosendale, per barrel of about 300 tbs net, $l.lo to $1.40. Other 
wt ^ <M k in111s« per barrel of about 300 lbs net, 81.00 to $1.*>0 . 4»i*oiiihI ('«i 1 ■ 
cinc’d plaster of Paris: selected, barrel of 300 tbs net, $2.00 to $2.25. 
Commercial, barrel of about 200 lbs net, $1.23 to $1.80. 

The, American Improved Cements Co, E»,vpl 1 office -16 .. ud 
St, Philadelphia, make 

“ Giant” Portland. Price per barrel of 400 tbs net, $2.10 in 1886 














G 74 


CEMENT, CONCRETE, ETC. 


appreciably deteriorate for six months ; butafter 14 or 16 months, Gillmore says it is unfit for use ii 
important works. But in lumps, kept dry, it will remain good for 2or 3years; and may be grouu 


as required for use. . . . .. , 

Good Portland cement is stated by good authority rather to improve by free exposure to the ai 


under cover; but whether this is correct or not, we cannot say. 

Restoration by rebnrning may be effected. 


^ the « 

ment is spread in a thin layer, on a red-hot iron plate, for about 15 minutes, its good qualities will l: 
in a great measure restored. The time should be ascertained by trial. If it has been actually wei 
and lumpy, or cemented into a mass, it should first be broken into small pieces, and then ground. 0 
these pieces may be first kiln-burnt at a bright red-heat for about hours; and then ground. 


Art. 5. For roughcasting", or stuccoing the outside of walls, very fet 



I860; and appears to be in perfect condition in 1880, 

Quantity required. A barrel of cement, 300 lbs; and 2 barrels of sand, (< J 
bushels, or 7}$ cub ft;) mixed with about a barrel of water, will make about 8 cub ft of mortar * 
sufficient for 

192 sq ft of mortar-joint inch thick — 21 H sq yards. 

288 ... % “ “ = 32 “ ** 

384 •• “ “ “ % " “ ~ 42 ^ “ “ 

768 “ “ ** “ H “ “ = 85H “ 11 

Or, to lay 1 cubic yard, or 522 bricks of 814 by 4, bv 2 ins, with joints % inch thick; or n cubic yar( 

of roughly scrabbled rubble Btonework. The quantity of sand may be increased, however, to 3 or ■ 

measures for ordinary work. 

Pointing mortar. Gen Gillmore recommends “1 part by weight of gooc 
eemeut powder, to 3 or parts of saud. To be mixed under shelter, and in quantities of only 2 oi 
3 pints at a time, using very little water, so that the mortar, when ready for use, shall appear rathei 
iuoohereut, and quite deficient in plasticity. The joints being previously scraped out to a depth ol 
at least % an inch, the mortar is put in by the trowel; a straight-edge being held just below the joint 
if straight, as an auxiliary. The mortar is then to be well calked into the joint by a calking 
iron and hammer; then more mortar is put in, and calked, until the joiut is full. It is then rubbci 
aDd polished under as great pressure as the mason can exert. If the joints are very fine they shoulc 
be enlarged by a stonecutter, to about y'E inch, to receive the pointing. The wall should be well we 
before the pointing is put in, and kept in such condition as neither to give water to, nor take it from tb< 
mortar. In hot weather, the pointing should be kept sheltered for some days from the sun, so as no 
to dry too quickly.” Why not finish joints at once, without subsequent pointing? Author. 

Art. 6.* Color is no indication of strength in hyd cements. The find 
they are ground the belter. At least 90 per ct should pass through i 

sieve of 50 meshes per lineal inch, of Wire No 35 Arner wire gauge (.0056 inch thick); or 2500 meshei 
per sq inch. Weight is a good indication when equally well ground. A flat 
cake of good cement paste placed in water as soon as it admits of so doiug safely, and left in it for t jp 

week, should show no cracks. New eemeut is not as good as when a few 
weeks old. The term Setting does not imply that the cement has hardened 

to any great extent, but merely that it has ceased to be pasty and has become brittle. Quick setting 
cements may do this sufficiently to allow small experimental samples to be lifted and handled care 
fully within five to thirty minutes; while others may require from one to eight or more hours 

Stow Hotting does not indicate inferiority, for many of the very 

best are the slowest setting. A layer of very quick setting cement may partially set. especially il l 
warm weather, before the masonry is properly lowered and adjusted upon it, and any disturbance 
after setting has commenced is prejudicial. Such are to be regarded with suspicion, and sub 
mined to longer tests than slow ones. Still, quick setting ones are best in certain cases, as when 
exposed to running water, &c. They may be rendered slower by adding a bulk of lime paste equal to J 
or 15 per ct of the cement paste, without weakening them seriously. As a general rule cements 

set and harden better in water tlian in air. especially in warm 

weather. If, however, the temp for the first few days does not exceed 55° to 65* Pah, there seems to 
lie no appreciable difference in this respect; but in warm air cement dries Instead of setting, and thus 
loses most of its strength. In hot weather every precaution should be used against this. 

The time reqd to attain the greatest liardnenn is many years, but 

after about a year the increase is usually very small and slow, especially with neat cement. More I 
over, any subsequent’increase is a matter of little importance, because generally by that time, anr 
often much sooner, the work is completed and exposed to its maximum strains. Sand retards 
setting, and weakens the cement paste. Rut although with sand the strength of the mortar may 
never attain to that of the neat paste, yet it increases with age in a greater proportion; so that a 
neat paste which at the end of a year would be but twice os strong as in 7 days, may with sand yield 
a mortar which at the end of a year will be 3, 4, or 5 times as strong as it was in 7 days. Good Port¬ 
lands neat usually hare at the end of a year from 1.5 to 2 times their strength at the end of 7 days;! 
and the American natural cements, Rosendale, Louisville, Gnmberland, &c, from 2.5 to 3.5 times; 
but inasmuch as Portlands average (roughly speaking) about 5 or 6 times the strength of the others 
In 7 days, they still average about 2.5 to 3 times as strong in a year or longer. Cements of the same 

cla8S differ much in their rapidity of burdening. One may at the end of 

a month gain nearly one-half, and another not more than one-sixth of its increase at the end of a 
year, at which time both may have about the same strength, lienee, tests for 1 week or 1 mouth are 
by no means conclusive as to their filial comparative merits. 

There seems to be a period occurring from a few weeks to several months after having been laid, at 
which cement and its mortars for a short time not only cease from hardening, but actually lose 
strength. They then recover, and the hardening goes on as before. It has been suggested that 
this opluion has originated in some oversight of the experimenters, but the writer believes it to be 
founded on fact. In his expts with various hvd cemeuts of the consistence of mortar, even without 
saud, the writer detected no change of bulk In setting. 


* Sec “ Mr. Eliot C. Clarke ”, p 678. 








CEMENT, CONCRETE, ETC. 675 


h Art. 7.* Mr. Wm. W. Maday, €. E. (see his very instructive paper in 
wifTrans. Aru. Soc. C. E., Dec, 1877), found that in the testing: of cements the 

temperature of the air and water had far more influence than had before been suspected. Thus neat 
>UPortland moulded in air at 3i>° Fah. and kept 6 days in water at 40°, had a tensile strength of but 
15<i lbs per sq inch, while that kept in water of 70° had 299 tbs, or nearly twice as much. Other bars 
^moulded in air at 60°, after 6 days in water of 40' ) , broke with 113 tbs tensile per sq inch, while those 
"in water at 70° required 254 tbs, or about 2.25 times as much. But at the end of only 20 days the 
strengths of these last were as 212 to 336 tbs, or as 1 to 1.6; the weaker one having in that time gained 
0 , rapidly on the stronger. As longer time would of course bring them still nearer to an equality, the 
ul Wtimate effects of temperature within certain limits are fortunately not so important in actual’prac 
tice as the lirst expts might lead us to infer. Work must go on notwithstanding changes of tempera 
i ture. but we must take care that our mortar shall at all times be strong enough even under their most 
injurious influences. Cements in open air are certainly more or less injured by drying instead of 
letting, as the temp exceeds about 65° to 70°. But if mixed only in small quantities at a time, and 
'I* quickly laid in masonry of dampened stone, so as to be sheltered from the air, the injury is much 
lll !reduced. The sand and stone shouid both be damp, not wet, in hot weather, and a little more water 
may be used in the cement paste: also if possible not only the mortar while being mixed, but the 
[5 masonry also should then be shaded Mr Maclay found that 6 day specimens of neat Portland broken 
■Jiirect from the water were much stronger than if first left 24 hours to dry in the shade at tolerably 
high temps. But the reverse occurs with such U. S. natural cements as Rosendale, Sc, the strength 
»f some being largely increased by such drying. Experiments in Europe with Portlands kept 3 months 
n water, seemed to show the weakest period for such to be at 2 days' exposure, when the strength was 
>ut half as great as when first taken from the water. But on the 4th day they were even stronger than 
>t first; and the strength then increased with time as if there had been no interruption. 

udj The effects of col«l, although it retards the setting, do not appear to he 

ri erious otherwise, if the cement mortar even freezes almost as quickly as the masonry is laid with 
t, it does not seem to depreciate appreciably. The writer has found this to be the case also with lime 
j nortar, even when a few hours after freezing the temp became so high as to soften the frozen mortar 
w igain. But although the mortar of either lime or cemeut may not be thereby injured, the work, espe- 
/' dally iu,thin brick walls, may be ruined and overthrown. Thus if soon after the mortar through the 
e . r utire thickness of such a wall be frozen, the sun shines on one face of it so as to soften the mortar 
0 !>f that face, while the mortar behind it remains hard, it is plain that the wall will be liable to settle 
Et Ut the heated face, and at least bend outward if it does not fall. The writer has observed that coat- 
T ugs of cement applied to the backs of arches on the approach of wiuter, and left unprotected, were 

Jmtirely broken up and worthless on resuming work the next spring. The 
. heating; of sand and cement in freezing weather seems to be a bad prac- 

I! ice. especially if to be placed in cold water. But for use out of water Mr Maclay says they may be 

heated to 50° or 60°. Cold water lor mixing; is probably no farther inju¬ 

rious than that it retards the setting. All cements when mixed with sand to a proper consistence for 

nortar will fall to pieces if placed in water before setting has commenced. 

W 'ortlauds do so even without sand ; but U. S. natural cements of good quality do not. 

lit 

to Art.. 8.* Strength of cements. The strength as before stated is much 

1st iffected by the temp of the air and water, as also by the degree of force with which the cement is 
n iressed into the moulds; by the extent of setting before being put into the water, and of drying when 
akeu out; and still more by the consideration of whether or not it sets while under the influence of 
; iressure, which increases the strength materially. On this account cements in actual masonry may 
. under ordinary circumstances give better results than in door expts. These causes, together with the 

"degree of thoroughness of the mixing or gauging, the proportion of water 

* ised, and other considerations may easily affect the results 100 per ct, or even much more. Hence, 
r J, he discrepancies in the reports of different experimenters. 

Rem.* Portlands require more water than the common U. S. 

ements, and shrink less in mixing. See next Art. Also, mortars require more than concrete, espe- 

ially when the last is to be well rammed, in which case it should be merely 

• - • ■ - - 1 -*-j !-*— " 1 —' J If more water is present, the consol!- 


ice 

4 

^Itoiaflao as barely to cohere when pressed into a ball by hand. 

nu latiou by ramming is proportionally imperfect. To assure himself that 
m he quality of cement furnished is equal to that contracted for, the engineer 

to!hould reserve the right to bore with a long auger into any part of each barrel, and to reject every 
o, ,arrel of which the sample drawn out does not satisfy the stipulations On works using large quan¬ 
ts, jities, there should be one person specially detuiled to this duty. One advantage of very strong 
■it ements is their economy, even at a higher cost, in allowing the use of a larger proportion of the 
heaper iugredients, sand, gravel, and broken stone. Almost any common U..S- cement, if of good 
, ,uality , will with 1.5 or 2 measures of sand give a mortar strong enough for most engineering pur- 
1 1 loses ; but a good Portland will give one equally strong with 3 or 4 meas of sand, and will, therefore, 
“ ,e equally cheap at twice the price; beside requiring the handling, storing, and testing of only half 
*' he number of barrels. , 

After what has been said it is plfiin that great latitude must be allowed in attempting to prepare a 
able of approximate average strengths. The writer can pretend to nothing more than the following, 
rhich is deduced from reliable reports, aided by a few experiments of his own on transverse strengths, 
i summary of which last forms the last column. 

If one measure of cement slightly shaken be mixed to a paste with about .35 meas of water if a 
ommon U. S. cement, or about .40 meas if Portland, in the shade, and in a temp of from 60° to 90°, 
his paste will occupy about .7 meas if common, and about -H6 if Portland, when well pressed into 
'ooden moulds by the fingers (protected from corrosion by gloves of rubber or buckskin). If then 
Uowed from 30 minutes to some hours (according to its setting properties) to set; then removed from 
he moulds, and at the end of 24 hours total, placed iu water of the above limits ot temp for 7 days, 
nd broken at once when taken from the water, the samples w ill generally exhibit about the following 
trengths. Those for compression are supposed to be cubes ; and those for transverse strength in the 
tble were beams 1 inch square, and 12 ins clear span, loaded at the center. 


* See “ Mr. Eliot C. Clarke ”, p 678. 









676 


CEMENT, CONCRETE, ETC. 


Table A. Average Ultimate Strengths of neat Cements aftei 
6 days in water, and broken directly from the water. 


Portlands, artificial, either foreign, or the 
•• National " of Kingston, N.Y. 
“ Saylor's natural. Coplay, Penn.. 
U. S. common hydraulic cemeuts . 


Tensile. Ibs 

Compres. tbs 

Compres tons 

Transv. 1" > 

per sq in. 

per sq in. 

per sq ft. 

1" X 12". D>! 

200 to 350 

1400 to 2400 

90 to 154 

25 to 45 

170 to 370 

1100 to 1700 

71 to 109 

26 

40 to 70 

250 to 450 

16 to 29 

3 to 7 


II 


All below the lowest of these should be rejected ; the average of the table may be considered fair 

and all above the highest superior. After only 24 hours in water tlv 

strength of the common U. S. cements averages about half that for 6 days, but with considerabl 
variations both ways. In like manner at the end of a year neat Portland 
average from 1.5 to 2 times as strong as in 6 days; and our common cemeuts from 2.5 to 3.5 times 
The KiOiidon board of works require that Portlands after 7 days in wate 
shall have at least 35 lbs transverse, and 350 tbs tensile strength. Some have reqd 500 or more ten 
sile to break them. For Portlands the writer found the transverst 

strengths of several well known English brands moulded as before described, to be 26 to 40 lbs after 
days In water; National Portland of Kingston, N. Y., 40 and 46; Saylor's Portland (only 2 trials) 2' 
lbs. Toeptler, Gruwitz, Co, of Stettin, Germany, warrant all thei 
Portland (kuowu as the “ Stern ” brand) equal to 569 lbs tensile after 7 days in water. Some of it ha 
b.orne 760 lbs. 

Mr. J. Herbert Sliedd, as Engineer of the Water Works and Sewers ol 

Providence, H. I., rejected all Rosendale which when mixed to a stiff paste, and allowed 30 min in ai 
to set, and theu put into water for only 24 hours, broke with less tension than 70 lbs. At first h‘ 
found some that broke with 10 to 15 tbs; some that would not set at all in water; and but little tha 
bore 30 tbs. Now samples frequently bear 100 tbs or more; but that usually sold still rarely exceed 

40 to 50, and frequently scarce half as much. The Sewer department ol 
St. Louis. Missouri, requires all Louisville, Kentucky, cement to hear at lea$' 

4 <) lbs tensile after 24 hours in water. Some of it now shows as high as 100 or more; and 60 or 7( 
would have heeu adopted as the mininum, but for the fear that it would have encouraged the makinj 
of too quick setting cement. Most of that sold will probably not exceed 30 tbs. 


Art. 9. Cement mortar is cement mixed with water and sand only 

The writer found that for making cement pastes of about equal consistency and fit for mortar b;I' 
themselves, the English Portlands, slightly shaken in the measure, required an average of about . » 
of their own bulk of water; aud the U. S. common cements about .35. The Portland pastes whei 
thoroughly mixed and slightly pressed by hand into a box shrank about one-eighih of their bulk a: 
dry shaken cement; and the others about one-fourth ; or iu other words the common U. S. cement: 
shrink about twice as much as the Portlands ; and these are about the proper proportions to assumi 
in estimating the quantity of cement for theoretically filling the voids in sand. 

But when sain# is added, more water is reqd. It is impossible to laj 

down rules for all cases, but as a very rough average, mortar will require an addition equal to abou 
.2 of the bulk of dry sand; varying of course with the weather, &c. Trial on the work iu baud i; 
better than rules. 

Any addition of sand weakens eement. especially as regards ten 

sinn ; as it does also lime mortar. Rut economy requires its use. Sand also retards the setting, s< 
that cement which by itself would set in half an hour, may not do so for some days if mixed with t 
large proportion of sand. This weakening effect will of course vary with different cements, and witt 

many circumstances inferable from Art. 7, Ac. As a roue'll average tin 

following is perhaps not far from the truth as regards either tensile or transverse strength when no 
rammed. See p 682. 


Sand. 

0 


1 


2 

3 

4 

5 | 6 ! 

Strength. 

1 

% 

'A 

.4 


.3 

l 4 

i 


8 


| i % 


Tensile Strength of Cement Mortars,* 


of medium coarse sea-beach sand, and good Rosendale, and English Portland ce 
ments; being averages of about *25000 experiments in the years 1878 to 1882. The 
area of breaking section was 2.25 sq ins. The proportions of Sind and cement wen 
by measure. The mortar was rammed into the moulds, and the specimens wen 
immersed in water as soon as they would bear handling, and so remained for 1 day 
or 1 week, or for 1, 6, or 12 months. The strengths are in lbs per sq inch. 


* Prom experiments by Mr. Eliot C. Clarke, C. E ; of Boston. See also p 678. 


































CEMENT, CONCRETE, 


ETC 


677 


Neat. 


1 D. 

1W. 

1M. 

6M. 

1 Y. 

71 

92 

145 

282 

290 

Cement 1. 

Sand 2. 


22 

49 

105 

169 

| . ; 


Neat. 


102 

303 

412 

468 1194 

• 

Cement 1. 

Sand 2. 


126 

163 

279 

323 


Rosendale. 

Cement 1. Sand 1. 


1W. 


56 


1M. 


116 


6 M. 


180 


1 Y. 


236 


Cement 1. Sand 3. 


12 


25 


65 


121 


Cement 1. Sand 1.5. 


Portland. 

Cement 1. Sand 1. 


160 


225 


347 


387 


Cement 1. 
95 I 130 


Sand 3. 


198 


257 


11V. 

1M. 

6M. 

1 Y. 

41 

90 

135 

210 

Cen 

lent 1. 

10 

Sand 5. 

36 80 

Cen 

lent 1. 

Sand 

1.5. 

Cen 

55 

lent 1. 

78 

San 

116 

d 5. 

145 


The crushing strength. For each proportion of sand we may take the 
trength preceding it in the table, p 676. Moreover the crushing strength with sand increases with 
ge much more rapidly than the tensile; and the more so the greater the proportion of sand. 

As a general rule with cements of good quality we shall have mortars fit for most engineering pur- 
>oses if we do not exceed from 1 to 1.5 measures of dry sand to 1 of the common cements ; or from 2 
o 3 of sand to 1 of Portland. 

The shearing strength of neat cements averages about one-fourth of the 

ensile. 

The adhesion of cements to bricks or rough rubble, at dif¬ 
ferent ages, and whether neat or with sand, may probably he taken at an average of about three- 
ourlhs of the cohesive or tensile strength of the cement or mortar at the same age. If the bricks and 
itoue are moist and entirely free from dust when laid, the adhesion is increased ; whereas if very dry 
md dusty, especially in hot weather, it may be reduced almost to 0. The adhesion to very hard, 
nnooth bricks, or to finely dressed or sawed masonry is less. 

The voids in sand of pure quartz like that found on most of our sea- 

ihores, when perfectly dry and loose, occupy from .303 of the mass in sand weighing 115 lbs per cub ft, 
;o .515 in that weighing 80 tbs. Usually, however, such dry sand weighs say from 105 lbs with voids of 
364 : to 95 lbs, with voids .424; the mean being 100 lbs, with voids .394.* But the wet sand in mortar 
jccupies about from 5 to 7 per cent less space than when dry ; the shrinkage averaging say 6 per ct 
or jV part; thus making the voids .304 of the 105 lb sand when wet; and .364 of the 95 lb ; the mean 
of whioh is .334. But to allow for imperfect mixing, &c, it is better to assume the voids at .4 of the 
dry sand. Moreover, since the cements, as before stated, shrink more or less when mixed with water, 
and worked up into mortar, it would be as well to assume that to make sufficient paste to thoroughly 
fill the voids, we should not use a less volume of dry common cement, slightly shaken, than half the 

bulk of the dry sand; or than .45 of the bulk if Portland. The bulk of the 
■nixed mortar will then be about equal to or a trifle less than that of the dry 

sand alone. 

The best sand is that with grains of very uneven sizes, and sharp. The 
more uneven the sizes the smaller are the voids, and the heavier is the sand. It should be well 
washed if it contains clay or mud, for these are very injurious to mortar or concrete. 


* If greater accuracy is desired pour into a graduated cylindrical 

measuring-glass 100 measures of dry sand. Pour this out, and fill the glass up to 60 measures with 

i/jiwater. Into this sprinkle slowly the same 100 measures of dry sand. These 

will now be found to fill the glass only to say 94 measures, having shrunk say 6 per ct; while the 
water will reach to say 121 measures : of which 121—94 ~ 27 measures will be above the sand : leaving 
50—27 — 33 measures filling the voids in 94 measures of wet sand; showing the voids in the wet sand 

to be = .351 of the wet mass. If the sand is poured into the water 

hastily, air is carried in with it, the voids will not be filled, and the result will be quite different. 

Since a cubic foot of pure quartz weighs 165 lbs, it follows that 

if we weigh a cubic foot of pure dry sand either loose or rammed, then as 165 is to the wt found, so is 
1 to the solid part of the sand. And if this solid part be subtracted from 1, the remainder will be the 
voids, as below. 


Wt in lbs per cub ft dry 
Proportion of solid 
Proportion of voids 


80 

85 

90 

95 

100 

105 

110 

115 

.485 

.515 

.546 

.576 

.606 

.636 

.667 

.697 

.515 

.485 

.454 

.424 

.394 

.364 

.333 

.303 







































































678 CEMENT, CONCRETE, ETC. 


The common (not Portland) cements, when used as mortar tor brickwork, often disfigure it, especi¬ 
ally near sea coasts, and in damp climates, by wliite efflorescences whic! 
sometimes spread over the entire exposed face of the work, and also injure the bricks. This als 
occurs in stone masonry, but to a much less extent, and is confined to the mortar joints, and injure 
only porous stone. It is usually a hydrous carbonate of soda or of potash often containing other salt; 
Gen'l Gillmore recommends as a preventive to add to every 300 lbs (l barrel) of the cement powdei 
100 lbs of quicklime, and from 8 to 12 lbs of any cheap animal fat. The fat to be well incorporate 
with the quicklime before slacking it preparatory to adding it to the cement. This addition will rt 
tard the setting, and somewhat diminish the strength of the cement. It is also said by others tbs 
linseed oil at the rate of 2 gallons to 300 lbs of dry cement, either with or without lime, will in a 
exposures prevent efflorescence, but like the fat it greatly retards setting, and weakens, nee also || 
673. 

Mr. Eliot C. Clarke, to whom we are already indebted for tables on pp 67' 
and 682, has published, in Trans Am Soc C E April 1885, the results of a series of ex 
periments made for the Boston Main Drainage Works. From his paper we condense L 
as follows, by permission. . jpi 

Variations in shade in a given kind of cement may indicate diffs in the characte t 
of the rock or degree of burning. Thus, with Rosendale, a light color generally in 
cated an inferior or under-burned rock. A coarse-ground cement, light in color ant f 
weight, would be viewed with suspicion. 

Finer sieves than No 50 (about 50 meshes to the lineal inch) should be used , l; , 
No 120 (120 meshes) was used in the experiments. 

The highest strength was obtained by the use of just enough water to thor 
oughly dampen the cement. An excess of water retards setting. American cement 
needed more water than Portland ; fine-ground more than coarse; quick-settingmor 
than slow. Neat Rosendale, a year old, was strongest with 35 per cent water. Nea 
Portland, same age, with 20 per cent. 

The finer the Hand, the less is the strength. 

Salt, either in the water used for mixing, or in that in which the cement is laid 
retards setting somewhat, but has no important effect upon the strength. 

Adding clay gives a much more dense, plastic, water-tight paste, useful for plas 
ter or for stopping leaks. Half a part of clay did not seem to weaken mortar mate 
rially, except in the case of sample blocks exposed to the weather for 2^4 years afte i 
a week’s hardening in water. 

A year’s saturation in fresh or salt water, and in contact with oak, hart <r - 

pine, white pine, spruce or ash, did not affect the mortars. 


With sand, fine-ground cements make the strongest mortar; but when teste 
neat, coarse-ground cements are strongest. This is especially the case wit! 
Portlands. 

Good results were obtained from mixing cements. A mortar of half a par 
each of Rosendale and Portland, and two parts sand, was stronger, at 1 wk. 1 mo, 
mos and 1 year, than the average of two mortars, one of 1 part Rosendale and one o 
1 part Portland; each with 2 parts sand. Mixtures of Roman (quick setting) an* 
Portland (slow) set about sis quickly as Roman alone, and were much stronger. 

Portland resisted abrasion best when mixed with 2 parts sand; Rosondal 
with 1 part. A little more or less sand rapidly reduces! the resistance in both cases. 

Cements expand in setting ; but not more than 1 part in 1000 of any giveil 
dimension. 

Art. 10. Cement concrete or Beton, is the foregoing cement morta 

mixed with gravel or broken stone, brick, oyster shells, &c, or with all together. In concrete as i 
mortar, it is advisable on the score of strength that all the voids be tilled or more than filled. Thos 
of broken stone of \olerably uniform size aud shape are about .5 of the mass ; with more irregularit 
of size and shape they may decrease to .4. Those of gravel vary like those of sand, aud had like 
better be taken at .5 when estimating the dry cement. VV T e shall then have as follows. 


For 1 cub yd of concrete of slonc, gravel, and sand, withon 

voids. 


1 cub yd broken stone with .5 of its bulk voids, requiring 
0.5 cub yd gravel “ .5 “ “ “ 

0.25 cub yd sand “ .6 “ “ “ 


.5 cub yd gravel. 

.25 cub yd sand. 

.125 (or %) cub yd dry cement. 


It is probable that mistakes have occurred from inadvertently assuming that because 
voids in a broken mass, constitute a certain proportion of the bulk of said mass ; therefore, the origin; 
solid has s welled in only that same proportion. Thus, if a solid cubic yardof stone be broken into sms 
irregular pieces, which have among themselves about the same proportions of large aud small ones, i 
usually occurs in quarrying, or in railroad rock cuttings; and if these be loosely thrown iuto a hea| 
the .47 of this heap, or rather less than half of it. will be voids. But it does not follow, therefore, th; 
the original solid cub yd has swelled only .47. or nearly one-half, or makes only l $4 cub yds of broke 
stone ; although many young engineers would probably consider this a very full allowance; and woui 
suppose that they were quite just to the company, if they counted for the contractor one solid yni 
of excavation for every 1 \4 yds of fragments. Now. it is plain that if .47 of the broken heap are void 
the remaining .53 must be stone. But these .53 constituted the original solid cubic yard ; and th< 
still remain equal to it in actual solidity. Hence we must say as follows : If .53 of the brokeu uia 
occupies oue cub yd of actual space, how much space will the whole mass occupy; or, 


Of the 

brokeu mass. 


Cub yd 
of space. 


Entire 
broken mass. 


Cub yds 
of space. 


1 X 1 


. -y 









CEMENT, CONCRETE, ETC. 


679 


is in 

?n 

«■ 

* 

ter 

in' 


» Hence, we see that a solid cab yd of stone, when so broken, swells to 1.9, or nearly 2 cub vds; and 
ence a proper allowance to a contractor, would be 1 cub yd solid, for every 1.9 cub yds of piece’s • or 
alsot» e yds of pieces must be divided by 1.9 for the solid yards. 

ures we know that a cubic yard of auy stone, breaks to, say 1.9 yds, then to find the proportions of 
salts, eids, and solid, in the broken mass, proceed thus: The solid part of the broken mass must occupy 1 
Ser, ul) yd of spape; aud the questiou is what part of 1.9 yds does this 1 yd constitute. The answer is 

— =■ .511; therefore, 5.'1 hundredths of the brokeu mass is solid; aud of course the remaining 47 huu- 
toi redths are voids. 

If a cubic foot solid weighs N lbs ; but when broken up, or ground, only n lbs per cub ft, then n 
ivided by N, will be the proportion of solid in the brokeu mass. 

If the broken stone is loosely piled up, it will occupy a little less space, say about 1.8 cub vds; in 
'hich case the voids will be .14 ; and the solid, .56 of the mass. We will here venture to express our 
oubts whether hard rock when blasted and made into embankment, settles to less than 1 % yds for 
very solid yd. Mr Ellwood Morris gives as the result of certain embankment of hard sandstone, 
tade under his supervision, an increase of bulk of y 5 ^; or in other words, that I cub yd of rock in 
lace, made 1-^, or 1.417 yds of embankment. Tins'corresponds to very nearlv .7 solid; and .3 
oids ; while 1 % yds to 1 solid, corresponds to .6 solid; and .4 voids. The rough sides of rock excava- 
ons, make it difficult to measure them with accuracy; and we cannot but suspect that something 
f this kind has interfered with the results obtained by Mr Morris. He, however, may be right, aud 
•e wrong. 

By some careful experiments of our own, an ordinary pure sand from the sea shore, perfectly dry, 
nd loose, weighed 97 fts per cub ft; and its voids were .41, and the solid .59 of the mass. By thorough 
aaking, and jarring, it could be settled the .1333 part, (halfway between and >*,) and no more. 

; then weighed 112 lbs per cub ft; and its voids were then .32; and the solid. .68 of the mass. 
us Another pure quartz sand, of much finer grain, perfectly dry and loose, weighed but 88 tbs per cub 
OftJ.; the voids were .466; and the solid .534 of the mass. By thorough shaking and jarring it could 
eatje reduced; like the former, only the .1333 part: it then weighed 101.6 lbs per cub ft; and its voids 
ere .384 ; and the solid .616. Another, consisting of the finest sifted grains, of the last, weighed 82 
|»s per cub ft; so that its voids, and solid, each were very nearly .5 of the mass. This could be com- 
. I acted about % part; and then weighed 98}^ lbs per cub ft. 

HO, The first, or coarsest of these sands, when quite moist, but not wet, perfectly loose, weighed but 86 
■s per cub ft; or 11 lbs less than when dry. It could be rammed in thin layers until it settled one- 
IjjJfth part; and then weighed 107*s£ lbs per cub ft. Voids .348. solid .652. 

The second sand, similarly moist and loose, weighed but 69 lbs per cub ft; or 19 tbs less than when 
,ry. It could be rammed in thin layers, until it settled 14 part; and then weighed I03**> Bis per cub 
tat. Voids .373, solid .627. 

None of these sands when dry, and loose, if poured gently into water to a depth of 15 inches, set¬ 
tled more than about one-fifteenth part; the coarsest one, considerably less. 

Here the . 125 cub yd of dry cement constitutes oue-eighth of the single mass; or one-fifteenth of 
^jll the dry ingredients as measured separately. 

p 1 cub yd of concrete of broken stone and Mind without 

voids. 

cub yd broken stone, with .5 of its bulk voids requiring | .5 cub yd sand, 
i, ij|j cub yard sand, “ .5“ “ “ “ j .25 cub yd dry cement. 

i The strength of concrete is affected by the quality of the broken stone, 

well as by that of the cement, the degree of ramming, &c. Cubes of either of the above with Port- 
|ind, as well as one composed of l meas of good Portland to 5 of sand only, well made, and rammed, 
klejiouid either in air or in water require to crush them at different ages, not less than about as follows. 


Ase in months.1 3 6 9 IS 

Tons per s«| ft .15 40 65 85 100 

Under favorable conditions of materials, workmanship and weather, the strengths may be from 50 
i 100 per ct greater. For transverse strength as beams see p 682. 

If not rammed the strength will average about one-third part less. 
rit j With common U. S. cements, if of good quality from .2 to .3 of the 

iei|.rength of Portland concrete may be had. 

Slow setting' cements are best for concrete, especially when to he 

mined. 

It may not be amiss to state here that when masonry is baelced by 
loncrete the two are liable in time to crack apart from unequal settlement, 
specially if the ramming has not been thorough; also that in variable climates 
ast iron cylinders filled with concrete are frequently split horizon- 

lUy by unequal expansion and contraction. In such structures it is safest to consider the cyls as 
lere moulds for the concrete: and to depend upon the last only for sustaining the load. 

The concrete for the New York City docks consists of 1 measure 

iM f either English or Saylor's Portland, 2 of sand, 5 of broken stone (hard trap). That made of Kng- 
u»l sh Portland, after drying a few days, and then being immersed 6 weeks, requited about 30 tons per 
11 q ft to crush it. Saylor’s would probably require the same. At (he Missis¬ 
sippi Jetties, (see “South Pass Jetties” by Max E. Schmidt, C. E., Trans Am 

jet oc C E, Aug 1879) Saylor's Portland 1; sand 2.76; gravel 1.46; broken stone 5. 

iii In the. foundations of the Washi litflon Monument at Washington, D.C., 

an 1880) English Portland (J. B. White & Bros) 1 ; sand 2 ; gravel 3 ; broken stone 4; and according to 

ili Govt. Report, has a crushing strength of 155 tons persq ft when 7.5 months old, 
le Yt Croton Dam, N. Y., (1870) Rosendale I; sand 2; brokeu stone 4.5. Some 

** t the same work, and deposited under water, had 6 meas of stone; and at the eud of a year had be 
ome so hard that it was found necessary to drill and blast a portion that had to be removed. 


A 







680 CEMENT, CONCRETE, ETC. 


Lime witll cement weakens all of them, but General Q. A. Gillmore, on 
best authority, repeatedly states that even in important concrete work in either the air or water, (pr. 
vided the water does not come into contact with it until settingtakes place), from .25 to even .o of th 
neat cement paste of the U. S. common cements may he replaced by lime paste without serious dim II 
nution of either strength or hydraulicity; and with decided economy. It retards the setting w hie 
is often of great advantage, especially with quick setting cements which at times cam^ot on that at » 
count be advantageously used without some lime. 

M<M 1 I<I 0 <I blocks of Portland concrete of even 50 tons wt can generally b » 

handled and removed to their places in from 1 to 2 weeks. — 

Ritnimin^ of concrete, when properly done, consolidates the mass alwni 1 J 

5 or fi per ct, rendering it less porous, and very materially stronger. The rammers are like thos £ 
used in street paving, of wood, about 4 ft long, ft to 8 ins diam at foot with a lifting handle, and sho 
with iron; weight about 35 lbs. Thej are let fall 6 or 8 ins. The men using them, if standin « 
on the concrete, should wear iudia-rubber boots to preserve their feet from corrosion by the cement o 

Kiuniniiig cannot be done under water, except partially, when th. a 

concrete is enclosed in bags. A rake may, however, be used gently for levelling concrete under water 

Blake’s Stone Crusher (Co, New Haven, Connecticut), is useful fo , 

breaking the stone more cheaply than bv hand on a large work. The two sizes best adapted to thi , 
purpose cost about $000 and $1200; break 6 to 7 cub yds per hour; and require steam-engines of abou , 
8 to 10 horse power to run them properly. According to Mr. J. J. K. Croes, C. E. (see “ Constructioi ( 
of Croton Dam," Trans. Am. Soc. C. E., Feb, 1875), a machine will require about as follows: , 
engine man. 1 or 2 men to break the larger stones to a size that will enter the machine, 1 driver t . 
horse-cart, 1 man to feed the stone into the machine, 2 to keep him supplied with stone, 1 at th | 
screen, 2 wheeling away the broken stone to the stone-heap, 1 or 2 to receive it at the heap. _ Say 1 | 

or 12 men in all. Tlie size of the broken stone for concrete is gen i 

erally specified not to exceed about 2 ins on any edge: but if it is well freed from dust by screening o i 
washing, all sizes from .5 to 4 ins on any edge may be used, care being taken that the other ingredi . . 
euts completely fill the voids. 

Concrete is groo<I for bringing np an uneven foundation t< 
a level before starting the masonry. By this means the number of horteont; 
joints in the masonry is equalized, and unequal settlemeut is thereby prevented. 

Concrete inay readily be deposited under water in the usm 

way of lowering it, soon after it is mixed, in a V shaped box of wood or plate-iron, with a lid thi 
may be closed while the box descends. The lid however is often omitted. This box is so arrange 
that on reaching bottom a pin may be drawn out by a string reaching to the snrface. thus permittir 
one of the sloping sides to swing open below, and allow the concrete to fall out. 'J he box is the 
raised to be refilled. In large works the box may contain a cubic yard or more, and should be su 
pended from a travelling crane, by which it cau readily be brought over any required spot in ti 
work. The concrete may if necessary be gently levelled by a rake soon after it leaves the box. 1 
consistency and strength will of course be impaired by falling through the water from the box : an 
moreover it cannot be rammed under water without still greater injury. Still, if good it will in du' 
time become sufficiently strong for all engineering purposes. Coucrete has been safely deposited ii 
the above manner in depths of 50 ft. 

The Tremie, sometimes used for depositing concrete under water, is a box 

of wood or of plate iron, round or square, and open at top and bottom ; and of a length suited to th. 
depth of water It may be about 18 ins diam. its top, which is always kept above water, is hopper 
shaped, for receiving theconcrete more readily. It is moved laterally and vertically by a travellin; 
craue or other device suited to the case. Its lower eud rests on the river bottom, or on the deposite. 
concrete. In commencing operations, its lower end resting oh the river bottom, it is first entirely 
tilled with concrete, which (to prevent its being washed to pieces by falling through the water in th. 
tremie) is lowered in a cylindrical tub, with a bottom somewhat like the box before described, whicl 
can be opened when it arrives at its proper place. After being filled it is kept so by throwing frest 
concrete into the hopper to supply the place of that which gradually falls out below, as the tremie i> 
lifted a little to allow it to do so. The wt of the filled tremie compacts the concrete as it is deposited 
A tremie had better widen out downwards, to allow the concrete to fall out more readily. See “ Gill 
more on Cements.” 

The area upon whicl* it is deposited must previously be stirrotinde. 

by some kind of enclosure, to prevent the concrete from spreading beyoud its proper limits; and t 
serve as a mould to give it its intended shape. This enclosure must be so strong that its sides mu; 
not be bulged outwards by tbe weight of the concrete. It will usually be a close crib of timber o 
plate-iron without a bottom: and will remain after the work is done. If of timber it may require ai 
outer row of cells, to be filled with stone or gravel for sinking it into place. Care must be taken t 
prevent the escape of tbe concrete through open spaces uuder the sides of the crib or enclosure. T 
this end the crib may be scribed to suit the inequalities of the bottom when the latter cannot readii i 
be levelled off. Or iusidc sheet piles will be better in some cases; or an outer or inner broad flap oi 
tarpaulin may be fastened all around tbe lower edge of the crib, and be weighted w ith stone or grave 
to keep it in place on the bottom. Broken stone or gravel or even earth (the last two where there i 
no curreut) heaped np outside Of a weak crib will prevent the bulging outwards of its sides by th. 
pressure of the concrete. After the concrete has been carried up to within some feet of low water 
and levelled off, the masonry may be started upon it by means of a caisson (page 636); or by men ii 
diving dresses. Or if tbe concrete reaches very nearly to low water, a first deep course of stone ma; 
be laid and the work thus brought at once aliove low water witiiout any such aids. 

The concrete should extern! out. from 2 to 5 feet (according t< 
the case) beyond the base of the masonry. All soft. 1 Bl 11 < I should he removed 
before depositing concrete. Bags partly lilted Will* Concrete, and merely throw i 

into the water may be useful in certain cases. If the texture of the hags is slightly open, a portioi 
of the cement will ooze out. aud biDd the whole into a tolerably compact mass. Such bags, bv the ai. 
of divers, may be employed for stopping leaks, underpinning, and various other purposes, that mat 
suggest themselves. Such bag3 may be rammed to some extent. 

Tarpaulin may be spread over deep seams in rock to preveul 

the loss of concrete; aud in some cases, to prevent it from being washed away by springs. 






CEMENT, CONCRETE, ETC. 


681 


■ om 
pm 
■tthf 

dig. 

aich 

:i '-ac 

' be 


it* 

'sue 

hg; 

iiag 

■iilM 

the 

iter, 

1 for 

this 

'l)W 

-''Job 
: 1 
■trio 
- lie 
: id 
en- 

;er 
; edi 

:1« 


There is much room for judgment in the various applications of 

concrete ; especially under water. 

Concrete has been used in very large masses; as in the founda¬ 
tion of a graving dock at Toulon, France; where it was deposited to a thickness of 15 feet, over an 
area of 400 ft by 100 ft; forming, as it were, a single artificial stone of that size. It was deposited 



.. - . conforming i 

shape. The last deposits of concrete were then faced with masonry. Walls of buildings are also fre 
quently built of cement concrete deposited between planks as a mould ; and which are moved upward 
as the building goes on. Flues may be made in these walls by ramming concrete around a tube, which 

can afterwards be lifted out; and be used for the. next course above. The 
dome of the Pantheon, at Rome, 142 ft diam, and now nearly 2000 years 

old, is of concrete. The St. It. viaducts Pont Napoleon, and Pout 

d’Alma, at Paris, have arches of 115 and 141 feet span, of concrete. 

With regard to the mixing of concrete, Gen Gillmore gives the 

method pursued and described by Lieut Wright. The gravel and pebbles being first separated by 
screening, into different sizes, “ the concrete was prepared by spreading out the gravel on a platform 
of rough boards, in a layer from 8 to 12 ins thick ; the smaller pebbles at the bottom, and the larger 
ones on top. The mortar was then spread over the gravel as uniformly as possible. The materials 
were then mixed by 4 men ; 2 with shovels, and 2 with hoes; the former faciug each other, and always 
working from the outside of the heap, to the center. They then went back to the outside, and re-' 
peated this operation, until the whole mass was turned. The men with hoes worked each in conjunc¬ 
tion with a shoveller, and were required to rub each shovelful well into the mortar, as it was turned 
and spread, or rather scattered on the platform by a jerkiug motion. The heap was turned over a 
second time, in the same manner, but iu the opposite direction ; and the ingredients were thus thor¬ 
oughly incorporated ; the surface of every pebble being well covered with mortar. Two turnings 
usually sufficed, and the concrete was then carried to the foundation in which it was to be used. The 
success of the operation depends, however, entirely upon the proper management of the hoe and 
shovel; and although this may be easily learned by the laborer, yet he seldom acquires it without the 
particular attention of tbe overseer.” It is bard work. 


■in' 

itn 

aged 

in? 

tin 

» 

It 


lit 

: '0I 

o tbe 
«■ 
inf 

ted 
:re!j 
the 
■ ich 

eit 

ill 

A 

m 

ied 

lio 

PIT 

■or 


Or simple machinery is sometimes employees for incorporat¬ 
ing the ingredients of concrete, when large quantities are required. A machine that has been much 
used successfully in Germany, consists simply of a cylinder about 13 ft long, aud 4 ft diam, open at 
both ends; and lined on the inside, which is perfectly smooth, with sheet iron. It is inclined 6 or 8 
degrees with the horizou. This cylinder is made to revolve 15 or 20 times per min, by means of a 
simple leather strap or band arouud its outside; and to which motion is given by a locomotive, which 
at the same time worked a heavy mill for mixing the mortar. This simple machine easily turns out 
from 105 to 130 cub yds of concrete iu 10 hours ; and when worked iu connection with a mortar mill, 
at a trifling expense." 

“ When concrete is deposited in water, especially in the sea, a pulpy gelatinous fluid exudes from 
the cement, and rises to the surface. This causes the water to assume a milky hue; hence the term 
laitance, which French engineers apply to this substance. As it sets very imperfectly, and 
with some varieties of cements scarcely at all, its interposition between the layers of coucrete, even 
in moderate quantities, will have a teudency to lessen, more or less sensibly, the continuity and 
strength of the mass. It is usually removed from the enclosed space by pumps. Its proportion is 
greatly diminished by reducing the area of concrete exposed to the water, by using large boxes, say 
from 1 to 1*4 cub yds capacity, for immersing the concrete.” 

Weight of good concrete 130 to 160 !bs per cub ft, dry. 

Cost off concrete $5 to $9 per cub yard if roughly deposited ; and $9 to $15 
if first made into blocks; depending on size, cement, locality, wages, &c. 

M. F. Coignet’s beton. The artificial stone which bears this engineer’s 
name has for several years been used in France with perfect success, not only for 
dwellings, depots, large city sewers, &c, but for the^piers and arches of bridges, 
light-houses, &c. Bridge arches of 116 ft span;and of low rise, have been built of 
it. It is composed of 5 measures of saud, 1 of sifted dry-slaked lime, aud from % 
to 34 measure of ground Portland cement. Or of sand 6, cement 1, lime 341 &c. 


AD 

a to 

lily 

Sff 1 

nel 
eis 
lie 
ter, 
j in 
a»J 

to 

»<l 

ion 

aiii 

*f 

'lit 


These are first well mixed together dry, and then placed in a mixing-mill: at the same time 
sprinkling them with .3 to .4 measure of water, so as to moisten them slightly, without wetting 
them. They are then thoroughly incorporated by mixing, until they form a stiff pasty mass, 
slightly coherent. This is then placed in a mould, in successive thin layers, each of which is well 
compacted by blows of a 16 lb rammer. The top of each layer may be scored or cross-cut, to make 
the next one unite betier with it. Owing to the small proportion of water, it sets soon; and may 
generally be taken from the mould in from a few hours to a few days, depending on the size of the 
block • and left to harden. River sand is the best, inasmuch as it requires less lime and cement 
than pit sand, to make equally good stone.(?) The cement should be a rather slow-setting one; and 
both it and the lime should he screened, to exclude lumps. About 1*4 bushels, or 1% cub ft of 
the dry materials, make 1 cubic foot of finished stone, weighing about 140 H>s; resisting 100 to 
150 tons per square foot at 3 months old. 250 to 400 in 2 years. Arches of it are made no thicker 
than brick ones. An arch, pier, wall, foundation, &c. may be built of it. as one stone, instead of 
in separate blocks. In sewers the centers may be struck within 10 to 15 hours after the arch is 
finished; and the water may be admitted within a week or less. The distinctive features of Coignet's 
beton are - the verv small proportion of water; the thorough incorporation of the ingredients; and 
the consolidation of the separate layers bv ramming. It is difficult for a person who has never seen 
the process to credit the rapidity, facility, and economy with which blocks of good stone can be 
made by it ’ Its cost, as compared with perfectly plain dressed granite, does not exceed one-half; 
while for ornamental work it compares even far more favorably. Hence the Ooignet beton, or artifi¬ 
cial stone is nothing more than good, well-prepared mortar, mixed with eery little water; and well 
rammed into moulds, in successive layers. A mixture of 1 measure of hydraulic cement, and 3 
measures of sand, similarly treated, has been successfully used in the U. S., for some years, in 
buildings of all kinds. Ornamental work can be furnished at *4, the price of stone; aud will answer 
equally well. For full information, see Gillmore’s “ Coignet Beton." 





682 CEMENT, CONCRETE, ETC. 


Transverse Strength of Concrete Beams.* 

Averace results of 24 beams, 10 ins square, made of good Rosendalo and Englisl 
Portland cements, pit sand and screened pebbles, few exceeding 1 inch diam. Th 
beams were buried for 6 months, in a pit 4 ft deep, in gravelly soil, exposed to tli 
rain, snow, Ac. A first set of beams all broke on being taken from the moulds afte 
7 or 8 days, although carefully handled. To avoid this, the bottom of the pit itsel N 
was rammed to a smooth, hard surface: immediately upon which a new set wa 0 j 
made by ramming the concrete into 2 inch planed plank moulds without bottonif j 
The moulds were removed after 24 hours, and when all were done the earth wa 8 
filled in over the undisturbed beams. Very little of the soil adhered to them. Thei f 8 
wt in all cases when tested was about 150 lbs per cub ft, or 520 lbs wt of 5 ft clea |j 
span of beam; one half of which, or 260 lbs, must be deducted from the cen break) ri 
loads of the 5 It spans below; and 124 lbs from the 2 ft 4.5 ins ones. The coefticien 
or Constant C is the cen breakg load in lbs for a beam 1 inch square, and 1 f f 
clear span, like those in table p 493; and like them is found by the lormnlaat to] t! 
ef p. 492. Its use is shown by the formula at foot of p 492. « 

i 

tl 


Proportions of mate¬ 
rials by measure. 

Center Breaking 
load in lbs, including half 
wt of beam. 

- --- 

Constant c 

1 

Cement. 

Sand. 

Pebbles. 

Span 2 ft 4.5 ins. 

Span 5 ft. 


Roseudale 1 

2 

5 

1782 

690 

3.7 

“ 1 

3 

7 

all broke in 

handling 

>v 

Portland 1 

3 

7 

3926 

1995 

9.8 

“ 1 

4 

9 

3648 


8.1 

“ 1 

6 

11 

2822 

1190 

6.2 


* This useful table and that on p 677 were kindly furnished us by Eliot C. 
Clarke, Esq., then Principal Ass’tin charge of the Improved Sewerage Works of 
Boston, Mass.; for which the experiments were made. See also “ Mr. Eliot C. 
Clarke ”, p 678. 
















RETAINING-W A LLS. 


683 


Hie 

ths 

>Het 

elf 

t was 1 
' m; 

j| *is: 

Tiieii i 




RETAINING-WALLS. 


^I* We here speak only of walls sustaining earth ; for those sustaining water, 
see pp 229 to 232, and 236. A retaining-wall is one for sustaining the pres 
}f earth, sand, or other filling or hacking , deposited behind it after it is built; in 
listinotion to a face-wall, which is a similar structure for preventing the fall of 
rnrth which is in its undisturbed natural position, but in which a vert or inclined 
face hits been excavated. The earth is then in so consolidated a condition as to exert 


, ar little or no lateral 


wall may generally be thinner than a 



Du¬ 


pres, and therefore the 

retaining one. 

This, however, will depend upon the nature and 
position of the strata in which the face is cut. If 
the strata are of rock, with interposed beds of clay, 
arth, or sand ; and if they dip or incline toward the 
wall, it may require to be of far greater thickness 
than ahy ordinary retaining-wall; because when the 
thin seams of earth become softened by infiltrating 
rain, they act as lubrics, like soap, or tallow, to fa- 
« silitate the sliding of the rock strata; and thus bring 
c * an enormous pres against the wall. Or the rock may 
be set in motion by the action of frost upon the clay 
seams; or, as sometimes occurs, by the tremor pro¬ 
duced by passing trains. Even if there be no rock, 
still if the strata of soil dip toward the wall, there 
will always be danger of a similar result; and addi¬ 
tional precautions must be adopted, especially when 
the strata reach to a much greater height than the 
wall. A vertical wall has both c o 
and d s vert. 

Experience, rather than theo- 
C. *\y. must bo our guide in the building of 
both kinds of wall. VVe recommend that 
d, the hor thickness a b , Fig 1, at the base of a 
vert or nearly vert retaining-wall c d b a, 

which sustains a backing of either sand, gravel, or earth, level with its top c d, 
as in the tig, should not be less than the following, in railroad practice, when the 
foundations are not more than about three feet deep. 

When the backing; is deposited loosely, as usual, as when 
dumped from carts, cars, &c. 

Wall of cut-stone, or of first-class large ranged rubble, 

in mortar . a.b .35 of its entire vert height d b. 

“ good common scabbled mortar-rubble, or brick. .4 “ “ “ “ 

“ well-scabbled dry rubble .5 “ “ “ “ 

With good masonry, however, we may take the height d s instead of d b, and then 
the above proportions of d s will give a sufficient thickness at the ground-line o s. 
See Table, p 690. 

When the backing; is somewhat consolidated in hor layers, 

each of these thicknesses may be reduced, but uo rule can be given for this. 

The offset o e, in front of the wall, is not included in these thicknesses. 

When, however, the backing is a pure clean sand, or gravel, we should use only the full dimen¬ 
sions; inasmuch as the tremor, caused by passing trains, would neutralize any supposed advantage 
from ramming materials so devoid of cohesion. Such sand may be rammed with much advantage 
for the purpose of compacting it in foundations; but a diff principle is involved in that case. When 
it is done even with cohesive earths, with a view of saving masonry in retaining-walls, it is probable 
that the expense will generally be found quite equal to that of the masonry saved. See Bern 4, p 691. 

The base ah in Fig 1, is of the height hd. In the foregoing thicknesses at base, the back d b 
of the wall is supposed to be vert; and the face ca either vert, or battered (sloped or inclined back¬ 
ward) to an extent not exceeding about 1% inches to a foot; which limit it is rarely advisable to ex¬ 
ceed in practice, owing to the bad effect of rain, &c, upon the mortar when the batter is great. The 
base of a vert wall need not in fact be as thick as one with a battered face; but when the batter does 
not exceed 1.5 inches to a foot, the diff is very small. See Table, Art 7 

Rf,m. 1. A mixture of Maud, or earth, with a large proportion 

of round bowlders, paving pebbles, Ac, will weigh considerably more than the materials ordinarily 
used for backing; and will exert a greater pres against the wall ; the thickness of which should be 
increased, say about one-eighth to one-sixth part, when such backing has to be used. 

Rf.m. 2. The wall will be stronger if all the courses of masonry be laid 
with an inclination inward, as at neb; especially if of dry masonry, 
or if time cannot be allowed (as it always should be, when practicable) for the mor¬ 
tar to set properly, before the backing is deposited behind it. The object of inclin- 




















684 


RETAINING-WALLS, 




ing the courses, is to place the joints more nearly at right angles to the direction 
/P, Figs <5, 7, and 8, of the pres against the back of the wall; and thus diminish 
the tendency of the stones to slide on one another, and cause the wall to bulge. See 
Art 19 of Force <fcc, p 314. When the courses are hor, there is nothing to pre¬ 

vent this sliding, except the friction of the stones, one upon the other, when of dry 
masonry; or friction and the mortar, when the last is used. But if, as is frequently 
the case, (especially in thick and hastily built walls,) this has not had time to harden 
properly, it will oppose but little resistance to sliding. But when the courses are 
inclined, they cannot slide, without at the same time being lifted up the inclined 
planes formed by themselves. In retaining-walls, as in the abuts of important 
arches, the engineer should place its little dependence as possible upon mortar; but 
should rely more upon the position of the joints, for stability. 

An objection to this inclining of the joints in dry (without mortar) waits, is that rain-water, falling 
on the battered face, is thereby carried inward to the earth backing: which thus becomes soft, and 
settles. This may be in a great measure obviated by laying the outer or face-courses hor; or by 
using mortar for a depth of only about a foot from the face. The top of the wall should be protected 
by a coping c d, Fig 1, which had better project a few ins in front. After the masonry has been 
built up to the surface of the ground, the foundation pit should be filled up; and it is well to con¬ 
solidate the filling by ramming, especially in front of the wall. 


tin 


The hack d b of the wall should be left rough. In brickwork it 

would be well to let every third or fourth course project an inch or two. This increases the friction 
of the earth against the back, and thus causes the resultant of the forces acting behiud the wall to 
become more nearly vert; and to fall farther withiu the base, giving increased stability. It also con¬ 
duces to strength not to make each course of uniform height throughout the thickness of the wall; 
but to have some of the stones (especially near the back; sufficiently high to reach up through two or 
three courses. By this means the whole masonry becomes more effectually interlocked or bouded 
together as one mass: and therefore less liable to bulge. Very thick walls may consist of a facing 
of masonry, and a backing of concrete. 


( !i 

\\ 

I 

i 

ti 

I 
!! 

lit 

I u 

II 

I tl 

II 


Rkm. 3. It is the pres itself of the earth against the back, that creates the friction, which in turn 
modifies the action of the pres ; as the wt or pres of a body upon an iucliued plaue produces friction I 
betw’eeu the body and the plane, sufficient, perhaps, to preveut the body from sliding down it. A re- . 
taining-wall is overthrown by beiug made to revolve around its outer toe or edge e, Fig 1, as a ful¬ 
crum, or turniug-poiut; but in order thus to revolve, its back must first plaiuly rise; and in doing 
so must rub against the backing, and thus encouuter and overcome this friction. The 
friction exists the same, whether the wall stands firm or not; as in the case of tbe 1 
body on an inclined plane ; the only diff is that in one case it prevents motion ; and ' 
in the other only retards it. 

Wliore deep freezing occurs the back of the wall should 

be sloped forwards for 3 or 4 ft below its top as at c o, which should be quite smooth 
so as to lessen the hold of the frost and prevent displacement. 



Rem. 4. When the wall is too thin, it will generally fail 
by bulging outward, at about % of its height above the 
ground, as at a, in Fig 2. A slight bulging in a new wall 
does not necessarily prove it to be actually unsafe. It is 
generally due to the newness of the mortar, and to the 
greater pres exerted by the fresh backing; and will often 
cease to increase after a few months. It need not excite 
apprehension if it does not exceed inch for each foot in 
thickness at a. See Remark 3, Art 7, p 691. 

Art. 2. The young engineer need not in practice concern himself particularly about the precise 
sp ora v op his backing, or about the angle op slope at which it will stand ; for the material which 
he deposits behiud his wall one day, may be dry and incoherent, so as to slope at 1 to 1 ; the next 
day rain may convert it into liquid mud. seekiug its own level, like water; the next it may be ice, 
capable of sustaining a considerable load, as a vert pillar. 

Moreover, he cannot foretell what may be the nature of his backing; for. as a general rule, this 
must consist of whatever the adjacent excavation may produce from time to time ; sand to-day. rock 
to-morrow, &c. Retaining-walls are therefore usually built before the engineer knows the character 
of their backing: so that in practice, these theoretical considerations have comparatively but little 
weight. Theory, uncontrolled by observation and oommon sense, will lead to great errors in every 
departmeut of engineering ; but, on the other hand, no amount of experience alone will compensate 
for an ignorance of theory. The two must go haud-in-hand. 



Again, the settlement of the backing under its own wt, aided 

by the tremors produced by heavy trains at high speed; its expansion by frost, or 
by the infiltration of rain; the hydrostatic pressure arising from the admission of 
the latter through cracks produced in the backing during long droughts ; as well as 
its lubricating action upon it, (diminishing its friction, and giving it a tendency to 
slide,) &c,, exert at times quite as powerful an overturning tendency as the legitimate 
theoretical pres does. The action of these agencies is gradual. Careful observation 
of retaining-walls year after year, will often show that their battered faces are lie- 
coming vertical. Then they will begin to incline outward; and eventually the wall 
will fail. Theory omits loads that may come on backing increasing its pres. 








RETAINING-WALLS. 


685 


tit 

"i* 

•see 

l'» 

■*7 

% 

uen 

ire 

i 

ut 

»nt 


3 


it 

i 

to 

'► 

Jl, 

If 

i 

■«. 


a. 

'0 


S 

it 

it 

d 


i 

4 

1 




Assuming the theoretical views advanced by Professor Moseley to be correct as 
theories, the thicknesses which we have recommended in Art 1, for mortar walls, 
correspond to from 7 to 14 times; and for dry walls about 10 to 20 times, the pres 
assigned by him; and we do not consider ours greater than experience has shown 
to be necessary. See Table 3. Retaining-walls designed by good engineers, but in 
too close accordance with theory, (which assumes that a resistance equal to twice 
the theoretical pres is sufficient,) have failed; and the inference is fair that many of 
those which stand have too small a coefficient of safety. 

The fact is, (or at least so it appears to us,) there must be defects in the theoretical assumptions of 
some of the most prominent writers who give practical rules on this subject. Thus Poucelet, who 
certainly is at their head, states that his tables, for practical use, give thicknesses of base for sus¬ 
taining 1 JL times the theoretical pres ; and this he considers amply safe. Yet, for a vert wall of cit 

granite, his base for sustaining dry sand level with the top, as in Fig 1, is .35 of the vert height; 
and for brick, .45. But the writer found that when not subject to tremor , a wooden model of a vert 
wall weighing but 28 lbs per cub ft, aud with a base of .35 of its height, balanced perfectly dry sand 
sloping at 1)4 to 1, and weighing 89 Bis per cub ft. 

Now, THE RESISTANCE OF SIMILAR WALLS, OF THE SAME DIMENSIONS, 
varies as their specific gravities ; and, since granite weighs about 135 
lbs per cub foot, or (> times as much as our model, it follows, we conceive, 
that a wall .of that material, with a base of .35 of its height, must have 
a resistance of 6 times any true theoretical pres, instead of only 1.8 
times; and that his brick wall must have about 5 times the mere bal¬ 
ancing resistance. Our experiments were made in an upper room of a 
strongly built dwelling ; and we found that the tremor produced by pass¬ 
ing vehicles in the street, by the shutting of doors, and walking about 
the room, sufficed to gradually produce leaning in walls of considerably 
more than twice the mere balancing stability while quiet: and it appears 
to us that the injurious effects of a heavy train would be comparatively 
quite as great upon an actual retaining-wall, supporting so incohesive 
a material as dry sand. 

Siuce, therefore, Poncelet’s wall is in this instance sufficiently stable 
for practice, it seems to us that his theory, which neglects the effect of 
tremors, &c, must be defective. He also gives -1 of the height as a suf¬ 
ficiently safe thickness for a vert granite wall supporting stiff earth; but 
we suspedt that very few engineers would be willing to trust to that pro¬ 
portion, when, as usual, the earth is dumped in from carts, or cars; espe¬ 
cially during a rainy period. If deposited, aud consolidated in layers, 
theory could scarcely assign any thickness for the wall; for the backing thus becomes, as it were, a 
mass of unburnt brick, exerting no hor thrust; and requiring nothing but protection from atmospheric 
influence, to insure its stability without any retaining-w all. It is with great diffidence, and distrust 
in our opinions, that we venture to express doubts respecting the assumptions of so profound an in¬ 
vestigator and writer as Poncelet; and we do so only with the hope that the views of more compe¬ 
tent persons than ourselves, may be thereby elicited. Our own have no better foundation than ex¬ 
periments with wooden and brick models, by ourselves ; combined with observation of actual walls. 

Art. 3. After a wall ab c o, Fig 3. with a vert back, has been proportioned by 
our rule in Art 1, it may be converted into one with an ofl'sctted 
back, as a i n o. This will present greater resistance to overturning; and yet con¬ 
tain no more material. Thus, through the center t of the back, draw any line i n ; 
from n draw n s, vert: divide i s into any even number of equal parts; (in the fig 
there are 4;) and divide *ti, into owe, more equal parts; (in the fig there are 5.) From 
the points of division draw hor, and vert lines, for forming the offsets, as in the fig. 

In the offsetted wall, the cen of grav is thrown farther back from the toe o, than 
in the other, thus giving it increased leverage and resistance; but within ordinary 
practical limits, the diff is very small; and since the triangle of supported earth is 
greater than when the back is vert, its prts is also greater; so that probably no ap¬ 
preciable advantage attends that consideration. The increase of thick ness 
near the base. diminishes, however, the 
leverage v a, Fig H. of the pres / P, of the 
earth against the back. The renter of pressure of 
this pres is in both cases at the vert height, meas¬ 
ured from the bottom ; and it is therefore plain that 
the farther back from the front it is applied, the shorter 
must v a become. Moreover, in the offsetted back, the 
direction of the pres becomes more nearly vert than 
when the back is upright. It is to these causes, rather 
than to the throwing back of the cent ol gray, that 
the offsetted wall owes its increase ot stability over 
one with a vert back. 

Art. 4. When, as in Fig 4, the backing: is higher than the 

wall, and slopes away from its inner edge d, at the nat ural slope d s, of 1% to 1, we 
are confident that the following thicknesses at base will at least be found sufficient 

























686 


RETA1NING-WALLS 


for vert walls with sand. They are deduced from the experiments just alluded tc 
and are but rude approximations, with no scientific basis. VVe should not have in 
sorted them, but for the fact that we know of no others for this case. See p 689. 

The first column contains the vert height s v , of the earth, as compared with th 
vert height of the wall; which latter is assumed to be 1; so that the table begin 
with backing of the same height as the wall, as in Fig 1. These vert walls may b 
changed to others, with battered faces, by Art 8; or without any such proceeding 
their faces may be battered to any extent not exceeding inches to a foot, or 1 ii 
8, without sensibly affecting their stability, without increasing the base. 

TABLE 1. (Original.) 


m 

m 

it 

1 In 


£*> S 

* o> o 
v A u 
o> v to 

5 -B «> 
w > 

*r « o 

° *3 

T - •* 

JB {S — 

.S?xs "3 

a> o> > 

■*= 5a 

Wall 

of 

Cut Stone, 
in 

Mortar. 

Good 

Mortar 

Rubble, 

or 

Brick. 

Wall 

of 

good dry 
Rubble. 

— 2 "o 

r.a>2 

js. £ 
a) 4, no 
<0 

" - ► 

oeB 
' 00 
* - 
.Sif^O c3 

0> p k 

JS ^ 

Wall 

of 

Cut Stone 
in 

Mortar. 

Good 

Mortar 

Rubble, 

or 

Brick. 

Wall 
of • 

good dry 
Rubble. 

a3 J J 

Thickness at Base, in 

parts of 

a »>.s 

1 5- 

Thickness at Base, in parts of 

® O <M 

H o o 


the height. 


,P O'M 

Ho© 


the height. 


i. 

.35 

.40 

.50 

2. 

.58 

.63 

.73 

i.i 

.42 

.47 

.57 

2.5 

.60 

.65 

.75 

1.2 

.46 

.51 

.61 

3. 

.62 

.67 

.77 

1.3 

.49 

.54 

.64 

4. 

.63 

.68 

.78 

1.4 

.51 

.56 

.66 

6. 

.64 

.69 

.79 

1.5 

.52 

.57 

.67 

9. 

.65 

.70 

.80 

1.6 

.54 

.59 

.69 

14. 

.66 

.71 

.81 

1.7 

.55 

.60 

.70 

25. 




1.8 

.56 

.61 

.71 

or more 

.68 

.73 

.83 


Art. 5. But when the slope n r. Fig 5, of 1% to 1, starts from the outer edge i 
of the wall, greater thickness is required. Poncelet gives the following for thii 
case, for dry sand. 

TABLE 2. 



Fig-. 5. 


Total depth of earth, 
compared with height 
of wall. 

Wall 

of 

Cut Stone 
in 

Mortar. 

Wall 

of 

Brickwork. 

Total depth of earth, 
compared with height 
of wall. 

Wall 

of 

Cut Stone 
in 

Mortar. 

1 

.35 

.452 

2.4 

.762 

1.1 

.393 

.498 

3 0 

.811 

1.2 

.439 

.548 

4.0 

.852 

1.3 

.485 

.604 

6.0 

.883 

1.4 

.532 

.665 

11.0 

.909 

1.5 

.579 

.726 

21.0 

.922 

1.6 

.617 

.778 

31.0 

.926 

1.7 

.645 

.824 

Infinite. 

.934 

1.8 

.668 

.847 



1.9 

.690 

.903 



2.0 

.707 

.930 






Wall 

of 

Brickwork. 


1.02 
1.11 
1.18 
1.25 
1 28 

1.31 

1.32 
1.34 


When the earth reaches above the top of the wall, as in Figs 4 and 5. the wall is Slir<*IlHrjC<*4l • 
and the earth that is above the top, is called the sukchakgk When the surcharge is carefully deposited 
above the wall, so as to slope back at a steeper angle than 1^ to 1, as say at 1 to 1, theory does not 
require the wall lobe as thick. Notwithstanding Poncelet’s high position, the writer cannot imagine 
that the base of a brick wall need be so great as ijj times its height for any height of sand whatever. 

Art. 6. On Hie theory of retaining--walls. Let b c a m, Fig 6, be 

such a wall, upholding backing or filling csmg; the upper surf c s of which is 
hor, and level with the top b c of the wall; and let m s represent the nat slope of the 
earth which composes the backing; m g being hor. 

Abundant experience on public works shows that this slope, whether for sand, gravel, or earth, 
when dry, may be practically taken at to 1 ; that is. l,t£ hor, to 1 of vert measurement; which 
corresponds to an angle am j/of 33° 41' with the hor: which is also about the angle at which bricks 
and roughly dressed masonry begiu to slide on each other. This angle, however, varies considera- 




























































RET AINING-WA LLS. 


687 


•n 


bljr; being greatly Influenced by the degree of dryness, or dampness, of the material; so that mode¬ 
rately damp sand or earth will stand at a slope of 1 to 1, or at an angle of 45°. Whatever it may be, 
■t 18 called the angle of nat slope of the material under consideration. In theoretical calculations 
or walls, it is safest to assume (as we have done throughout) that the backing is perfectly dry, since 


111 



Fio\7 


ts pres Is then greatest; unless it be supposed to be so wet as to possess some degree of fluidity. The 
.riangle cm a of earth above the nat slope ms, tends to slide down said slope, but is prevented from 
10 doing by the wall. 

It is assumed in all oases, that the wall is secured from sliding along its base,Art 9. p 692, that it is 
.hick enough to prevent failure by bulging; and that it will fail only by overturning, by rotating 
irouud its toe, a, as a fulcrum. The thickness necessary to insure safety against the last will also be 
mfficient to prevent bulging. Now referring only to Fig 6 with a vert back, if the angle o m s, con- 
ained between the natural slope m s, and a vert line to o, drawn from the inner bottom edge m of 
he wall, be divided by a line m t, into two equal angles, o m t, t m s, then the angle o m t is called 
the angle, and m t the slope, of maximum PRESSUKK. The triangular prism of earth, of which 
>mt is a section, or an end view, is called the prism of max pres; because, if considered as a wedge 
ictiug against the back of the wall, it would produce a greater pres upon it than would the entire 
.riangle c to s of earth, considered as a single wedge. For although the last is the heaviest, yet it is 
nore supported by the earth below it. Calculation shows that if we consider the earth o to s to be 
hus div into wedges by any line m t, the wedge that will press most against the wall is that formed 
vheu m t divides the angle otu,or the arc o i, into two equal parts. But see Art 11. 

Since mg is hor, and to o vert, the two form an angle of 90°: consequently the angle of max pres 
s plainly found by taking the angle smg of nat slope from 90°, and div the' rem by 2. Thus a nat 

slope of 114 to 1, or 33° 41', taken from 90°, leaves 56° 18'; and 


56° 18' 
2 


= 28° 9', the cor¬ 
responding: angle o in t of max pres. 

For ease of calculation, only one foot of the length of the wall, and of Its backing, is usually con¬ 
sidered. The number of cub ft of wall, or of backing, is then equal to that of the square feet in 
their respective proflles, or cross-sections. 

Now, according to Moseley, if we assume the particles of earth composing the 
backing to be perfectly dry, and devoid of cohesion, (or tendency to stick to each 
other,) which is very nearly the case in pure sand; and if we suppose the wall to be 
suddenly removed, then the triangle of earth cm.t, comprised between the slope mt 
of max pres, and the vert back c m of the wall, Fig 6, would slide down, under the in¬ 
fluence of a force which may be represented by y P, acting in a direction y P, at right 
angles to the face c to of the triangle of earth ; (or in other words, at right angles 
!to the back of the vert wall,) its center of force being at P, distant % way between 
m and c, measured from the bottom ; and its amount equal to either of the following: 

Perp pres ^ °f the triangle of earth cmtXot 
y P vert depth o m 1 


No 1. 


or 


No 2. 


Perp pres 

yP 


Wt of a single cub , 

ft of the backing ^ • * 


2 . 


See 

1-Art. 11, 
p 692. 


In view of the great uncertainty involved in the matter of the actual pressure of 
earth against retainiug-walls in practice (see Art 2. ), and in order to furnish 

a simple rule which, although entirely unsupported by theory, is still (in the writer’s 
opinion) sufficiently approximate for ordinary practical purposes, we shall assume 
that No 1 of the two foregoing formulas applies near enough to walls with in¬ 
clined backs c in, also, as Figs 7 and 8, (precisely as they are lettered,) at least 
until the back of the wall inclines forward as much as 6 ins 
hor, to 1 foot vert, or at an angle cmo of 26° 34'. What follows on 
retaiuiug-walls will involve this incorrect assumption, and 
must be regarded merely as giving safe approximation. 

Some appear to assume this perp pres to be the only one acting against the back 
of the wall; and hence arrive at erroneous practical conclusions. For when, in 
order to prevent this force from causing the triangle of earth to slide, we place a 
retaining-wall in front of it, then, instead of motion , the force will produce pres of 
the earth against the wall. But in producing pres, it 


















'688 


RETAINING-WALLS. 


A 


necessarily produces the new force of friction , between the pressed surfaces of th( 
earth and wall. That is, if a wall were to begin to overturn around its toe a as « 
fulcrum, its back c vi must of course rise, and in so doing must rub against the irl 
earth filling in contact with it; and this rubbing would evidently act to impede, thi i 
overturning. So long as the wall does not move, the same friction assists in pre¬ 
venting overturning. To ascertain the amount and effect of this friction, let y P, Figklo 
8, represent by scale, the force perp to the back c m; and supposed to have been pre 
vi on sly calculated by the foregoing formula No 1. Make the angle y P/ equal tcjtu 

the angle of wall friction,* draw yf at right angles to 
y P, or parallel to m c; make P x equal to y f, and com 
plete the parallelogram P yfx. Then will x P represent 
by the same scale, the ainount of the friction 
against the hack of the w:tll. Since the fric¬ 
tion acts in the direction of the back cm, (see end of Art 
62, of Force, etc, p 354), it may be considered as 
acting at any point P, in that line : (see Art 18, of Force 
p 314). Hence we have acting at P, two forces ; 

namely, the perp force y P, and the friction x P; conse¬ 
quently, by coinp and res of force, the diag/P of the 
parallelogram P yfx , if measured by the same scale, will on 
give us the amount of their resultant; which is the 
approx single theoretical force, both in 
ainount and in direction, xvhich the wall 
has to resist, including the wall friction. 

But this force,/ P, is also always equal to the perp 
force y P, mult by the nat sec of the angle y P f of 
the wall friction; (or divided by its nat cosine) and of[} 
course may be ascertained tLus; 



FiVS. 


Approx theoreti¬ 
cal pres t P 


wt of triangle v . v nat sec of angle y P f wt of v . 
emt AOi.x 0 / wa n f r i c tion _ c m t * 


vert depth o m 


cos y P f X o m 


Or finally, if it is assumed, as we do throughout, that the earth is perfectly dry (in l [ 
asmuch as its pressure is then the greatest) and that the angles of nat slope,' and j rt 
O' wall friction are then each 33° 41' or 1.5 to 1, then in Figs 6, 7 and 8, if the angle* 19 
c m o between the back c m and the vert o m does not exceed about 26° 34' we may 
assume 


AP,>r °prSfp retiCal = wt of triangle c m t x .643 


which includes the action of the friction of the earth against the back of the wall. 


it 


Rem. 1. When the back of the wall is offsetted or stepped, as 

in Fig 3, instead of being simply battered, as in Figs 7 and 8, the direction of the 
pres of the earth will be the same as if the back had the batter i n, on the principle 
given in Art 34, Fig 17, of Force in Rigid Bodies, p 326. 


Rem. 2. Xow to find both the overturning' tendency of the, 
earth, and the resistance of the wall against being overturned around its toe a as i 
a fulcrum, first find the cen of grav g of the wall (p 348), and through it draw a 
vert line g h. Prolong/P towards rand draw a v perp to it. By any scale mak( ; 
80 = wt of wall, and si = calculated pres/P. Complete the parallelogram sino .!. 
and draw its diagonal sn, which will be the resultant of the pres/P and of the wt 
of the wall; and should for safety be such that aj be not less than about one-fifthi 
of a m, even with best masonry a>ul unyielding soil. Otherwise the great pressure so, 
near the toe a may either fracture the wall or compress the soil near that point 
so that the wall will lean forward. In walls built by our rule, Art 1, or by table, 
p 690, aj will be more than one-fifth of a m. The pres / P if mult by its leverage 
a v will give the moment of the pres about «; and the wt of the wall mult by its 
leverage e a will give that of the wall. The wall is safe from overturning in pro¬ 
portion as its moment, exceeds that of the pres. It is assumed to be safe against 
sliding , breaking, or settling into the soil. See Art 13, p 231. 


be 


* This angle of xvall friction is that at which a plane of masonry must 

inclined to the horizontal so that dry sand or earth would slide down it. It is about the same as 


the nat slope, or 33° 41', or 1.5 to 1; and Its nat secant is 1.202, and its nat cos .832. 


















RETAIN ING-W ALLS. 


689 




r>\ B 




m 


Stem. 4. If the earth slopes downward from C, as 
A or B, instead of being hor as in Figs 6, 7, 8, use the wt of the 
rth c in n instead of c m t , m n being the slope of max pressure. 

A the point of application will still be at P (at one-third of 
c) as in 6, 7, 8; but in B it will be a little higher as explained 
low for Fig 9. 

Surcharged walls are those in which the earth backing 
tends above the tops of the walls. 

According to theory, when as in Fig 9, there is a surcharge 
5 k of backing, sloping away from c at its natural slope c v, 
e max pres against the wall is 
tained when the earth reaches to 
e level of d, where the slope mid 
max pres intersects the face of the 
t slope cv; so that if afterward the 
rth is raised to v, or to any greater 
ight, no additional pres is thereby 
Town against the back of the wall. 

also if the earth slopes from b, or 
>m between c and b, except that 
en the slope m d of max pres must 
tend up to meet this other slope. 

The approximate amount 

the oblique pres, when the wall is 
rcharged, (as in any of the Figs 4, 

9,) may be found on the same prin- 
ale as when the earth is level with 
i top; namely, instead of the trian- 
i cmt of earth, Figs 6, 7, 8, 9 find 
e wt of all the earth d s m t , Fig 4, 
m t r, Fig 5, or cd m, Fig 9 (if the 
rcharge reaches to d or v, or higher), 

tween the slope m d, Fig 9, m t , Figs 4 and 5, of max pres, the back of the wall, and 
e front slope; omitting any which, like den, Fig 5, rests on the top of the wall 
tid thus adds to its stability) when the slope starts in front of c. Having found 
is weight, then for dry backing the 



Pressure \ 
approximately j 


= Wt of the earth X .643, 


eluding the action of the friotiou of the earth against the back of the wall; near 
ough fin the writer’s opinion) for practical purposes in so uncertain a matter; 
it essentially empirical. 

The direction of the pressure thus found will be the same as when the 
rth is level with the top be; namely, as in Figs 6 and 7, first draw a line, as P y, 
rp to the back c m, whether vert or inclined. Then draw another line, as P/, 
aking the angle y P/ = the angle of wall friction, which we all along assume to 
33° 41' or 1.5 to 1. Then P f will give the direction of the pressure. But its 
tint of application will not always be at P (one-third of the height of the wall 
»ove m) as heretofore; for in ail cases it will be at that point P, or at some 
i«-lier one as h, where the back is cut by a line J P or e ft, Fig 9, drawn from the 
nof grav of the sustained earth (omitting any that rests immediately on the top 
and parallel to the slope md of max pres; and such a line will strike at one- 
ird the height of the wall only when the sustained earth tem or d cm forms a 
[>mplete~triaugle, one of whose angles is at the inner top edge c of the wall, 
i all other cases said line for a surcharge will strike above P. 

45 


it 

















690 


RETAINING-WALLS. 


Art. 7. Oil page 683, Fig 1, we recommend that the base o s at the groun 
line of well built vertical walls should not be less than .35, or .4, or .6 of t|j|( 
height d s above said line, depending on the kind of masonry. But a wall with 
battered (inclined) front or face as found by Art 8, (by which the followi 0 t 
table was prepared), will be as strong, and at the same time contain less mason ,|i 
than a vert wall, although the battered one will have the thickest base os. 

Table 3, of thicknesses at base o s, Fig 1 1, an<l at top e d % * 11 
walls with battered faces, so as to be as strong as vertica 1 
ones which contain more masonry. 

For the cub yds of masonry above © s per foot run of wall, mult t 

square of the vert height ds by the number in the column of cub yds. Th< 
add the foundation masonry below o s. See also Table, p 693. Also study Rems 
and 2, Art 8. 

(Original.) 


w 


All the walls below have the same strength 
as a vert one whose base os, fig 1=.35 
of its ht ds. 

All the walls below have the 
same strength as a vert oue 
whose base o s, fig l=r.4 
of its ht d s. 

All the walls below have ti) 
same strength as a vik 
oue whose base os, fig IP 
.5 of its ht ds. 


Cut stone. 

Mortar rubble. 

Dry rubble. 

Batter, in 
ins to a ft. 

Base, in 
pts of 
ht. 

Top, in 
pts of 
ht. 

C yds per 
ft run. 

Base, in 
pts of 
ht. 

Top, in 
pts of 
ht. 

C yds per 
ft run. 

Base, in 
pts of 
ht. 

Top, in 
pts of 
ht. 

C yds f 
ft run 

1 

0 

.350 

.350 

.01296 

.400 

.400 

.01482 

.500 

.500 

.0185:, 

y 2 

.352 

.310 

.01226 

.401 

.359 

.01407 

.501 

.459 

.0177 i L 

1 

.355 

.270 

.01158 

.403 

.320 

.01339 

.503 

.420 

.0170! | 

1 M 

.359 

.234 

.01098 

.408 

.283 

.01280 

.506 

.381 

.0164; 

2 

.364 

.197 

.01039 

.413 

.246 

.01220 

.510 

.343 

.01581 ,■ 

2 H 

.371 

.163 

.00989 

.419 

.210 

.01165 

.516 

.308 

.01521 f 

3 

.379 

.129 

.00941 

.425 

.175 

.01111 

.522 

.272 

.0147< ; 

3 M 

.389 

.096 

.00898 

.435 

.143 

.01070 

.528 

.236 

.0141* , 

4 

.400 

.066 

.00863 

.445 

.110 

.01028 

.537 

.204 

.01371 ii 

5 

.425 

.007 

.00800 

.468 

.051 

.00961 

.555 

.138 

.0128; 

Triangle 

.429 

.000 

.00794 

.490 

.000 

.00907 

.612 

.000 

.0113;; 


in 


* 


Moseley and others quote Gadroy, for a dry sand sloping at 21°. It would be better to cease fro 
circulating such evident mistakes. Dry saud will stand at no less angle for a savant than for an 
body else. For practical purposes, we may say that dry sand, gravel and earths, slope at 33° 41' 

1 'A to 1; as abundant experience on railroad embkts proves. Poncelet gives tables for walls to su 
port dry earth sloping at 1 to 1, or 45°; but as we do not believe in the existence of such earth, i 
omit such tables. Sand, gravel, and earths may be moistened to diff degrees, so as to stand at ai 
angle between hor and vert; and by moistening and ramming, the earths may be converted intocoi 
pact masses, exerting little or no pres; and may even so continue after they become dry ; being the 
in fact, a kind of air-dried brick. It is sometimes diflicult to know whether earth or sand is perfect 
dry or not; and an exceedingly small degree of moisture will cause them to stand at 1 to 1, in snu 
heaps, such as have probably been observed by the authorities on the subject. The writer found th 
fine sand from the sea shore, and under cover, would stand at lt^ to 1 during warm dry weather, at 
at 1 to 1 when the air was damp. Yet no diff whatever in its degree of moisture was'perceptible (' 
the feeling. Its susceptibility to dampness was of course owing to salt. A few handfuls pf drv ear ! 
may perhaps be coquetted into standing at 1 to 1 on a table; but so far as our observation extern^ 1 
when it is dumped in large quantities from carts and wheelbarrows, its slope is about 1U to 1 • ai » 
this we consider the proper oue to be used in practical calculations, where safety is the consider’ati 
of paramount importance. 


. ! es * *, he * ,a * s *°I ,e ' greater Is the pres ; and since tl 
,P e 18 least when the hacking is perfectly dry, (omitting of course its conditic 

Wlion SO Jlhsnl Ilfol V !/»*/ oa _11* u ’ ? i \ a - HO vinmiut 


confined mrelnhlll 1 °^ 1° l ,’ e . Como P artiaI, V fl'iid.) we have, on the scorS of safet 
^ t0 d M y bac V n S- As stated in Art 1, we cannot recommend dime 
sions less than those there given, when we consider the rough treatment to whit 
masonry is exposed on public works. 

In carrying n road along dangerous nreeiniees we nbnn 

the ce r nt > rif. t , e 0 .Tf ed r tlm £* to " lakf> wall8 ‘ w «^niagiim. for instance, th 

d. fU v» al { °u Ce ,°J a b f avy tra,n - whirling around a sharp curve, convex on tl 

langerous side, should not be overlooked in designing walls for such localities. Th 
force is hor; and is applied near the top of the wall; and, consequently its lovera- 
may be considered as equal to the height; whereas the theoretical pres of the ear; 
is oblique; and is applied at % of the height from the bottom; so that its levera> 
about the toe of the wall is very short. Moreover, the simple v'riqht of the train nr 
duces pres against the wall; as well as that of the backing. All such considerate 
are omitted by theorists. The dangerous pres caused by tremors, &c , cannot 1 
































RETAINING-WALLS. 


691 


.« 




43 


1 


11111 snmed to be applied at % °f the height from the bottom; nor indeed, can it be 
i 1 lculated at all. 

7 Rem. 2. Wharf walls are an instance where the thickness should be increased, 
'"■Utwithstanding that the pres of the water in front helps to sustain them. The earth 
J|lr hind such walls, is not only liable to he very heavily loaded when vessels are dis- 
arging; hut is apt to become saturated with water, especially below low-water 
vel; and thus to exert a very great pres against the walls. Moreover, the water 
cuts under the wall; and by its upward pressure virtually reduces its weight, and 
nsequently its stability. The same cause of course diminishes the friction of the 
til upon its base. Such walls are, therefore, very liable, to slide, if the foundation 
smooth, and horizontal; and have done so even when the foundation had a con- 
lerable inclination backward, as in Fig 1. See Art 9. 

iEM. 3. A retaiuing-wall is usually in greater danger for a few months after its completion, than 
er time has been allowed for the mortar to harden perfectly ; and for the backing to settle. When 
•re are suspicions of the safety of a new wall, it would be well to place strong temporary shores 
linst it, at about % to J4 of its height above groutid. In some cases, permanent buttresses of 
isonry may be built for the purpose. They should be well bonded into the wall. 
ediEM.’-i. The pres of the earth backing will be much reduced, if the first few feet of its height be 
vj ide up in thin hor layers, to be consolidated by being used by the masons instead of scaffolding; as 
‘jl: jwn at h. Fig 1. Frequently this can be doue without inconvenience; and at very trifling cost. 

Art. 8. To change a vert retaining'-wall. Into one with a 
itterert faee, which shall present an equal resistance 
gainst overturning; although requiring: less masonry. 

tis is sometimes termed a transformation of prolile. (Original.) 

Let aboi , Fig 10, be the vert wall. Mult its base 
', by 1.225 ; (1.22475 is nearer;) the prod will be the 
tse o e, of a triangular wall /> o e , possessing the 
me stability; and yet not requiring much more 
>an half the masonry of the vert one. See Rem 1. 

:iis being done, suppose a wall to be desired with a 
ce batter, of say 3 ins to a ft; or 1 in 4. From the 
>int n, where the face of the triangular wall inter- 
cts that of the vert one, step olf vert any 4 short 
[ual spaces ; and from the upper obe m, step oft one 
i!5 ace hor, to v. Through v and n draw the dotted 
Sae s t, which evidently will hatter 1 in 4. Then is 
s t o approximately the reqd wall; hut a little 
icker than necessary. To reduce it, from t draw 
“tie dotted line t b. Mark the point c, where this 
! ™ie intersects the face a i, of the vert wall; and 
7 , trough c draw d 1, parallel to s l. Then is b dl o 
4e reqd wall. Our fig is drawn in an exaggerated 
anner, so as to avoid confusion in the lines. Hie 
“Jtse o e of the triangular wall, would not in reality 
near so great as it is represented. 

'»(! 

'*4 Rem. 1 . The battered wall will in fact be safer than the vert 

me. The battered wall has the same moment of stability as the vert one ; and the 



\. Fio/0 


It will be*observed that as the base increases, the quantity of masonry diminishes. 



arth filliug, which would be at one-third of a n above n.) 
n d s to overturn the wall around its toe to, is the dist to «, 
1 easured from the toe or fulcrum to, and at right angles to 


ie direction f o s c of the pres ; and this leverage mult by 

*. . , . . _ _._4 .1 n r, /. .f <M. ... r. m on f Af 


ie force / o. gives the overturning tendency or moment of 
aid force. See <• Moments and leverage." Aga|n, 

t a ny. represent a triangular wall of the same stability 
s the other, as found by our rule. Here we still have the 
ien ime amount fo, and direction f o s c. of pres force against 
ie wall; but it now acts to overturn the wall any 
round the toe y ; and therefore, with the reduced leverage 
C Consequently, its overturning tendency is less than 
efore. Therefore, in ordinary language, we may say that 
ie wall is stronger than before, although its moment of 
ability, or standing tendency, has iu itself undergone no 
bange. ir the pres fo against the vert back were hor, as 
i the case of water, then its leverage would evidently be 
tie same in both walls; and the proportion between the 
verturuing moment of the pres, aud the moments of 
tabilitv of the two walls, would be constant. P 229. 

Rem 2 Tn attempting to reduce the masonry by adopt- 
ig a wall o b e. Fig 10. of a triangular section : or of one 
early approaching a triangle, special attention should be 
jven to the quality of the masonry near the thin toe e; 
'hich will otherwise be apt to crack, or fail under the prea. 
















692 


RETAINING-WALLS. 


Rem 3. 


Moreover, when common mortar is used without an admixture op cement, which it ne 
should he, in reiaimug-walls,where durability is au object, a great batter is obj 
tiouable; inasmuch as the rain, combined with frost, &c, soon destroys the ni 
tar. In such cases, therefore, the baiter should uotexceed 1 origins to a ft;s 
even then, at least the pointing of the joints, aud a few feet iu height of b> 
the upper and the lower courses of masonry, should be doue with cement, 
cement-mortar. We have observed a most marked di If iu the Corrosion of the in - 
tar, where, iu the same walls, with the same exposure, one poriiou has been bt 
with a vert face; aud another with a batter of but 1J-6 inch to a loot- Coium 
mortar will uever set properly, and continue iirm, wheu it is exposed to m< 
tore from the earth. This is very observable near the tops and bottoms 
abuts, retaining-walls, <fcc; the lime-mortar at those parts will generally 
fouud to be rendered eutirely worthless. A profile somewhat like Fig 12, hi 
at times prove serviceable, instead of the triangular. This is the form of l 
Gothic buttress ; which probably had its origin in the cause just spoken of. 

Art. 9. A retain) ng-w'all may slide, witlioi 
losing its verticality ; and, indeed, without any dang 
of being overturned. This is very apt to occur if it is built upi 
a lior wooden platform; or upon a level surf of rock, or cla 
without other means than mere friction to prevent sliding. This may be obviaf 



Flo 12- 


by inclining the base, as in Fig 1; by founding the wall at such a depth as to pr 
Vide a proper resistance from the soil iu front; or in case of a platform, by securii 
one or more lines of strong beams to its upper surf, across the direction in whit 
sliding would take place. On wet clay, friction may be as low as from .2 
l / A the weight of tne wall; on dry earth, it is about ^4 to ^; and on sand or grave 

about % to %. The friction of masonry on a wood< 
platform, is about ^ of the wt, if dry; and % if wt 

{'on Jlterforts, shown in plan at c c c, Fig 13, consist 

an increase of the thicknessof the wall, at iu back, at regular inti 
villa of its length. We couceive them to be but little better that 
waste of masonry. When a wall of this kind fails, it almost i 
variably separates from its counterforts: to which it is connect 
merely by the adhesion of the mortar; and to a slight extent, by t 




Ficj 13 


bonding of the masonry. The table in Art 7 shows that a very small addition to the base of a wall, 
attended by a great iucrease of its strength; we therefore think that the masoury of counterfoi 
would be much better, and more oheaply employed in giving the wall an additional thickness, alo 
its entire length: and for the lower third of its height. Counterforts are very generally used 
retainingwalls by F.uropean engineers; but rarely, if ever, by Americans. 

lillt tresses are like counterforts, except that they are placed in front of a wall instead of V 
hind it; aud that their profile is generally triangular, or nearly so. They greatly increase its streugt 
but. being unsightly, are seldom used, except as a remedy when a wall is seeu to be failing. 

I.aild*t ies, or long rods of iron, have been employed as a makeshift for upholding weak r 
taiuing-walls. Extending through the wall from its face, the land ends are connected with ancho 1 
of masonry, cast-iron or wooden posts; the whole being at some dist below the surface. 

Retaining- wall* witll curved protile* are mentioned here merely to ca , 
tion the young engineer against building them. Although sanctioned by the practice of some hit 
authorities, they really possess no merit sufficient to compensate for the additional expense and tro 1 
ble of their construction. 

Art. 10. Among military men, a retaining-wall is called a revetment. When tl 
earth is level with the top, a scarp revetment; when above it. a connterscar 
revetment, or a Uemi-revetmcnt. Wheu the face of the wall is battered, a slopiuy; and wheu the bai I 
is battered, a counter sloping revetment. The batter is called the tit In*. 


Art. 11. The pres against a wall Fig 6, from sand etc level with its top, is m 
diminished by reducing the quantity of sand, until its top width cs becomes less tha 


that (cl) pertaining to the angle c m t of maximum pres. The pres then begins to d 
minish, but in practice the diminution isnot appreciable until the width is reduced to ahoi 


one sixth of that (c s) pertaining to the angle cm s of natural sloj 
c t. The pres then begins to decrease rapidly as the w idth is iuitl 


>e, or about half o 


i llier reduced. 


Table 4, of contents in cub yard* for each foot in lengti 
of retaining*wall*, with a thickness at base equal to .4 of the vert heigh , 
it tlie hack is vert. If the hack is stepped according to the rule in Art 3, p CSS^tl 
proportionate thickness at base will of course lie increased. Face batter. iy in’ch< ' 
to a foot; or %th of the height. Back either vert, or stepped according to the ru! f*' 
in Art 3, Fig 3. The strength is very nearly equal to iliat of avert wall with a 
base of .4 its height. See table, p «90. Experience lias proved that such wall 
when composed of well-scabbled mortar rubble, are safe under all ordinary circun 
stances for earth level with the top. Steps or offsets, o e, at foot, Fig 1, are not hei » 
included. 


I 









STONE BRIDGES 


693 


' no 

' ahi. 

liq 


TABEE 4. (Original.) 


Utji 

■Cbo 

a *Hi 

'atflifi 

j 

Ht. 

Ft. 

Cub. 

Yds. 

Ht. 

Ft. 

Cub. 

Yds. 

Ht. 

Ft. 

Cub. 

Yds. 

Ht. 

Ft. 

Cub. 

Yds. 

Ht. 

Ft. 

Cub. 

Yds. 

Ht. 

Ft. 

Cub. 

Yds. 

Still, 

1 

.018 

10)4 

1.38 

20 

5.00 

2934 

10.9 

48 

28.8 

74 

68.5 

m 

hi 

.028 

11 

1.51 

hi 

5.25 

30 

11.3 

49 

30.0 

76 

72.2 

■'BIO 

2 

.050 

hi 

1.65 

21 

5.51 

31 

12.0 

50 

31.3 

78 

76 1 

11* 

A 

.078 

12 

1.80 

hi 

5.78 

32 

12.8 

51 

32 5 

80 

80.0 

If 

3 

.lKt 

hi 

1.95 

22 

6.05 

33 

13.6 

5*2 

33.8 

82 

84.1 

. nil 

A 

.158 

13 

2.11 

X 

6.33 

34 

14.5 

53 

35.1 

84 

88.4 

f it 

4 

.200 

hi 

,2.28 

23 

6.61 

35 

15.3 

54 

36 5 

86 

92.5 

u7. 

A 

.258 

14 

2.45 

34 

6.90 

36 

16.2 

55 

37.8 

88 

96.8 

on 

5 

.318 

\A 

/'I 

2.63 

24 

7.20 

37 

17.1 

56 

39.2 

90 

101.3 

H 

.378 

15 

2.81 

hi 

7.50 

38 

18.1 

57 

40.6 

92 

105.8 

#gi 

6 

.450 

hi 

3.00 

25 

7.81 

39 

19 0 

58 

42.1 

94 

110.5 

ipo 

fri 

.528 

16 

3.20 

hi 

8.13 

40 

20.0 

59 

43.5 

96 

115.2 

'kj 


.613 

hi 

3.40 

26 

8.45 

41 

21.0 

60 

45.0 

98 

120.1 


hi 

.703 

17 

3.61 

X 

8.78 

42 

22.1 

62 

48.1 

100 

125.0 


8 

.800 

hi 

3.83 

27 

9.12 

43 

23.1 

64 

51.2 

102 

130.1 

pn 

hi 

.903 

18 

4.05 

hi 

9 45 

44 

24.2 

66 

54.5 

104 

135.2 

rin 


1.01 

hi 

4.28 

28 

9.80 

45 

25.3 

68 

57.8 

106 

140.5 

lie 

X 

1.13 

19 

4.51 

34 

10.2 

46 

26.5 

70 

61.3 



ll 

ive 

10 

1.25 

hi 

4.75 


10.5 

47 

27.6 

72 

64 8 




se 

iti 
a«s 
an 
’tii 
» 
f U 

.a, 

for 


STONE BRIDGES. 


Art. 1. In an arch sts , Fig 1, the dist eo is called its span; ia its rise; t its 
rown ; its lower boundary line, eao, its soffit, or intrados ; the upper one, 
5 r , its back, or extraffos. The terms soffit and hack are also applied to the 
itire lower and upper curved surfaces of the whole arch. The ends of an arch, or 
iIm ie showing areas comprised between its intrados and extrados, are its faces ; thus 
le area sts a is a face. The inclined surfaces or joints, re, rn , upon which the feet 
the arch rest, or from which the arch springs, are the skcwbacks* Lines 
vel with e and o, at right angles to the faces of the arch, and forming the lower 
Iges of its feet, (see nn, Fig 2^,) are the springin;;' lines, or springs. The 
dlocks of which the arch itself is composed, are the arch-stones, or vonssoirs. 

• lie center one, ta, is the keystone; and the lowest ones, ss, the springers, 
he term archblcctc might he substituted for voussoir, and like it would apply to 
wrick or other material, as well as to stone. The parts tr,tr, are the haunches ; 
Ml id the spaces tri, tr b, above these, are the spiandrcls. The material deposited 
■™ i these spates is the spandrel tilling ; it is sometimes earth, sometimes ma- 
ti mry; or partly of each, as in Fig 1. 

h In large arches, it often consists of several parallel spandrel-walls. II, Fig V&, running lengthwise 
the roadway, or astraddle of the arch. They are covered at top either by small arches from wall to 
all, or by flat stones, for supporting the material of the roadway. They are also at times connected 
getlier by vert cross-walls at intervals, for steadying them laterally, as at tt, Fig 2The parts 
DO p e n, gpon, Fig 1, are the abutments of the arch ; en, on, the faces ; gp, gp, the backs ; and 
HU n, p n', the bases of the abuts. The bases are usually widened by feet , steps, or offsets, d d, for dis- 
].• ibuting the wt of the bridge over a greater area of foundation ; thus diminishing the danger of set- 
“ 3 ment. The distance t a in any arch-stone, is called its depth. 

311 

,,j The only arches in common t 1) 

Jsc for bridges, are the circular, 


'>ften called segmental); and 
le elliptic. 

Art. 2. To find tlie 
Icpth of keystone for 
irst-class cut-stone 
trclies, whether cir- 
nlar or elliptic.* 

Find the rad co, Fig 1, which 
.’ill touch the arch at o, a, and 
Add together this rad, and 
alf the span o e. Take the sq 
t of the sum. Div this sq rt 
y 4. To the quot add ^ of a 
t. Or by formula, 



* Inasmuch as the rules which we give for arches and abuts are entirely original and novel, it may 
ot be amiss to state that they are Dot altogether empirical; but are based upon accurate drawings 



































































694 


STONE BRIDGES. 


_ IT 

Depth of key _ V Rad + half span , 9 * f 

infect -4 + ' J j" 


For second-class work, this depth may be increased about J^th part; o 
for brick or fair rubble, about %rd. See table of Keystones, p 697. 

In large arches it is advisable to increase the depth of the archstones toward th 
springs ; but when the span is as small as about (50 to 80 or 100 feet, this is not at al 
necessary if the stone is good; although the arch will be stronger if it is done. Ii 
practice this increase, even in the largest spans, does not exceed from % to }/ 2 th 
depth of the key; although theory would require much more in arches of great l ist 


mi; 

r 


5 


Rem. To find the rad c o, whether the arch be circular or elliptic. Squai > 
half the span e, o. Square the whole rise i a. Add these squares together; div th - 
sum by twice the rise i a. Or it may be found near enough for this purpose by th 
dividers, from a small arch drawn to a scale. 



Amount of pressure sustained by archstones. In bridges oil' 
the same width of roadway ; if all the other parts bore to each other the same propor 
tion as the spans, the total pres would increase as the squares of the spans, while th* _ 
pressure per square foot would increase as the spans. Rut in practice the depth of th< 
archstones increases much less rapidly than the span; while the thickness of th* 
roadway material, and the extraneous load per sq ft, remain the same for all spans 
Hence the total pressures, at key and at spring, increase less rapidly than the square 
of the si>ans ; but more rapidly than the simple spans; as do also the pressures pe\ 
square foot. Thus in two bridges of the same width, but with spans of 100 and 200 ft 
with depths of archstones taken from our table page 607, and uniform from key t* 
spring; supposed to be filled up solid with masonry of 160 lbs per cub It. to a level ol 
about 15 inches above the crown, (including the stone paving of the roadway); witl 
an extraneous load of 100 lbs per sq ft; the pressures will be approximately as fol 
lows: 



Span lOO ft. 



Span 200 ft. 



AT KEY. 


AT SPRING. 

AT KEY. 

AT SPRING. 


For 1 ft in 


For 1 ft in 


For 1 ft in 


For 1 ft in 


width of 
its eutire 

Per sq ft. 

width of 
its entire 

Per sq ft. 

width of 
its entire 

Per sq ft. 

width of 
its entire 

Per sq ft 


depth. 


depth. 


depth. 


depth. 


Rise. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

Tons. 

k 

4‘2 k 

13?* 

58 

18** 

126 

29k 

179 

42 

1 

J 

36k 

12k 

57 

19 

112 

27k 

181 

44 

y* 

31k 

11 

57 H 

20 

97 

24k 

188 

47k 

k 

25 

o : 

61k 

22k 

80?* 

21 

207 

54>* 

k 

18 

6?* 

61 y 

*25 

57 k 

15k 

230 

61k 


It will be seen that with the same span, the pres at the key becomes less, while tha 
at the spring becomes greater, as the rise increases. Also that when the archstone 
are of uniform depth, the pres at either spring of a semicircular arch is about 4 time* 
as great as at the key ; whereas when the rise is but one-sixth of the span, the pres ai 
spring averages but about one-third greater than at the key. These proportions varj 
somewhat in different spans. 

The greater pres per sq ft at the springs may be reduced by increasing the depth of 
the archstones towards the springs. This however is not necessary in moderate spans 
inasmuch as good stone will be safe even under this greater pres. 

By using parallel spandrel n ails, see Fig 2^, p 698, or by partly fill 
irig with earth instead of masonry, the pres on the archstones may be diminished 
say, as a rough average, about £ part. 


and calculations made by the writer, of lines of pres. &c. of arches from 1 to 300 ft span, and of everj 
rise, from a semicircle to T * 5 of the span. Prom these drawings he endeavored to find proportion! 
which, although they might not endure the test of strict criticism, would still apply to all the case* 
with an accuracy sufficient for ordiuary practical purposes. 


































STONE BRIDGES 


695 


Table 1. Of some existing; arches, with both their actual and their 
, lculated depths (by our rule) of keystone. Where two depths are given in the column of keys the 
tallest is for first class cut-stone, aud the largest for good rubble, or brick. Those also which are 
t specified are of first-class cut-stone. C stands for circular, E for elliptic. For 2d class work, add 
; o out ^th part; aud for brick, or fair rubble, about }4th. 




































































696 


8TONE BRIDGES. 


The arch on the Botthbonkais Railway, is probably the boldest;* aDd the Cabin John arch, 1 
Capt, now Gen’l M. C. Meigs, U S Army, the grandest stone one in existence. Pwt y Pkvdd, i 
Wales, is a common road bridge, of very rude construction; with a dangerously steep roadway, 
was built entirely of rubble, in mortar, by a common country mason, in 1150: and is still in perfe 
condition. Only the outer, or showing arch-stones, are 2.5 ft deep; and that depth is made up of tv 
stones. The inner arcli-stoncs are but 1.5 ft deep ; and but from 6 to It inches thick. The stone qua 
ried with tolerably fair natural beds; and received little or no dressing in addition. The bridge is 
line example of that ignorance which often passes for boldness. Pont Napoleon carries a raiiros 1 
across the Seine at Paris. The arches are of the uniform depth of 4 ft, from crown to spring. Tin I 
are composed chiefly of small rough quarry chips, or spatrls; well washed, to free them front di 
and dust; and then thoroughly bedded in good cement; and grouted with the same. It is in fact a 
arch of cement-concrete. The Font i>b Alma, near it, and built in the same way, has elliptic a relit 
of from 126 to 141 ft span ; with rises of I the span. Key 4.9 ft. These two bridges, considering tl 
want of precedent in this kiud of construction, on so large a scale, most be regarded as very bold 
and ns reflecting the highest credit for practical science, upon their engineers, Darcel and Coucht 
Some trouble arose from the unequal contraction of the different thicknesses of cement. They sho' 
what may be readily accomplished in arches of moderate spaus, by means of small stone, and goo 
hydraulic cement when large stone fit for arches is not procurable. In Pont Napoleon the depth o 
arch is fes3 than our rule gives for second class cut-stone. 

Rem. Our engineers are usually too sparing 1 of cement. Itshoali 

be freely used, not only in the arches themselves, and in the masonry above them, as i 
protection from rain-soutane; but in abuts, wing-walls, retaining-walls, and all othe 
important masonry exposed to dampness. The entire backs of important brick arche 
should be covered with a layer of good cement, about an inch thick. The want of i 
otn be seen throughout most of our public works. The common mortar will l* 
found to be decayed, and falling down from the soffits of arches; and from the joint 
of masonry generally, within from 3 to 6 ft of the surface of the ground. The mois 
ture rises by capillary attraction, to that dist above the surf of the nat soil; oi 
descends to it from the artificial surf of embankments, &c ; therefore, cement-mortal 
should be employed in those portions at least. The mortar in the faces of Uatterec 
walls, even when the butter is but 1 to 1% inches per foot, is far more injured by rail 
and exposure, than in vert ones; and should therefore be of the best quality. Set 
Mortar, &c. 

We have, however, seen a quite free percolation of surface water through brick 
arches of nearly 3 ft in depth, even when cement was freely used. In aqueduct 
bridges, we believe that cement lias not been found to prevent leaks, whether thf 
arches were of brick, or even of cut-stone. May not this be the effect of cracks 
produced by settlement of the arch; or by contraction and expinsion under atmos¬ 
pheric influence? Cement at any rate prevents the joints from crumbling. 

Art. 3. Tlie keystones for large elliptic arctics by the best en 
gineers, are generally made about ]/ A part deeper than our rule requires ; or than is 
considered necessary for circular ones of the same span and rise; in order to keep the 
line of pres well within the joints; although the elliptic arch,with its spandrel filling. 


1 


k? 



n 


Figli 


has slightly less wt; and that wt has 
a trifle less leverage than in a circular 
one; and consequently it exerts less 
pres both at the key, and at the skew- 
back. See London, Gloucester, and 
Waterloo bridges, in the preceding 
table. 


Rem. Voting engineers are apt to affect shallow arch-stones; hot it would be far better to adopt 
the opposite course: for not only do deep oues make a more stable structure, but a thin arch is as 
unsightly an object as too slender a column. According to onr own taste, arch-stones fullv deeper 
than qur rule gives for first-class cut-stone. are greatly to be preferred when appearance is consulted 
Especially when an arch is of rough rubble, which costs atx>ut the same whether it is built up a 
arch, or as spandrel filling, it is mere folly to make the arches shallow. Stability and durability 
ahoold be the objects aimed at; and when they can be attained even to excess, without increased cost, 
it is best to do so. 


* Built like that at Soupes in the preceding table. 

















STONE BRIDGES. 


697 


a 

u 
. i 
* 

H; 

5 

Table 2. Depths of keystones for arches of first-class cut stone, 
>y Art 2. For second class add full one-eighth part; and for superior brick one- 
ourth to one-third part, if the span exceeds about 15 or 20 ft. Original. 

Span. 

Feet. 

1 

2 

Kit 

1 

3 

ie, in parts 

l 

? 

of the spar 

1 

L. 

1 

6 

1 

8 

1 

TO 

Jin 

t|| 


Key. Ft. 

Key. Ft. 

Key. Ft. 

Key. Ft. 

Key. Ft. 

Key. Ft. 

Key. Ft. 

fliss 

2 

.55 

.56 

.58 

.60 

.61 

.64 

.68 


4 

.70 

.72 

.74 

.76 

.79 

.83 

• 88 


6 

.81 

.83 

.86 

.89 

.92 

.97 

1.03 

•>M 

8 

.91 

.93 

.96 

1.00 

1.03 

1.09 

1.16 

che. 

10 

.99 

1.01 

1.04 

1.07 

1.11 

1.18 

1.26 

Qtv 

15 

1.17 

1.19 

1.22 

1.26 

1.30 

1.40 

1.50 

mi 

20 

1.32 

1.35 

1.38 

1.43 

1.48 

1.59 

1.70 

101 

25 

1.45 

1.48 

1.53 

1.58 

1.64 

1.76 

1.88 


30 

1.57 

1.60 

1.65 

1.71 

1.78 

1.91 

2.04 

914 

35 

1.68 

1.70 

1.76 

1.83 

1.90 

2.04 

2 19 

40 

1.78 

1.81 

1.88 

1.95 

2.03 

2.18 

2.33 

«a 

50 

1.97 

2 00 

2.08 

2.16 

2.25 

2.41 

2.58 

aer 

60 

2.14 

2.18 

2.26 

2.35 

2.44 

2.62 

2.80 

ae 

80 

2.44 

2.49 

2.58 

2.68 

2.78 

2 98 

3.18 

fit 

100 

2.70 

2.75 

2.86 

2 97 

3.09 

3.32 

3.55 


120 

2.94 

2.99 

3.10 

3.22 

3.35 

3 61 

3.88 

1)8 

140 

3.16 

3.21 

3.33 

3.46 

3.60 

3.87 

4.15 

Qtg 

160 

3.36 

3.44 

3.58 

3.72 

3.87 

4.17 


V* 

180 

3.56 

3.63 

3.75 

3.90 

4.06 

4.38 



200 

3.74 

3.81 

3.95 

4.12 

4.29 




220 

3.91 

4.00 

4.13 

4.30 

4.48 



tar 

240 

4.07 

4.15 

4.30 

4.48 





260 

4.23 

4.31 

4.47 

4.66 




iin 

280 

4.38 

4.46 

4.63 





lee 

300 

4.53 

4.62 

4.80 




* 


ek 

ret 

t| 

h 

» 

i» 

lie 

t 

a> 

nr 

s 

w- 

ei 

“J 

x 

i*r 

4. 

ft 

Hit; 

»’> 


Art. 4. To proportion tlie abuts for an arch of stone or 
>rick, whether eirenlar or elliptic. (Original.) 

The writer ventures to offer the following rule, in the belief that it will be found 
o combine the requirements of theory with those of economy and ease of applica- 
ion, to perhaps as great an extent as is attainable in an endeavor to reduce so com- 
ilicated a subject, to a simple and reliable working' rule for prac- 
ical bridge-builders. This is all that he claims for it. Notwithstanding its 
implicity, it is the result of much labor on his part. It applies equally to the smallest 
ulvert, and to the largest bridge; whatever may be the proportions of span and rise; 
,nd to any height of abut whatever. It applies also to all the usual methods of filling 
hove the arch; whether with solid masonry to the level vf. Fig 2, of the top of the 
rch ; or entirely with earth ; or partly with each, as represented in the fig: or with 
•arallel spandrel-walls extending to the back of the abut, as in Fig 2%. Although 
he stability of an abut cannot remain precisely the same under all these conditions, 
et the diff of thickness which would follow from a strict investigation of each par- 
icular case, is not sufficient to warrant us in embarrassing a rule intended for popu- 
ar use. by a multitude of exceptions and modifications which would defeat the very 
bject for which it was designed. We shall not touch upon the theory of arches, 
xcept in the way of incidental allusion to it. Theories for arches, and their abuts, 
•mit all consideration of passing loads; and consequently are entirely inapplicable 
n practice when, as is frequently the case, (especially in railroad bridges of moderate 
pans,) the load bears a iarge ratio to the wt of the arch itself. Hence the theoretical 
ine of thrust has no place in such cases. Our rule is intended for common practice: 
,nd we conceive that no error of practical importance will attend its application to 
ny case whatever; whether the arch be circular or elliptic. 

it gi ves a thickness of abut, which, without any backing: 
>1'earth behind it. is safe in itself, and in all cases, against 
he pres, when the bridge is unloaded. Moreover, in very large arches, 
n which the greatest load likely to come upon them in practice is small in comparison 
vith the wt of the arch itself, and the filling above it, our abuts would also be safe 
rom the loaded bridge, without any dependence upon the earth behind them ; but 
s the arches become less, and consequently the wt of the load becomes greater in 
•roportion to that of the arch, and of the filling above it, we must depend more and 
nore upon the resistance of the earth behind the abuts, in order to avoid the neces- 
ity of giving the latter an extravagant thickness. It will therefore be understood 
iroughout that , except when parallel spandrel walls are. used, our rules suppose that 
fter the bridge, is finished, earth wdl be deposited behind the abuts, and to the height 
" the roadway, as usual. 


























698 


STONE BRIDGES. 


lu small bridges and large culverts of first class railroads, subject to the jarring f 
of heavy trains at high speeds, the comparative cheapness with which an excess of „ 
strength can thus be given to important structures, lias led, in many cases, to the r 
use of abutments from one-fourth to one-half thicker tkau by the following rule. ,j 
JEf of rolls'll rubble add 6 ins to insure full thickness in every part. 

Thieles on of abut at spring 
in ft, when the height o s 
does not exceed 1 % times the 
base s p 

Mark the points n and y thus ascertained. Next, from the center i, of the span d 
or chord eo, lay off t h, equal to -fj part of the span. Join a A; and through n, and 1 
parallel to a A, draw the indefinite line gnp of the abut. Do the same with the f 
other abut. Make y m and ng each equal to half the entire height i t of the arch; 
and from g draw a straight line gx, touching the back of the arch as high up as pos¬ 
sible ; or still letter, as shown at tm, with a rad dt or dm, (to be found by trial,) ! 
describe an arc t m. Then gx or t m will be the top of the masonry filling above the '[ 
arch;* and this should t>e completed before striking the centers; before which, 
also, the embkt should be finished, at least up to y n. 


Rad in ft rise in ft . 

- 6 -+ — Tt~ + 2/L 



Now find by trial the point s, Fig 2, at which the thickness sp is equal to two- 



* Except when 
1 


THE RISE IS 


BUT ABOUT ^ OP THE SPAN, OR 

less ; in which case carry the 
masonry up solid to the level 
vtf, of the top of the arch. Or 
if the arch is a large one, ex¬ 
ceeding say about 60 ft span ; 
and especially if its rise is 
greater than about i of its 
span, it is better to economize 
masonry by the use of parallel 
interior spandrel-walls, 11, Fig 
'P4, carried up to vtf. Fig 2. 
Indeed, such interior walls may 
often he advantageously intro¬ 
duced in much smaller arches. 
When high, they are steadied 
by occasional cross-walls, as 11, 
Fig 2^. Their feet should he 
spread by offsets, as shown at 000 , so as to bear upon the whole snrf of the hack of the arch : thus 
equalizing the pres upon it. On top of the walls flagstones may be laid, or small arches may he 
turned from wall to wall, for supporting the ballast, &c, of the roadway. The spaces below are left 
hollow. In Fig 2*sj the dark part ui ic is supposed to be a section across an abutment; but omitting 
the second cross-wall, similar to < t. In R R bridges, put a spandrel-wall, 11 , under each rail. 

















































































STONE BRIDGES. 


699 


hirds of the corresponding vert height os, and draw sp. Then will the thickness 
>n or ey be that at the springing line of the given circular or elliptic arch of any 
■i8e and span ; and the line gp will be the back of the abut; provided its height os 
Joes not exceed times sp; or in other words, provided sp is not less than % of 
>s. In practice, o x will rarely exceed this limit; and only in arches of considerable 
•ise. But if it should, as for instance at oq, then make the base qu equal to sp, added 
:o one-fourth of the additional height sq; and draw the back u w, parallel to gp; 
ind extending to the same height, &c, as in Fig 2. If, however, this addition of ^ 
it's q should in any case give a base qu. less than one-half the total height oq, (which 
tvill very rarely happen in practice,) then make qu equal to half said total height; 
Irawing the back parallel to gp, and extending it to the same height as before. The 
idditional thicknesses thus found below sp, have reference rather to the pres of the 
jarth behind the abut, than to the thrust of the arch. In a very high abut, the inner 
line g p would give a thickness too slight to sustain this earth safely. 

When the height oh, Fig 2, of the abut is less than the thickness on at spring, a 
small saving of masonry (not worth attending to, except in large flat arches) may be 
affected by reducing the thickness of the abut throughout, thus: Make ok equal to 
on, and draw kl. Make oz equal to 3^ of on, and draw l z. Then, for any height 
oh of abut less than, on, draw b v, terminating in Iz. This bv will be sufficient base, 
if the foundations are firm. The back of the abut will be drawn upward from v, 
parallel to g p, and terminating at the same height as g or w. 

Rem. 1. All the abuts thus found will (with the provisions in Art 6) be safe, 
without any dependence upon the wing-walls; no matter how high the embkt may 
extend above the top of the arch. If the bridge is narrow, and the inner faces of 
the wing-walls are consequently brought so near together as to afford material as¬ 
sistance to the abuts, the latter may be made thinner; but to what extent, must 
depend upon the judgment of the engineer. 

We, however, caution the young practitioner to be careful how he adopts dimensions less than those 
given by our rule. There are certain practical considerations, such as carelessness of workmanship; 
newness of the mortar; danger of undue strains when removing the centers; liability of derange¬ 
ment during the process of depositing the earth behind the abuts, and over the arch; &c, which must 
not be overlooked; although it is impossible to reduce them to calculation. 

Whenever it can be done, the centers should remain in place until the embkt is finished; and for 
some time afterward, to allow the mortar to set well. But for more on this see Rem 4, p 713. 


Rum. 2. A good deal of liberty is sometimes taken, in reducing the quantity of masonry above the 
springing line of arches of considerable rise, and of moderate spans. When care is taken to leave 
the centers standing until the earth filling is completed above the arch, and behind its abuts, so that 
it may not be deranged bv accident during that operation ; and when good cement is used instead of 
common mortar, such experiments may be tried with comparative safety ; especially with culvert 
arches, in which the depth of arch-stones is great in proportion to the span. They must, however, be 
left to the judgment of the engineer in charge ; as uo specific rules can be laid down for them. They 
can hardly be regarded as legitimate practice, and we cannot recommend them. v\ e have known 
nearly semicircular arches, of 30 to 40 ft span, to be thus built successfully, wuh scarcely a particle 
of masonry above the springs to back them. Such arches, however, are apt to fall, if at any future 
Deriod the'earth filling is removed, without taking the precaution to first build a center or some other 
support for them. Even when the embkt can be finished before the centers are removed, we cannot 
recommend (and that only in small spans) to do less than to make n g. Fig 2, equal to X of the total 
height i t of the arch ; and from g so found, to draw a straight line touching the back of the arch as 
high up as possible. 

Rem 3. We have said nothing about l»atterli»ST <be faces of the abuts, 
because in the crossing of streams, the batter either diminishes the water-way; or 
requires a greater span of arch. Such a batter, however, to the extent of fiom 2 
to 11* ins to a ft, is useful, like the offsets, for distributing the wt of the structure 
and its embkt, over a greater area of foundation ; especially when the last is not 
naturally very firm ; or when the embkt extends to a considerable height above the 
arch In our tables, Nos 3 and 5, of approximate quantities of masonry in semi¬ 
circular bridges of from 2 to 50 ft span, the faces are supposed to be vert. 

Art. 5. Abutment-piers. When a bridge consists of several arches sus¬ 
tained bv piers of only the usual thickness, if one arch should by accident ot flood, 
or otherwise, be destroyed, the adjacent ones would overturn the piers; and arch 
after arch would then fall. To prevent this, it is usual in important bridges to make 
some of the piers sufficiently thick to resist the pres of the adjacent arches, in case 
of such an accident; and thus preserve at least a portion of the bridge from ruin. 
Such are called abutment-piers. 


Our formula of^-f. —** + 2 ft, for the thickness at spring; with the back battering as before, 

t the rate of J- of the span to the rise ; race vert; will of itself (without any modification for great 
. „ivo i 2 nlrfeetlv safe abut pier, for anv unloaded bridge ; and to any height whatever ; due 
gh !i had however to the consideration alluded to in the next Art. Thus, for an abut-pier 

ft<S K. h .l „ g „ Pi„ -ror of’anv ereater hei^ht; it is only necessary first to find the thickness o n at 
s high as o q, Fig 2. •^ liattered back a n />: extending it down to the base at B : with- 

sfaurj yss;»»*& >» «« 








700 


STONE BRIDGES. 


ma „ b e secure from the pres of the earth behind them; as well as from the pres of the arch ; a eon. 
Ration which does no?ap P ly to abu,pie.; la 



t apply to aout-piers ; m wmuu umj v..^ - , ;v ; 

But although the abut-pier thus found by our formula, would be abundantly 
safe vet its shape a h c o, Fig 3, is inadmissible. In practice it would be 
changed to one somewhat like that shown by the dotted lines ; having an equal 
degree of batter on both faces. This of course requires more masonry, with 
but little increase of stability; but that cannot be avoided. 

Whi n an abut-pieu is built in deep watku, or in a shallow stream sub¬ 
ject to high freshets, care must lie taken that water cannot find its way under 
the pier, and thus produce au upward pres, which will either diminish, or 
entirely counteract its efficiency as au abut, bee Remark 2, Art 4, ot Hy¬ 
drostatics. 

Art. 6. Inclination of the courses of masonry 
below the spring's of an arch. Although our iore- 
going rule gives a thickness of abut which cannot he overturned , 
or upset, by the pres of the arch, yet if the arch be*>f large span, 
and small rise, its great hor thrust may produce a sliding out¬ 
ward of the masonry near the level of the springs, if the stones 
are laid in hor courses; especially if the mortar has not set well. 

This danger, it is true, could be avoided by confining the courses together 
bv iron bolis and cramps; or by increasing considerably the thickness or the 
ovnon^p nf rinintr pit.hp.r nf those, leads to the cheaper expedient 


one 

fil 

ciei 

ri# 

|0C 


by iron bolts ana cramps ; ur uy —~ 

abuts; but the expense of doing either of these, leads to the cheaper expedient 
of inclining the masonry, as shown between o and w, Fig 4; the courses near o 
hnirw* Dtnmmp • QnH ormfltiniiv hAonminir less steep near a. 



ioury, as suuw u uctwccu u ouu x ig ■* , ^ 

being steelier; and gradually becoming less steep near n. 
By this process the arch is virtually prolonged into the body 

. t. „ f... tl>oi ii-lw.iv < ha In/tlinalimi rtf t.llA IdlVPr 


r»> Hits JMWC.ia V/UC mnuun; JA. -- - - 

of the abut, so far that when the inclination of the lower 
masonry ceases, as at n, the direction of the theoretical 
line of thrust, or of pres of the arch, (rudely represented 
by the dotted curved line o n) is nearly at right angles to 
the joints of the hor masonry below >»; and consequently, 
said thrust is unable to produce sliding at that point. Be¬ 
tween o and », the line of pres is everywhere so nearly at 
right angles to the variously inclined joints, as to preclude 
the possibility of sliding in that interval also. See Art 63 
of Force in Rigid Bodies.* The abut being thus safe 
throughout from both overturning and sliding, can fail 
only from defective foundations; or from the inferiority 
of the stone of which it is built; and which, if soft, may 
be crashed^^^^^K^HKKa 


This inclination of the masonry is as neces¬ 
sary in an elliptic arch, Fig 4]/£, as in a circular 


one. 





* This curved line of pressures is fouud in the manner directed at Rem 1, p 360, and at Fig 25, p 
231. Kaukine, Moseley, and others call it file line of resistance, and ap¬ 
ply line of pressures to another line which need not be introduced in a practical cousideration of 
abutments, walls, dams, &c. They however call any given point in our line a center of pressure, 
because at any part of the height of the abut such poiut shows where all the pressure or thrust may 
for many purposes be assumed to be concentrated. The perversion of commou technical terms is 
reprehensible. Said other Hue had better been called the line of resultants. We, both here 

and elsewhere, call the one to which we refer file line of pres sure, 
or of thrust, simply because bridge masons have no idea of any other line 
curving through an abut, and inasmuch as the pressure or thrust is greatest in that line they very 
properly so term it. Again, we have said above and elsewhere that the bed-joints should be neatly 
perp to this line of thrust. Theory properly requires them to be at ri$;iit 
an tiles to the resultants which cut the bed-joints at this line of 

thrust. Still we know that on account of the friction of masonry we may with perfect safety vary as 
much as about 30° from a right angle to these resultants, without depending at all on the stretigth 
of the mortar; and in using our rule of thumb ou thexiext page for drawing in¬ 

clined bed-joints, we shall always be fur within the limitof :’.0° : and therefore fully safe from sliding. 
Our rules do not call for this liue, nor for auythiug more than the span, rise, and radius of the arch. 












































STONE BRIDGES. 


701 


Ijr 


I he elliptic form is plainly unfavorable for uniting the arch-stones with the inclined masonry near 
the springs, so as to receive the thrust properly ; or about at right angles to its resultaut. In ordi¬ 
nary cases this difficulty may be overcome by malting the joints of only the outside or showing arch¬ 
stones to conform to the elliptic curve; as between e and u; while the joints of the inner or hidden 
ones, may have the directions shown between g and u, nearly at right angles to the line of thrust. It 
will rarely happen, however, that the young engineer will have to construct elliptic arches of suffi¬ 
cient magnitude to require either this, or any equivalent expedient. For spans less than 50 ft, with 
^ rises not less than about ^ of the span, nothing of the kind is actually necessary, if the mortar is 

* igood, and has time to harden, t 

In order to incline the masonry of any abut with sufficient accuracy, it would 
be necessary first to trace the curved line of pres of the given arch, as directed in 
Art 72 of Force in Rigid Bodies, so as to arrange the bed joints about at right, angles 
to it at every point of its course; but we offer the following process as sufficing for all 
ordinary practical purposes; while its simplicity places it within the reach of the com¬ 
mon mason. In actual bridges the direction of the actual thrust changes as the load 
s passing; therefore, in practice no given degree of inclination of the abut masonry 
;an conform to it precisely during the entire passage. Consequently, any excess of 
refinement in this particular-, becomes simply ridiculous; especially in small spans. 

Rule for inclining- tlie beds of the masonry in the abuts. 

Add together the rad cm, Fig 4; and the span of the arch. Div the sum by 6. To 
the quot add 3 ft. Make o t , on the rad, equal to the last sum. Then is t a central 
point, toward which to draw the directions of the beds, as in the fig. Draw t s hor, 
and from t as a center, describe the arc oy\o being the center of the depth of the 

* springers. From y lay off on the arc the dist yn, equal to one-sixth part of ty: draw 
1 t n a. It will never he necessary to incline the masonry below this t n a. Neither 

need the inclination extend entirely to the face m i of the abut; hut may stop at e, 
about half-way between i and n. From e upward, the inclination may extend for¬ 
ward to the line e m. 


t The feet of both elliptic and semicircular arches are always made hor ; but it is plain from Fig 
4 %, that this practice is at variance with correct principles of stability in the case of the ellipse. It 
is the same in the semicircle. In ordinary bridges of the latter form, the vert pres, or weight resting 
on each skewback, is (roughly speaking) usually about from 3^ to 4 times the hor pres on the same; 
and the total pres is about 4 times as great as the pres on the keystone. Therefore, theoretically, the 
skewback should usually be about 4 times as deep as the keystone ; and its bed, instead of being hor, 
should be inclined at the rate of about 1 vert to 4 hor. 






702 


STONE BRIDGES 



When the arch is hat, 



this inclination may become so steep, especially in the upper parts, that 
struts, or shores of some kind, must be used for preventing the ma¬ 
sonry from sliding down, until the completion of the arch secures it 
from doing so. The hor courses betweeu the face m i, and the line 
o e, will aid somewhat in this respect. 

This method should be applied to all very large arches wh 
rise is one-third, or less, of the span. As before remarked, it 
is not actually necessary in arches not exceeding about 50 ft span, 

and not Hatter than l of the span. Indeed, if the earth filling can 
be deposited before the centers are removed, these limits may be con¬ 
siderably extended without danger. Still, since a certaiu degree of 
inclination is attended with very little trouble or expeuse, we would 
recotumeud for even such arches, a process somewhat like the follow¬ 
ing : From half the span lake the rise. Div the retn by 3. Make o t, 
Fig 5, equal to the quot. Draw t n, and o m, hor. Div the angle 
s o m into two equal parts, by the line o a. Incline the masoury so 
as to be parallel to o a, as far dowu as t n. The inclined courses 
may extend out to the face o t, or not, at pleasure. 


IV 

tin 


Rem. 1. To find the length (ab, Fig 7) 

from face to face of a culvert. From 

the height h t of the embkt, take the above ground, height n a 
of the culvert; the rent will be the height h o of the embkt 
above the culvert. Then the reqd length a b is plainly equal 
to the top width id of the embkt, added to the two dists as, 
cb, which correspond to its steepness of side-slopes. Thus, if 
the side-slope is, as usual, 1 ^ to 1, then as and cb will each be 
equal to IJsj times oh; or the two together will be 3 times o h. 
So that if the width i d is 14 ft, and A o 5 ft, the length a b will be 

14 + <5 X 3) = 14 + 15-29 ft. 

Art. 7. The following tables. S. 4. and 5. of quantities, will 

be found useful for expediting preliminary estimates; for which purpose chiefly they 
are intended ; lienee no pains have been taken to make them scrupulously correct, 
but rather a little in excess of the truth. The first column of Table 3 contains the 




total vert height oc, Fig 6, from the 
crown o of a semicircular arch, to 
the foundation or base gm of its 
abut. The other columns give ap¬ 
proximately the number of cub yds 
contained in each running foot, or 
foot in length of the culvert or 
bridge, measured from end to end 
(face to face) of the arch proper; 
and including only the arch and its 
abuts, as shown in Fig 1; or in the 
half section oprngy in Fig 6; in¬ 
cluding footings to the abuts, but 
omitting the wing-walls (ten), and 
the spandrel-walls (*), Figs 6 and 
2^. At the foot of each column is the approximate content in cub yds of the two 
spaudrel-walls by themselves; one over each face of the arch. 


These spandrel-walls are calculated on the supposition that their thickness at base, at their junc¬ 
tion with the wing-walls, where their height is greatest, is equal to of their height at that point: 
except where that proportion gives a less thickness at top than 2}4 ft: and that they extend 2 ft (o a) 
above the top o of the arch. At the top of the arch, they are all supposed to be 2*^ ft thick at top; 
that being assumed to be about the least, thickness admissible in a rubble wall in such a position. 
Both the back and the face are supposed to be vert. The contents of these spandrel-walls will vary 
somewhat, however, even in the same span, with the height of the abut and the arrangement of the 
wings. They, however, constitute so small a proportion of the entire contents given in Table 5. that 
this consideration may be neglected in preliminary estimates. They are so firmly bonded into the 
masonry of the wings at their highest points, and so strongly connected by mortar with the backing 
of the arch at their bases, that they require no greater thickness however high the emb may be. 

The contents of the four winsr-walls, of which njw b , Fig 6, is one, 
will be found in a table (No. 4) immediately following that for the body of the cul¬ 
vert. We have also added a table (No. 5) for complete semicircular culverts of 
various lengths, including their spandrel and wing walls. 









































STONE BRIDGES. 


703 


w Rem. 1. Although the thickness of wing-walls increases in all parts with their 

* height, they are not made to show thicker at nj than at ti. Fig (5; hut (as seen in the 
fig) are offsetted at their back tn, a little below their slanting upper surf ij, so as 
to give a uniform width for the steps or flagstones, as the case may be, with which 

* they are covered. In the fig the covering is supposed to be of flagstones ; but steps 
11 are preferable, being less liable to derangement. To prevent the flagstones from 

sliding down the inclined planeji, the lower stone i should be deep and large, and 
laid with a hor bed. The flags are sometimes cramped together with iron, and bolted 
down to the wall. Steps require nothing of that kind, as seen at s, Fig 11. 

Rem. 2 . The tables show the inexpediency of too much con- 
"• trading’ the width of water-w ay, with a view to economy, by adopting 
a small span of arch, when a culvert of greater span can be made, of the same total 
height. 


i 'l | 


be 


* For th e wings must be the same, whether the span be great or small, provided the total height is 
the same in both cases ; and since the wings constitute a large proportion of the entire quantity of 
masonry, in culverts of ordinary length, the span itself, within moderate limits, has comparatively 
little effect upon it. Thus, the total masonry in a semicircular culvert of 3 ft span, 8 ft total height, 
and 60 ft long between the faces of the arch, is, by Table 5, 151)4 cub yds; while that of a 5 ft span, 
of the same height and length, is 152.4. A semicircular bridge of 25 ft span, 24 ft total height, and 
40 ft between the faces of the arch, contains 1031 cub yds; while one of 35 ft span, of the same height 
and length, contains 1134 yds; so that in this case we may add nearly 50 per cent to the water-way, 

by increasing the masonry of the bridge but qqyth part. 

Rem. 3. Partly for the same reason, and partly because the culverts for a 
double-track road are not twice as longas those for a single- 
track one, the quantity of culvert masonry for the former w ill not average more 
than about from % to x /a P art more than that for the latter; so that it frequently 
becomes expedient to finish the culverts at once to the full length required for a 
double track, although the embkts may at first be made wide enough for only a 
single one, with the intention of increasing them at a future time for a double one. 


Thus, the average size of culverts for a single track may be roughly taken at 6 ft span, 30 ft long 
from face to face, and 10 ft total height; and such a one contains, by Table 5, 140 cub yds. For a 
double track, it would require to be about 12 feet longer; and we see by Table 3 that this will add 
2.67 X 12 = 32 cub yds; making a total of 172 yds instead of 140; thus adding rather less than 
part. When the culverts are under very high embkts, aud consequently much longer, the addition 
for a double track becomes comparatively quite trifling. 

Table 3, of approximate numbers of eub yds of masonry 
per foot run, contained in the arches and abutments only, as 

shown in Fig 1 (omitting w ings, and the spandrel-w alls over the faces of the arches) 
of semicircular culverts and bridges, of from 2 to 50 ft span, and of different total 
heights, h t, Fig 1, or a c, Fig 6. It will be seen that in many cases, a bridge of larger 
span contains less masonry than one of smaller span, when their total heights are the 
same. There is a liberal allow ance for footings or offsets at the bases cf the abuts. 


TABLE 3. (Original.) 


Total 

Height. 

Span 

2 rt. 

Span 

3 ft. 

Span 

4 ft. 

Span 

5 ft. 

Span 

6 ft. 

Span 

8 ft. 

Span 

10 ft. 

Spar. 
12 ft. 

Span 

15 ft. 

Feet. 

Cub. y. 

Cub. y. 

Cub. y. 

Cub. y. 

Cub. y. 

Cub. y. 

Cub. y. 

Cub. y. 

Cub. y. 

2 

-42 









3 

go 

J53 

.67 







4 


83 

.87 

.92 

.97 





r. 

99 

1 04 

1.08 

1JL5 

1.21 





g 

1 

1 l )H 

1 28 

1.37 

1.46 

1.58 

1.69 



7 

1.62 

1.59 

1.55 

1.64 

1.72 

1 85 

1.97 

2.12 


8 

2.01 

1.96 

1.91 

1.95 

1.99 

2.13 

2.26 

2.38 


9 

2.45 

2.38 

2.31 

2.29 

2.27 

2.42 

2.56 

2.65 

3.02 

10 

2.94 

2.85 

2.76 

2.72 

2.67 

2.77 

2.87 

2.93 

3.34 

11 


3.38 

3.26 

3.19 

3.12 

3.16 

3.19 

3.23 

3.67 

12 


3.98 

3.82 

3.72 

3.62 

3.57 

3.52 

3.55 

4.01 

13 



4.42 

4.29 

4.17 

4.10 

4.02 

8.86 

4.36 

14 



5.08 

4.90 

4.77 

4.67 

4.57 

4.41 

4.72 

15 




5.57 

5.42 

5.30 

5.17 

5.01 

5.09 

lg 




6 30 

6.12 

5.97 

5.82 

5.56 

5.69 

17 





6.87 

6.70 

6 52 

6.26 

6.34 

18 

1 



7.69 

7.48 

7.27 

7.01 

7.04 

19 






8.32 

8.07 

7.71 

7.69 

20 






9.20 

8.92 

8.56 

8.49 

21 







9.82 

9.46 

9.34 

22 

23 







10.8 

10.3 

10.2 








11.3 

11.1 

24 








12.3 

12.1 










13.2 

2b 








14.2 


1 





, 

1 



Contents of the two spandrel-walls, over the two ends of the arch, in cub yds. 

I 2.9 I 3.7 | 44 | 5.2 | 5.8 | 7.9 | 9.8 | 12. | 16. 































































704 


STONE BRIDGES. 




TABLE 3. (Continued.) 


Total 

Span 

Span 

Total 

Span 

Total 

Span 

Height. 

20 ft. 

25 ft. 

Height. 

35 ft. 

Height. 

50 ft. 

Feet. 

Cub. v. 

Cub. v. 

Feet. 

Cub. y. 

Feet. 

Cub. y. 

12 

13 

4 60 


20 

10.5 

27 

18.0 

4.98 


21 

11.0 

28 

18.7 

14 

5.37 

6.10 

22 

11.6 

29 

19.4 

15 

5.77 

6.41 

23 

12.2 

30 

20.1 

16 

6 18 

6.76 

24 

12.7 

31 

20.9 

17 

6.60 

7.16 

‘25 

13.3 

32 

21.6 

18 

7.03 

7.61 

26 

13.8 

33 

22.4 

19 

7.47 

8.10 

27 

14.5 

34 

23.1 

20 

8.12 

8.60 

28 

15.1 

35 

23.9 

21 

8.82 

9.02 

29 

15.7 

36 

24.7 

22 

9.57 

9.72 

30 

16.3 

37 

25.5 

23 

10.4 

10.4 

31 

17.0 

38 

26.3 

24 

11.3 

'1.2 

32 

18 l 

39 

27.1 

25 

12.2 

12.1 

33 

19.2 

40 

28.0 

26 

13.1 

13.0 

34 

20.4 

41 

28.8 

27 

14.1 

14.0 

35 

21.7 

42 

30.0 

28 

15.2 

15.0 

36 

23.0 

43 

31.5 

29 

x6.3 

16.1 

37 

24 3 

44 

33.0 

30 

17.4 

17.2 

38 

25.7 

45 

34.6 

31 

18.6 

18.4 

39 

27.2 

46 

36.3 

32 

19.9 

19.6 

40 

28.7 

47 

38.1 

33 

21 2 

20.9 

41 

30.2 

48 

39.8 

•74 

22.6 

22.2 

42 

31.8 

49 

41.6 

35 

2+.0 

23.6 

43 

33.5 

50 

43.6 

36 

25.4 

25.0 

44 

35.2 

51 

45.5 

37 

26.9 

26.5 

45 

36.9 

52 

47.4 

38 

28 5 

28.0 

46 

38.7 

53 

49.4 

39 

30.1 

29.5 

47 

40.6 

54 

51.6 

40 

31.7 

31.2 

48 

42.5 

55 

53.7 

41 


32.8 

1 49 

44.4 

56 

55.9 

42 

.. 

34.5 

50 

46.4 

57 

58.1 

43 


363 



58 

60.4 

44 


38.1 



59 

62.7 

45 


40.0 



60 

65.1 

Contents of the two spandrel-walls, over th» two ends of the arch, iu cub yds. 

28. | 42. || | 85. II | 195. 


Art. 8. The following; table of contents of wing-walls, or wings,-will, 
like the preceding one, be useful in making preliminary estimates. The wings 
no, no, shown in plan at Fig 8, are supposed to form an angle aoc, of 120°, with the 
face, or end n o of the culvert. Their outer or small ends n n, are all assumed to be of 
the dimensions shown on a larger scale at E. Thickness at base at every part equal 
to of the height of the wall at said part: except when that proportion becomes 
too small to allow the width or thickness at top to be 2.5 ft; in w hich case it is en¬ 
larged at such parts sufficiently for that purpose. See Remark 2. This happens only 



when the height mm, Fig E, of the wing, becomes less than 9 ft. Batter of face, 1^ 
ins to a ft; or 1 in 8. Back vert; but offsetted, if necessary, for a short dist below r 
the top, so as to give a uniform showing top tliick ness of 2]4 ft. The masonry is 
supposed to be good well-scabbled mortar rubble. The height given in the first 
column is the greatest one; or that at on, (or wj. Fig 6.) where the wing joins the 
face of the culvert. In the table no allowance is made for footings (offsets or steps) 
at the base ol the wings; as these are frequently omitted in wings on good founda- 
























































STONE BRIDGES. 


705 


tions. In taking out quantities from the table, bear in mind that the height of the 
wings is usually a little greater than that of the culvert itself. 

Table 4. of approximate eonieiits, in cub yds, of the l'onr 
Wing-walls of a culvert, or bridge. (Original.) 

The heights are taken where greatest; as aty to, Fig 6 


Height 

Length 

Cub. yds. 


Height 

Length 

Cub. yds. 

of 

of 

in 


of 

of 

in 

wing. 

one wing. 

4 wings. 


wing. 

oue wing. 

4 wings. 

Feet. 

Feet. 



Feet- 

Feet. 


6 

1.73 

4.04 


30 

43.3 

818 

7 

3.16 

8.85 


32 

46.8 

997 

8 

5.20 

14.6 


34 

50.3 

1192 

9 

6.93 

21.5 


36 

53.7 

1414 

10 

8.66 

30.2 


38 

57.2 

1661 

11 

10. 4 

40.9 


40 

60.7 

1928 

12 

12.1 

53.7 


42 

64.2 

2220 

14 

15.6 

85.2 


44 

67.6 

2552 

16 

19.1 

128 


46 

71.1 

2912 

18 

22.5 

183 


48 

74.6 

3306 

20 

26.0 

247 


50 

78.0 

3741 

22 

29.5 

329 


55 

86.7 

4942 

2+ 

32.9 

426 


60 

95.3 

6404 

26 

36.4 

541 


65 

104 

8131 

28 

39.8 

672 


70 

113 

10155 


To reduce cub yds to perches of 25 cub ft. mult by 1.080. 
To reduce perches to cub yds, mult by .926, or div by 1.08. 


The contents for heights intermediate of those in the table may be found approximately by simple 
proportion. 


".r 


Rem. 1. It is not recommended to actually prolong all wings until their dimen¬ 
sions become as small as shown at E, iti Fig 8. In large ones it will generally be 
more economical to increase their end height m m, a few feet. The contents, how¬ 
ever, may be readily found by the. table in that case also. Thus suppose the height 
jf the wings at one end to he 30 ft, and at the other end 8 ft: we have only to sub¬ 
tract the tabular content for 8 ft high, from that for 30 ft high. Thus, 818 — 14.6 = 
303.4 cub yds required content. 


I 

of 

dI 

tni 

a- 


Rem. 2. It might he supposed that inasmuch as the wings of arches often have to 
sustain the pressure from embankments reaching far above their tops, they Mould, 
like ordinary retaining-walls, be made much thicker in that case. But the . ct that 
■hey derive great additional stability from being united at their high enas to the 
I tody of the bridge or culvert, renders such increase unnecessary when proportioned 
by Our rule; no matter how far the earth may extend above them; as shown by 
ibundant experience. 


Relying upon tbis aid. we may indeed, when the earth does not extend above the top, reduce the 
hose at o to one third of the ht, as shown at o t; and by dotted line t. s. Expedience shows that we 
may also do the same even when the earth reaches to a great height above the top; provided that 
the wings, instead of being splayed or flared out, as at o n, o n. merely form straight prolongations 
of the abutments of the arch, as shown by the dotted lines at o g to. In this case the pressure of tbe 
earth against the wings is less than when they are splayed. We have known the thickness at o 
to be reduced iu such cases to less than one-third the height, when the wings were 15 ft high, aud 
die height of the embankment above their tops 16 feet iu one case, and 36 ft in another. In another 
nstance, similar wings 25*^ ft high, and with 29 ft of embankment above their top, had their bases 
it o rather less than 3 of the height. In all these cases, the uniform thickness at top was 2.5 feet; 
packs vertical. We mention them because this particular subject does not seem to be reducible to 
Vny practical rule. The last wall appears to us to be too thin ; especially if the earth is not deposited 
in layers: and after allowing the mortar full time to set. The labor, however, required in compact- 
ng the earth carefully in layers, may cost more than is thereby saved in the masonry. The young 
oractitioner must bear this in mind when he wishes to economize masonry by such means; and also 
hat the thin wall may bulge, or fail entirely, if the earth backing is deposited while the mortar is 
imperfectly set. 


46 





















706 


STONE BRIDGES. 


Tabic 5. Approximate contents in cubic yawls, of com 
plete semicircular culverts ami bridges of from 2 to 50 fee 
span; including the 2 spandrel walls; and the 4 wings; all proportioned by th 
foregoing directions; and taken from the two preceding tables. The height in th 
second column, is from the top of the keystone to the bottom of the foundation. Th 
wings are calculated as being 2 ft higher than this, including the thickness of th 
coping. The wings are frequently carried only to the height of the top of the arch 
thus saving a good deal of masonry. Table 4, of wings alone, will serve to make tb 
proper deduction in this case. 

The several lengths are from end to end, or from face to face, of the arch prope: 
Tim contents for intermediate lengths may be found exactly; and those tor intei 
mediate heights, quite approximately, by simple proportion. In this table, as i 
No. 3, it will be observed that when the heights are the same in both cases, a large 
span frequently contains less masonry than a smaller one. A semicircular culvei 
or bridge contains less masonry than a flatter one, when the total height is the sam 
in both cases; therefore, the first is the most economical as regards cost; but it dot I 
not afford as much area of water-way; or width of headway. 


(Original.) 



fci 

A • 

A . 

A ; 

A . 

A • 

A 

A W 

A 


. 

A ^ 

A - 

S3 

ed 



<-»«_» 

s-» *“* 

oo-=« 

•-» 


tC 1*4 


ty* 


tfj — 



O. 

QG 

*5 

go 

S=> 

g® 

S3 — 

a — 


§3 

s o 

0) C'l 

e o 

c © 

0) © 

2 © 

o> x. 

gf 

33 

n ' * 




-1-° 


- 

>j 





Ft 

Ft. 

Cub.Y. 

Cub.Y. 

Cub. Y. 

Cub. Y. 

Cub.Y. 

Cub.Y. 

Cub Y. 

Cub.Y. 

Cub.Y. 

Cub.Y. 

Cub.Y. 

Cub.Y. 


5 

27 

32 

42 

52 

72 

92 

112 

132 

152 

172 

192 

212 


6 

37 

43 

56 

69 

94 

120 

146 

171 

197 

222 

248 

274 

2 

7 

40 

57 

73 

89 

122 

154 

187 

219 

251 

284 

316 

349 


8 

63 

73 

93 

113 

153 

193 

233 

‘ITS 

313 

353 

393 

433 


10 

101 

116 

145 

175 

234 

291 

351 

410 

469 

527 

586 

645 


5 

28 

34 

44 

54 

75 

96 

117 

138 

158 

179 

200 

221 


6 

38 

44 

57 

70 

95 

121 

146 

172 

198 

223 

249 

275 

3 

7 

49 

57 

73 

89 

121 

153 

184 

216 

247 

280 

312 

343 

8 

63 

73 

93 

112 

152 

191 

230 

269 

308 

348 

387 

426 


10 

101 

115 

143 

172 

229 

286 

343 

400 

457 

514 

571 

628 


12 

149 

169 

208 

248 

328 

407 

487 

567 

646 

726 

806 

885 


5 

30 

35 

46 

57 

78 

100 

122 

143 

165 

186 

208 

229 


6 

38 

45 

58 

70 

96 

122 

147 

173 

198 

224 

250 

275 


7 

49 

57 

73 

88 

119 

150 

181 

212 

243 

274 

305 

336 

4 

8 

63 

73 

92 

111 

149 

188 

2-26 

264 

302 

340 

379 

417 


10 

100 

114 

141 

161* 

224 

279 

335 

390 

445 

500 

555 

611 


12 

147 

166 

204 

243 

319 

395 

472 

548 

625 

701 

777 

854 


14 

209 

234 

285 

336 

437 

539 

641 

742 

844 

945 

1047 

1149 


6 

41 

47 

61 

75 

102 

130 

157 

184 

212 

239 

267 

294 


7 

52 

60 

76 

93 

125 

158 

191 

224 

257 

289 

322 

355 

5 

8 

63 

75 

94 

114 

153 

192 

231 

270 

304* 

348 

387 

426 

10 

100 

114 

141 

168 

223 

277 

331 

386 

440 

495 

549 

603 


12 

116 

165 

202 

239 

314 

388 

46$ 

537 

611 

686 

760 

835 


14 

207 

231 

280 

329 

427 

525 

623 

721 

819 

917 

1015 

1113 


7 

53 

62 

79 

96 

131 

165 

200 

234 

268 

303 

337 

372 


8 

66 

76 

96 

116 

156 

196 

236 

276 

316 

356 

34*6 

436 

A 

10 

100 

113 

140 

167 

220 

274 

327 

380 

434 

487 

541 

594 


12 

146 

164 

200 

216 

308 

381 

453 

526 

598 

670 

743 

815 


14 

206 

219 

277 

325 

420 

516 

611 

706 

802 

897 

993 

1088 


16 

281 

311 

373 

434 

556 

679 

801 

923 

1046 

1168 

1291 

1413 


7 

57 

67 

85 

104 

141 

178 

215 

252 

289 

326 

363 

400 


8 

70 

81 

102 

124 

166 

209 

251 

24*4 

337 

379 

422 

464 

8 

10 

104 

118 

145 

173 

228 

284 

334* 

395 

450 

505 

561 

616 

12 

147 

165 

200 

236 

308 

379 

450 

522 

593 

664 

736 

807 


14 

206 

210 

276 

323 

416 

510 

603 

696 

74*0 

883 

977 

1070 


n> 

281 

310 

370 

430 

549 

H69 

788 

908 

1027 

1146 

1266 

1385 


18 

367 

405 

480 

554 

704 

854 

1003 

1153 

1302 

1452 

1602 

1751 


8 

74 

85 

108 

131 

176 

221 

266 

311 

357 

402 

447 

492 


10 

107 

121 

150 

179 

236 

294 

351 

408 

466 

523 

581 

638 

10 

12 

148 

166 

201 

236 

306 

377 

447 

518 

588 

658 

729 

799 

14 

207 

229 

275 

321 

412 

504 

595 

686 

778 

869 

961 

1052 


16 

280 

309 

368 

426 

542 

679 

775 

891 

1008 

1124 

1241 

1357 


18 

, 366 

402 

475 

548 

693 

839 

984 

1129 

1275 

1420 

1565 

1711 


10 

110 

125 

154 

183 

242 

301 

359 

418 

476 

535 

594 

652 

12 

12 

151 

168 

204 

239 

310 

381 

452 

523 

594 

665 

736 

807 

14 

206 

228 

272 

317 

405 

493 

581 

669 

758 

846 

934 

1022 

16 

279 

306 

362 

418 

529 

640 

751 

862 

974 

1085 

1196 

1301 


18 

364 

399 

469 

540 

680 

820 

960 

1100 

1241 

1381 

1521 

1661 


20 

470 

512 

598 

684 

865 

1026 

1197 

1368 

1540 

1711 

1882 

2053 





































































































































STONE BRIDGES 


707 


m 

ti, 

tfc 

ft 

th 

cl 

tl 

■ Pff 
tei 
»1 
P 
m 
m 
lot 



- 

i, 

r 


Table 5 — (Continued.) (Original.) 


0 

8 

GG 

Height. 

•2 - 
Si 

Length. 

20 Ft. 

Length. 

30 Ft. 

1 

Length. 

40 Ft. 

JC j 

1 

Length, i 

j 80 Ft. 

1 

Length. 

100 Ft. 

Length. 

120 Ft. 

Leugth. 

140 Ft. 

Length. 

160 Ft. 

XL ** 

*-* [V. 

c 9 

4) CG 

►J 

Length. 

200 Ft. 

Ft. 

Ft. 

Cub.Y. 

Cub.Y. 

Cub. Y. 

Cub.Y. 

Cub.Y. 

Cub.Y. 

Cub.Y. 

Cub.Y'. 

Cub.Y. 

Cub.Y. 

Cub.Y'. 

Cub.Y. 


12 

162 

182 

222 

262 

342 

422 

502 

583 

663 

743 

823 

903 


U 

215 

239 

286 

333 

427 

522 

616 

711 

805 

899 

994 

1088 

15 

IS 

285 

313 

370 

427 

541 

654 

768 

882 

996 

1110 

1223 

1337 

18 

369 

404 

474 

515 

686 

826 

967 

1108 

1249 

1390 

1530 

1671 


20 

473 

515 

600 

685 

855 

1024 

1194 

1364 

1534 

1704 

1873 

2043 


22 

595 

646 

748 

850 

1054 

1258 

1462 

1666 

1870 

2074 

2278 

2482 


14 

237 

264 

317 

371 

478 

586 

693 

801 

908 

1015 

1123 

1230 


IS 

304 

335 

397 

458 

582 

706 

829 

953 

1076 

1200 

1324 

1447 

20 

18 

381 

416 

486 

556 

697 

838 

978 

1119 

1259 

1400 

1541 

1681 

20 

479 

520 

601 

682 

844 

1007 

1169 

1332 

1494 

1656 

1819 

1981 


22 

598 

646 

741 

837 

1028 

1220 

1411 

1603 

1794 

1985 

2177 

2368 


24 

739 

795 

908 

1021 

1247 

1473 

1699 

1925 

2151 

2377 

2603 

2829 


16 

327 

360 

428 

496 

631 

766 

901 

1036 

1172 

1307 

1442 

1577 


18 

403 

441 

517 

594 

746 

898 

1050 

1202 

1355 

1507 

1659 

1811 

25 

20 

500 

543 

629 

715 

887 

1059 

1231 

1403 

1575 

1747 

1919 

2091 

22 

614 

663 

760 

857 

1051 

1246 

1440 

1635 

1829 

2023 

2218 

2412 


24 

751 

807 

919 

1031 

1255 

1479 

1703 

1927 

2151 

2375 

2599 

2823 


26 

909 

974 

1104 

1234 

1494 

1754 

2014 

2274 

2534 

2794 

3054 

3314 


28 

1085 

1160 

1310 

1460 

1760 

2060 

2360 

2660 

2960 

3260 

3560 

3860 


22 

685 

743 

859 

975 

1207 

1439 

1671 

1903 

2135 

2367 

2599 

2831 


24 

817 

880 

1007 

1134 

1388 

1642 

1896 

2150 

2404 

2658 

2912 

3166 

35 

26 

969 

1033 

1181 

1309 

1585 

1861 

2137 

2413 

2689 

2965 

3241 

3517 

28 

1130 

1205 

1356 

1507 

1809 

2111 

2413 

2715 

3017 

3319 

3621 

3923 


30 

1327 

1408 

1571 

1734 

2060 

2386 

2712 

3038 

3364 

3690 

4016 

4342 


32 

1549 

1639 

1820 

2001 

2363 

2725 

3087 

3449 

3811 

4173 

4535 

4897 


35 

1946 

2054 

2271 

2488 

2922 

3356 

3790 

4224 

4658 

5092 

5526 

5960 


30 

1494 

1594 

1795 

1996 

2398 

2800 

3202 

3604 

4006 

4408 

4810 

5212 


32 

1711 

1819 

2035 

2251 

2683 

3115 

3547 

3979 

4411 

4843 

5275 

5707 


34 

1956 

2071 

2302 

2533 

2995 

3457 

3919 

4381 

4843 

5305 

5767 

6229 

50 

36 

2228 

2350 

2597 

2844 

3338 

3832 

4326 

4820 

5314 

5808 

6302 

6796 

38 

2519 

2650 

2913 

3176 

3702 

4228 

4754 

5280 

5806 

6332 

6858 

7384 


40 

2835 

2975 

3255 

3535 

4095 

4655 

5215 

5775 

6385 

6895 

7455 

8015 


42 

3197 

3347 

3647 

3947 

4547 

5147 

5747 

6347 

6947 

7547 

8147 

8747 


45 

3818 

3991 

4337 

4683 

5375 

6067 

6759 

7451 

8143 

8835 

9527 

10219 


50 

5063 

5281 

5717 

6153 

7025 

7897 

8769 

9641 

10513 

11385 

12257 

13129 


Art. 9. Especial pains should he taken to secnre an nnylelding 1 fonn« 
elation for culverts ami drains nmler high embklsj otherwise 
the superincumbent weight, especially under the middle of the embkt, may squeeze 
them into the soil below, if soft or marshy; and thus diminish the area of water¬ 
way, or at least cause an ugly settlement at the midlength of the culvert. Also, in 
soft ground, the embkt may press the side walls closer together, narrowing the 
channel. This may be prevented by an inverted arch, or a bed of masonry, between 
the walls. A stratum from 3 to (1 ft thick, of gravel, sand, or stone broken to turn¬ 
pike size, w'ill generally give a sufficient foundation for culverts in treacherous 
marshy ground ; or quicksand, with but a moderate height of embkt. It should ex¬ 
tend a few feet beyond the masonry in every direction, and should be rammed: the 
sand or gravel being thoroughly w'et, if possible, to assist the consolidation. Piling 
will sometimes be necessary. If the masonry is built upon timber platforms, or a 
smooth surface of rock, care must be taken to prevent it from sliding, from the pres 
of the earth behind it. This same pres may even overthrow' the piles, if they are 
not properly secured against it. 


Art 10. Drains. 

Drains of the dimen¬ 
sions in Fig 11, con¬ 
tain 1 perch, of 25 
cub ft; or .926 of a 
cub yd, per ft run. 

They are frequently 
built of dry aeabbled 
rubble, and paved with 
spawls. When there is 
much wash through 
them, with a consider¬ 
able slope, it is better to 
continue the foundation 






























































































708 


STONE BRIDGES. 


solid clear across. This is ofteu done without those causes, inasmuch as the additional masonry Is a 
mere trifle; aud the exeavatiou of a single broad fouudatiou-pit is less troublesome thau that of two 
narrow ones. A deep llag-stoue/at the eutrauce, aud others at short dists of the length, may be in¬ 
troduced in both drains aud culverts, to protect from uudermiuing. 

These draius extend under the entire width of the embkt, from toe to toe; and may terminate in 
steps, as in the side view at S. They are of course better when built with mortar, with au admixture 
of cement to prevent the water when full from leaking into and softening the embankment. 

Sometimes two or three such drains may be placed parallel to each other, instead of a culvert. 
When two are so placed, they contain only times the masonry of one ; still their use will generally 
involve no saving of masonry over a culvert. A man can crawl through Fig 11 to clean it. 

Art. 11. The drainage of the roadways of stone bridges of several 

arches, is generally effected by means of open gutters, which descend slightly from 
the crowns of the arches, each way, until they reach to near the euds of the re¬ 
spective spans. 

There they discharge into vertical iron pipes built into the masonry. The upper ends of tlie 
pipes should be covered by gratings. When inconvenience would result from the water falling upon 
persons passing under the arches, these pipes may be carried down the entire height of the piers; 
but when such is not the case, they may extend only to the soffit, or under face of the arch ; allowing 
the water to fall freely through the air from that height. 

Table B, of approximate content*, in cub yds. of a solid 
pier of masonry, 6 ft by 22 ft on top; and battering 1 inch to a ft on each of 
its 4 faces. The contents of masonry of such forms must he calculated by the prismoidai formula; 
and not by taking the length and breadth of the pier at half its height as an average length and 
breadth, as is sometimes done. This incorrect method would give only 6492 cub yds as the content 
of the pier 200 ft high; instead 7178 yds, its true content. High piers may for economy be built hol¬ 
low, with or without interior cross-walls for strengthening them, as the case may require; and the 
hatter is generally reduced to % inch or less to a foot. Hollow piers require good well-bedded ma- 

taar J*- (Original.) 


lit. 

Ft. 

Lgth 

at 

base. 

Bdtli 

at 

base. 

Cubic 

yards 

lit. 

Ft. 

Lgth 

at 

base. 

Bdtli 

at 

base. 

Cubic 

yards 

lit. 

Ft. 

Lgth 

at 

base. 

Bdth 

at 

base. 

Cubic 

yards. 

6 

23. 

7. 

32.5 

52 

30.67 

14.67 

, 537 

128 

43.33 

27.33 

2759 

7 

.17 

.17 

38.6 

54 

31. 

15. 

570 

130 

.67 

.67 

2848 

8 

.33 

.33 

44.9 

56 

.33 

.33 

605 

132 

44. 

28. 

2940 

9 

23.5 

7.5 

51.3 

58 

.67 

.67 

641 

134 

.33 

.33 

3032 

10 

.67 

.67 

58. 

60 

32. 

16. 

679 

136 

.67 

.67 

3126 

11 

.83 

.83 

64.8 

62 

.33 

.33 

717 

138 

45. 

29. 

3222 

12 

24. 

8. 

71.7 

64 

.67 

.67 

757 

140 

.33 

.33 

3320 

13 

.17 

.17 

79. 

66 

33. 

17. 

798 

142 

.67 

.67 

3420 

14 

.33 

.33 

86.4 

68 

.33 

.33 

840 

144 

46. 

30. 

3521 

15 

24.5 

8.5 

94. 

70 

.67 

.67 

884 

146 

.33 

.33 

3623 

16 

.67 

.67 

102 

72 

34. 

18. 

928 

148 

.67 

.67 

3728 

17 

.83 

.83 

110 

74 

.33 

.33 

973 

150 

47. 

31. 

3835 

18 

25. 

9. 

118 

76 

.67 

.67 

1021 

152 

.33 

.33 

3944 

19 

.17 

.17 

127 

78 

35. 

19. 

1070 

154 

.67 

.67 

4056 

20 

.33 

.33 

135 

80 

.33 

.33 

1120 

156 

48. 

32. 

4168 

21 

25.5 

9.5 

144 

82 

.67 

.67 

1171 

158 

.33 

.33 

4284 

22 

.67 

.67 

153 

84 

36. 

20. 

1224 

160 

.67 

.67 

4402 

23 

.83 

.83 

163 

86 

.33 

.33 

1278 

162 

49. 

33. 

4520 

24 

26. 

10. 

172 

88 

.67 

.67 

1334 

164 

.33 

.33 

4610 

25 

.17 

.17 

182 

90 

37. 

21. 

1392 

166 

.67 

.67 

4763 

26 

.33 

.33 

192 

92 

.33 

.33 

1451 

168 

50. 

34. 

4887 

27 

26.5 

105 

202 

94 

.67 

.67 

1510 

170 

.33 

.33 

5014 

28 

.67 

.67 

212 

96 

38. 

22. 

1569 

172 

.67 

.67 

5143 

29 

.83 

.83 

223 

98 

.33 

.33 

1631 

174 

51. 

35. 

5275 

30 

27. 

11. 

234 

100 

.67 

.67 

1695 

176 

.33 

.33 

5409 

31 

.17 

.17 

245 

102 

39. 

23. 

1761 

178 

.67 

.67 

5545 

32 

.33 

.33 

256 

104 

.33 

.33 

1829 

ISO 

52. 

36. 

5680 

33 

27.5 

11.5 

268 

106 

.67 

.67 

1899 

182 

.33 

.33 

5820 

34 

.67 

.67 

280 

108 

40. 

24. 

1968 

184 

.67 

.67 

5962 

35 

.83 

.83 

292 

110 

.33 

.33 

2041 

186 

53. 

37. 

6106 . 

36 

28. 

12. 

304 

112 

.67 

.67 

2115 

188 

.33 

.33 

6252 

38 

.33 

.33 

329 

114 

41. 

25. 

2191 

190 

.67 

.67 

640C 

40 

.67 

.67 

356 

116 

.33 

.33 

2269 

192 

54. 

38. 

6552 

42 

29. 

13. 

383 

118 

.67 

.67 

2346 

194 

.33 

.33 

6704 

44 

.33 

.33 

411 

120 

42. 

26. 

2424 

196 

.67 

.67 

6859 

46 

.67 

.67 

441 

122 

.33 

.33 

2504 

198 

55. 

39. 

7016 

48 

30. 

14. 

472 

124 

.67 

.67 

2587 

200 

.3.3 

.33 

7178 

50 

.33 

.33 

504 

126 

43. 

27. 

2672 

202 

.67 

.67 

7339 


























































BRICK ARCHES. 


709 


Art. 12. Rrick Arches. Since even good brick fit for large arches has 
fjir less crushing strength than good granite or limestone, and is inferior even to 
good sandstone, while its weight does not differ very materially from stone, it is 
plain that it cannot be used in arches of as great span as stone can. Some of 
those already built, and which have stood for many years, have a theoretical co¬ 
efficient of safety of hut about 3; whereas the authorities direct us not. to trust even 
stone with more than one-twentieth of its crushing load. This last, however, ap¬ 
pears to the writer to be one of those hasty assumptions which, when once ad¬ 
mitted into professional books, are difficult to be got rid of. It is his opinion that 
with good cement, and proper care in striking the centers, one-tenth of the ulti¬ 
mate strength is sufficiently secure against even the abnormal strains caused by 
the settling at crown, and rising at the haunches when the centers are struck. It 
is useless to attempt to fix limits of safety for bad materials poorly put together. 

Rem. 1. The common practice of buildiug brick arches in a series of con¬ 
centric rings, as at a c ee, Fig 12, with no other bond between them than 

that afforded by the mortar, is censured by 
authorities, on the ground that the line of 
pressure in passing from the extrados to 
the intrados tends to separate the rings, 
and thus weaken the arch by, as it were, 
splitting it longitudinally. The reason 
for using these rings, instead of making 
the radial joints continuous throughout 
the depth m n of the arch, as at b, is to 
avoid the thick mortar-joints at the back of 
the arch, and shown in the Fig. If the 
center of an arch built as at b be struck 
too soon, the soft mortar in these thick 
joints will be so much compressed as to cause great settlement at the crown, 
throwing the arch out of shape, and creating such inequality of pressure as 
might even lead to its fall, especially if flat. As a compromise between rings 
and continuous joints, they are sometimes employed together, so as to get rid of 
some of the long radial joints ; and at the same time to break at intervals 
the continuity of the rings. Thus in Fig 12. which is supposed to be brick-and- 
a-half deep, beginning at the abutment a, we may lay half-brick rings as far as 
say to e o e; then cutting away the brick o to the line e e, we may lay from 
e'etorona block of bricks with continuous radial joints, the same as at b; and 
then start again with three rings; and so on alternately. A still better, but 
more expensive, mode would be to fill e e, m n with a regular cut-stone voussoir. 

The proper intervals for changing from rings to blocks will depend upon the 
number of the rings and the depth c a of the arch ; reference being also had to 
reducing the amount of brick cutting as much as possible. 

These points can be best decided on from a drawing of a portion of the arch 
on a scale of 3 or 4 ins to a foot. Generally the rings are made only half-brick, or 
about 4 to 4.5 ins thick, as at a c; and in Brunei’s Maidenhead viaduct of two ellip¬ 
tic brick arches of 128 ft span, and 24.25 ft rise; the boldest brick arches yet at¬ 
tempted; but which have been estimated to have a co-efficient of safety of but 
three against crushing at the crown. 

So many others of from 70 to 100 ft span have been successfully built, entirely in 
rings of either half or whole brick thick, as to justify us in attaching but little weight 
to the above theoretical objection, provided first class cement be used, and time 
allowed it to become nearly or quite as hard as the bricks themselves, before 
striking the centers. Under such circumstances we should not object to a series 
of rings even 1.5 bricks thick, laid alternately header and stretcher, as at b. 

If the bricks were voussoir-shaped. that is, a little thicker at one 
end than the other, then rings a whole-brick thick could be used without any in¬ 
crease in thickness of mortar-joint at the back of each ring. Still with more 
than one ring, the radial joints would not be continuous, as at b, but broken as at 
ac. Such bricks however would be more expensive to make; and moreover, in 
order fully to answer the intended purpose, they would have to be made of many 
patterns, so as to conform to the many radii used in arches; and even to the 
radii of the different rings, when the depth of the arch required several of them. 

See foot-note, p 671. 








710 


BRICK ARCHES 


Rem. 2. Wot the bricks before laying. See last paragraph of p 870. 

Rom. 3. When the ends or faces of a brick arch are to be finished with out- 
stone vonssoirs. these had better not be inserted until some time after the 
completion of the brickwork, the hardening of the mortar, and a partial easing 
of the centers; lest they be cracked or spawled by the unequal settlements of them¬ 
selves and the bricks. 


Rem. Rrick arches, from their great number of joints are apt to settle 
much more than cut stone ones when the centers are removed, and thereby to 
derange the shape of the arch, and at times, without due care, even to endanger 
its safety, especially if it be large and Hat. When the span exceeds about 30 to 85 
ft, and particularly if flat, use only brick of superior quality in good ccnicut 
mortar. With even best materials and work we advise the young engineer not 
to attempt brick arches for railroad bridges of greater spans than about the fol¬ 
lowing. Considerably larger ones than some of them have been built, and have 
stood; but their coels of safety are not in all cases satisfactory. In this table the 
rise is in parts of the span. 


R. 

s. 

R. 

s. 

R. 

S. 

R. 

S. 

R. 

S. 

.5 

100 

l A 

88 

.225 

68 

Ye 

50 

1 .134 

35 

.4 

97| 

.29 

82 


60 

.155 

45 

1 * 

30 

.36 

931 

M 

75 

.183 

55 

1 | 

40 



On the Filbert Street Extension of the Penna R R. in Pliila, 

are four brick arches of 50 ft 1 inch span, and with the very low rise of 7 ft. They 
are *2 ft 6 ins thick, except on their showing faces, where they are but 2 ft. The 
joints are in common mortar, and about J4 inch thick. These four arches, about 
200 yards apart, with a large number of others of 26 ft span, form a viaduct. Tho 
piers between the short spans are 4 ft 3 ins thick. Those at the ends of the 50-ft 
spans, 18 ft 6 ins. The springing lines of all tho arches are about 6 to 8 ft above the 
ground. One of the 50-ft arches settled 3 ins upon prematurely striking the 
centers; but no further settlement has been observed, although the viaduct has, 
since built (1880) had a very heavy freight and passenger traffic, at from 10 to 20 
miles per hour, lioadbed, about 100 ft wide, giving room for 9 or 10 tracks. 
































CENTERS FOR ARCHES. 


711 


>'th» 

ring 

I'm. 


CENTERS FOR ARCHES. 


>y t» 
n.sr 
to 31 

eut 

■ tuit 

ifol 

liave 


Art. 1. A center is a temporary wooden structure (built lying flat, on a full 
size drawing, on a fixed platform, under cover or not) for supporting an arch 
while it is being built. It consists of a number of trusses or frames,/,/, Fig. 1, 
placed from 1 to 6 ft apart from cen to cen, and covered with a flooring l, l, of 
rough boards or planks, usually laid close, and called the sheeting or lag¬ 
ging, immediately upon which the archstones are laid. In Fig 3, the lag¬ 
ging is not laid close. There is no great economy in placing the frames very 
far apart, on account of the greater required amount of lagging, the thickness 
of which increases rapidly. For the thickness of lagging see Rem 9, p 719. The 
frames are of many designs. Thus Figs 7 and 9, pp 554, 556, are often used for 
small spans (say 15 to 25 ft), their upper timbers supporting throughout their 
length planks on edge, with their upper edges trimmed to conform to the curve 
of the arch. Fig 14, p 570, cov¬ 
ered in the same way, is some¬ 
times used for still longer spans, 
say 25 to 40 feet; also Fig 28, p 
594; Fig 31, p 595 ; and F'ig 35, p 
598, for still longer ones. 



The centers rest by the ends of 
their chords, c, upon wooden 
striking wedges w, Fig 1, 
supported Dy standards com¬ 
posed of posts p, whose tops are 
connected by cap-pieces o; 
and whose feet rest on string¬ 
ers s; the whole being braced 
diagonally as shown. 

If the ground is very firm, and 
the arch light, the standards may 
rest on it, with the interposition 
of adjusting-blocks, n, be¬ 
low the stringer, to accommodate 

irregularities of the surface of the ground, as in the Fig. These blocks should 
be somewhat double-wedge-shaped, so that by driving them the standard may 
be raised at any point in case it should settle a little into the ground. But for 
heavy arches the standards must rest on a much firmer foundation, such as short 
blocks of brickwork sunk a few feet into the ground, or some other device 
adapted to the case. Frequently projecting offsets or footings, or at times re¬ 
cesses, are provided in the masonry ot the abutments and piers for this express 
purpose; and with a view to this it is well to design the center at the same time 
as the arch. Knowing the wt of the arch the 
proper dimensions of the posts may readily be 
found by table p 459, etc. Up to spans of 50 or 60 
ft a single row of posts (one under each end of 
each frame) will suffice; but for much larger ones 
two or three rows, 2 or more feet apart may be¬ 
come expedient, as in the lower Fig 2. 

The striking or lowering-wedges 

before alluded to are for striking or lowering the 
center after the completion of the arch. They 
consist of pairs of wedge-shaped blocks, w w, at A, 

Figs 2, of hard wood, from 1 to 2 ft long, about half 
as wide, and a quarter or more as thick, (sufficient 
to lower the center from say 2 to 6 or more inches, 
according to span and other circumstances,) rest¬ 
ing on the cap o, of the standard, while the chord 
c of the frame rests on them. When the end of a 
frame is supported by two or more posts p, as at B, 

Fig 2, instead of upon one, the striking-wedges are 
sometimes made as there shown; and where B v 
is one long wedge at right angles to the abutment, 
and acting as four wedges which may all be low¬ 
ered together by blows against the end B. 

Up to spans of 60 or 80 ft, all the frames may rest on but two wedges like B v 
























































712 


CENTERS FOR ARCHES. 


each so long as to reach transversely across the entire arch. Then all the 
frames can be lowered at one operation, as described near end of Art 9. 

If we had to consider only the friction of dry wood against dry wood, the taper J 
of these wedges might be as steep as 1 vert to 3 hor, without any danger of their 
sliding upon each other of their own accord; and they would then require very 
moderate blows to start them, or even to entirely separate them, when the center 
had finally to be lowered, lint it is of the utmost importance, especially in Inigo ^ 
arches, that the centers should be lowered very slowly, otherwise 
the momentum acquired by so heavy a body as the arch in descending suddenly 
oven but 2 or 3 ins, might possibly affect its shape, or even its safety. 

Therefore the wedges should not have a taper steeper than about 1 in 6 or 8.for 
arches of less than about 50 ft span; or than 1 in 8 or 10 for larger spans. \ ertical 
lines at equal dists apart should be drawn on the long sides of the wedges as a 
guide for lowering them all to the same extent at a time ; and this should not ex¬ 
ceed in all about half an inch a day in intervals of about an eighth of an inch, for 
50 ft spans; or about .1 to .25 of an inch per day in all, for spans over 100 It. 
Slowness is especially to be recommended lii bricK strobes, not only because 
their greater number of joints exposes them to greater derangement of shape, 
but because even good brick has much less than the average crushing strength 
of good granite, limestone, or sandstoue, and therefore is far more liable than 
they to crack, or even to crush (as the writer has seen) when the strains are 
thrown almost entirely upon their edges, as described in Art 3. i or more on brick 

arches, see p 709. . 

At Gloucester Bridge, England, of first class cut stone, span 150 ft, rise 
35 ft, the centers were entirely struck within the very short space of 3 hours; and 
the crown of the arch descended 10 ins! At Grosvenor Bridge, England, 
of first class cut stone, span 200 ft, rise 42 ft, such care was taken in casing the 
centers that the crown of the arch settled but 2.5 ins. This case however was 
marked by two or three peculiarities, all of which contributed to this favorable 
result. Namely, the center instead of being a series of frames supported as usual 
by their ends, and of course involving an appreciable, although small, degree of 


sagging or settlement, consisted 
essentially of vertical and in¬ 
clined posts or struts, see Fig 3, 
looting on four temporary piers 
of masonry, 7 or 8 feet thick, built 
in the river, parallel to the abut¬ 
ments, and as long as they. These 
piers supported six frames (or 
rather six series) about 7 ft apart 
cen to cen, of such struts, footing 
on cast iron shoes. Fig 3 shows 
half of one series. Each frame 
or series consisted of four fan-like 
sets of posts, all in the same ver- 



temporary piers 
8 feet thick, built 



tical plane. The long horizontal pieces seen extending from side to side of the 
arch were bolted to the struts to increase their stiffness; and other pieces for the 
same purpose united the six series transversely. Here each strut sustains its own 
share of the weight of the archstones, and transfers it directly to the unyielding 
foundation of the pier; whereas in the usual trussed centers, the entire load rests 
upon the frames, and is finally transferred to the comparatively unstable support 
of the posts at their ends. 

The tops p of the posts of a series varied about from 5 to 8 ft apart cen to cen; 
and were connected by a continuous curved rib, rr, of two thicknesses of 4 inch 
plank, bent to conform approximately to the curve of the arch. On tikis rib were 
placed pairs of striking-wedges w like Fig 2, about 16 ins long, 10 to 12 ins wide, and 
tapering 1.5 ins, so near together (varying about from 2.5 to 3.5 fteen to cen) that 
there was a pair under each joint of the archstones, a a. On these wedges, and ex¬ 
tending over all six of the frames, were the lagging pieces l, 4.5 ins thick. 

This peculiar arrangement of the striking-wedges and lag¬ 
ging has, in large spans, great advantages over the usual one of placing them only 
at the ends of the frames. In the last the entire center and the entire arch are 
lowered together, without giving an opportunity to rectify any slight derange¬ 
ments of shape or inequality of bearing that may have occurred in the arch during 
its construction, This center, designed by Mr. Trubshaw, admits of lowering 
either the whole equally, or any one part a little more or less than the others, 
lie had much experience in large arches, and stated that during the striking he 
found that he had an arch under better control, or could humor it better, by keep¬ 
ing the haunches a little down, and the crown a little up, until near the end of 
the operation. 
















CENTERS FOR ARCHES. 


713 


'fin 


top 

tlieii 


'■liter 


fir' 

ulj 

•for 
iica; 
sa» 
i ex- 
ii.lor 


ise 
ad 
“dt 
. the 
fas 


Rem. 1. Instead of piers of masonry for supporting the feet of the 
posts, wooden cribs or piles may often be used if the arch is over water. 

The principle of supporting- even trussed frames by struts 
at points of the chord as far from the abutments as circumstances will admit of 
■rj (in addition to those at the very ends) should always be applied when possible, 
yin order to reduce their sagging to a minimum. Steps or offsets in the 
masonry of the abutments and piers may be provided tor receiving the feet 
ot such struts, when they are inclined. 

Kem. 2. Screws may be used instead of wedges for lowering centers. At 
the Pont d’Alma, Paris, ellipse of 141.4 ft span, and 28.2 ft rise, the frames were sup¬ 
ported by wooden pistons or plungers, the feet, of which rested on sand con¬ 
fined in plate-iron cylinders 1 ft in diam and height, and having near 
a the bottom of each a plug which could be withdrawn and replaced at pleasure, 
' thus regulating the outflow of the sand and the descent of the center. This de¬ 
vice succeeded perfectly, and is well worthy of adoption under arches exceeding 
ft about 60 ft span. When much larger than this the driving of the wedges on 
i® striking requires heavy blows, and becomes a somewhat awkward operation, re- 
ipe,quiring at times a battering-ram, even when the wedges are lubricated. In rail- 
$i|road cuttings crossed by bridges, the earth under the arch has been 
made to serve as a center, by dressing its surface to the proper curve, and then 
are embedding in it curved timbers a few feet apart, and extending from abut to abut, 
"kjfor supporting the close plank lagging. 

Kem. 3. All centers must yield or settle more or less under the wt 
of the arch, especially when supported only near their ends; and since the arch 
itself also settles somewhat not only when the centers are struck, but for some 
time after, it is advisable to make them at first a little higher than the finished 
arch is intended to be. This extra height, when the supports are at the ends, 
may be from 2 to 4 ins per 100 ft of span for cut stone arches (according to time 
* of strikiug, character of masonry, workmanship, etc.), and about twice as much in 
brick ones. 

Rem. 4. The proper time for striking centers is a disputed 
point among engineers, some contending that it should be done as soon as the 
arch is finished and sufficiently backed up; and others that the mortar should 
first be given time to harden. It is the writer’s opinion that inasmuch as in 
cut-stone arches the mortar joints should be very thin; and since, in such, the 
mortar is at best of very little service, it is of no importance when they are struck; 
provided the masonry backing, and the embkt up to y n Fig 2, p 698, have been com¬ 
pleted ; but that in brick or rubble, the numerous joints of both of which require 
much mortar, (which for hardness should consist largely of cement,) 3 or 4 months, 
or longer, if possible, should be allowed it to harden sufficiently to prevent undue 
compression and consequent settlement when the centers are struck. The con¬ 
tinuance of the centers need not interfere with traffic over the bridge. 

Art. 3. The pressure of archstones against a center is very trifling until after 
the arch is built up so far on each side that the joints form angles of 25° or 30° 
with the horizontal. Theoretical discussions on this pressure make no allowance 
for accidental jarrings in laying the archstones, or by the accumulation ot material 
ready for use, laborers working on it, &c. Without going into any detail, we merely 
advise on the score of safety not to assume it at less than about the following pro¬ 
portions or ratios to the weight of the entire arch, namely, in a semicircular arch 
.47; rise .35 span, .61; rise .25 span, .79; rise .2 span, .86; rise .167 span, or less, 1, 
or equal to the wt of the arch. This gives the pressure of a semicircular arch 
upon its centers rather less than half its wt. The wt of tlie centers 
themselves when supported only near the ends must be considered as part 
of the load borne by them. 

Art. 3. We have seen that as an arch aaa is being gradually built upward on 
both sides, after passing the points e, e, Fig 4, where its joints form angles a s e, of 
about 30° with the horizontal a a, the arch begins to press more and more 
upon the centers; thereby tending to flatten them at the haunches, as shown at h 
in the dotted line; and consequently to raise them at the crown, as shown at c. 
But as the building goes on still higher, the added stones press much more heavily 
upon the centers than those below had done, and thereby tend to a final derange¬ 
ment of the centers just the reverse of that caused by the lower ones; namely to 
depress them at the crown a, as at o; and consequently to raise the haunches as 
at n; and this the more because the upper stones actually tend to lift or ease the 
lower ones from the lagging. In some cases where this tendency has been in¬ 
creased by forcing- the keystones into place by too hard driving, the lagging 
under the haunches could be drawn out without any trouble before the centers 
were eased at all. On striking the centers this tendency to sink at crown and 




714 


CENTERS FOR ARCHES 


rise at haunches is very apt to exhibit itself more or less dangerously in the arch- i 
stoues themselves, as iu Fig 5, causing those near the crown to press very hard 
together at the extrados, and to separate from each other at the intrados; while 
near the hauuches the reverse takes place. Heuce the angles of the stones are 
frequently split and spawled off near c and h by this unequal pressure. These 



derangements are of course much more likely to be serious in high arches than 
in flat ones, especially if their spandrels are not sufficiently built up before 
lowering the centers. 

In the Grosvenor bridge, before alluded to, of 200 ft span, this dangerous excess 
of pressure near c and h was prevented by covering the skewback joint of the 
springing course at each abutment with a wedge of lead 1.5 ins thick at the in¬ 
trados of the arch, and running out to nothing at the extrados. Beside this a 
strip 9 ins wide of sheet lead was laid along the intrados edge of every joint until 
reaching that point at which it was judged that the line of pressure' would pass 
from the intrados to the extrados; after which similar strips were laid along the 
extrados edges of the joints, up to the crown. Hence when the centers wero 
struck, this excess of pressure merely compressed the lead, and was thus enabled 
to distribute itself more evenly over the entire depth of the joints. See Trans 
Inst Civ Eng London, vol i. 

At the bridge at Neuilly, France (of 5 elliptic arches of 120 ft span, 
and 30 ft rise), the centers were so radically defective in design that the arches 
sank 13.25 ins at crown during the time of building; and 10.5 ins more during 
and immediately after the striking; or say 2 ft in all Their construction made 
the striking very tedious and hazardous; greatly endangering the lives of the 
workmen and the existence of the arches. Some of the joints at the extrados 
at the haunches opened an inch each ; and those at the intrados of the crown .25 
of an inch. By the exercise of great care and humoring in lowering the centers, 
these openings were much reduced. 


Rom. 1. Chamfering the edges of the arehstones diminishes 

the danger of their spawling off from unequal pressure; as does also the scrap* 

ing out of the mortar of the joints for an inch or two in depth be¬ 
fore striking the centers. 



Rem. 2. It is evident that in order to prevent, or at least to diminish the 
alternate derangements of the center, those of its web members which at first 
acted as struts near the haunches, Fig. 4, to prevent them from sinking as at 
h , must afterwards act as ties to prevent them from rising as at n; while those 

which at first acted as ties near 
the crown a, to prevent it from 
rising as at c, must afterwards 
act as struts to prevent it from 
sinking as at o. In other words, 
the principle of counter- 
bracing must be attended 
to as well in a frame or truss 
for a center, as in one for a 
bridge. If the web members 
are on the Warren or simple 
triangle system, as in Fig 23, 
p 589, this may be effected by 
making each member a tie- 
strut; or the Pratt, or the 
. , Howe system, Fig 35, p 598, 

may be used. * * 


Art. 4. From the foregoing it is plain that a simple unbraced wooden 






















CENTERS FOR ARCHES. 


715 


irch, or curved rib is, on account of its great flexibility, about as unfit a form 
i 3 c , j chosen for a center, except for very small spans, where a great propor- 
“ onal depth of rib can be readily secured. Still the writer has seen it. used for a 
, at-stone semicircular arch of 35 ft span, with archstones 2 ft deep. Fi°- 6 shows 
*1 “ e r . ib r and the arch, a a, drawn to a scale. Each rib consisted of two thicknesses 
f 2 inch plank in lengths of about 6.5 ft, treenailed together so as to break joint 
s at B. Each piece of plank was 12 ins deep at middle, and 8 ins at each end; 
tie top edge being cut to suit the curve of the arch The treenails were 1.25 ins 
l diam; and 12 of them showed to each length. These ribs were placed 17 ins 
part from cen to cen, and steadied together by a bridging piece of inch board, 13 
is long, at each joint of the planks, or about 3.25 ft apart. Headway for traffic 
eing necessary under the arch, there were no chords to unite the opposite feet 
f the ribs. The ribs were covered with close board lagging, which also assisted 
i steadying them together transversely. As the arch approached about two- 
nirds of its height on each side, the ribs began to sink at the haunches, as at h, 
ig 4; and to rise at the crown, as at c. This was rectified by loading the crown 
•ith stone to be used in completing the arch ; which was then finished without 
irtlier trouble. 

A still more striking? example of the use of a simple unbraced 
ooden rib, was in the old National Turnpike bridge over Wills Creek, Virginia, 
his bridge, of which one arch with 


* 



s center is shown in Fig 7 drawn 
j> a scale, consisted of two elliptic 
at stone arches 26.5 ft wide across 
.» aadway, and of 60 ft span, and 15 
rise. The archstones were 3 ft 
eep at crown, and 4 ft deep at 
tewbacks. Each frame of 
lie center was a simple rib 6 
is thick, composed of three thick- 

esses of 2 inch oak plank in different lengths (about 7 to 15 ft) to suit the curve, 
nd at the same time to preserve a width of about 16 ins at the middle of each 
;ngth, and 12 ins at each of its ends. The thicknesses were well treenailed to¬ 
gether, breaking joint and showing from 10 to 16 treenails to a length. 

Here, as in Fig 6, there were no chords, owing to the violence of the floods in 
lie creek. These ribs were placed 18 ins from cen to cen, and steadied against 
ne another by a board bridging-piece 1 ft long, at every 5 ft. These were of 
ourse assisted by the lagging. 

When the archstones had approached to within about 12 ft of each other near 
tie middle of the span, the sinking at the crown, and the rising at the haunches 
ad become so alarming that pieces of 12 X 12 oak, 00, were hastily inserted at 
itervals, and well wedged against the archstones at their ends. The arch was 
lien finished in sections between these timbers, which were removed one by one 
s this was done. 

Rem. 1. Suds instances of partial failure are very instructive, 
t is indeed by such, rather than by theoretical deductions, that the proper dimen- 
ions are arrived at in a vast number of cases pertaining to engineering, ma- 
hinery, Ac.* Thus we might with entire confidence of no serious mishap, apply 
ibs of the foregoing dimensions to spans only half as great. 

Rem. 2. Assuming the rib-planks to be 12 ins wide, it would, as a matter of 
etail, be better to make them about 10 ins wide at the ends instead of the 8 ins 
Fig 6 making top curve 2 ins. To secure this, their lengths, depending on the 
'aclius of the rib, must not exceed those in the following' table: 




Rad 

of Arch. 

Greatest Length. 

Rad 

of Arch. 

Feet. 

Feet and Ins. 

Feet. 

5 

2 “ 5 

30 

10 

3 “ 4 

35 

15 

4 “ 2 

40 

20 

5 “ 0 

45 

25 

5 “ 9 

1 

50 


Greatest Length. 


Feet and Ins. 


6 

7 

7 

7 

8 


4 

0 

6 

10 

2 


# The young engineer should make and preserve full notes in detail of all such as may fall within 
s notice; and if the professional journals would do the same thing in regard to failures which are 
instantly occurring, they would greatly increase the value of their papers. 
























716 


CENTERS FOR ARCHES. 


If cut VA times as long as this table, they will be very approximately 8 in 
wide at ends ; or each will on top curve 4 ins. 


ir 


2 


a 

=t= 


Art. 5. In eases where all possible headway is essentia 

during the building of the arch, as in the two foregoing ones, the writer wool 

suggest the expedient rudely illustrated b 
Fig 8; namely to plaee the center 
above the arch, instead of belov 
it; and after the arch is completed in secj 
tions, a a, instead of lowering 1 the cen | 
ters, to take them apart. The cen 
ters might resemble in principle Fig 35^3 
p 598. 

Fig 8 is a transverse section through par 
of the center, and of the arch a a. Her i 
rc,rc, r c, are frames of the center say 5 or ; 
ft apart; and of any depth and constructioi 


a 

=£= 


Fig 8. 


2 


e 


whatever that may be necessary to insure absolute safety, and 11 is the lagging 
Having built the arch from abutment to abutment in a series of sections a, a, a, ne 
eessarily separated say a foot or more by the deep frames, we may take the center 
apart, and then till in the narrow intermediate sections upon a lagging suspended 
by iron rods from the already completed sections. Good concrete might be uset 
for these narrow sections. In some cases it might be well to use deep plate 
iron ribs of I section, resting the lagging on the lower flange. Part of th< 
web might be left remaining embedded in the masonry; and the upper part anc 
both flanges removed after the arch is finished. 


Art. 6. Centers with hor chords c c Fig 9 are objectionable (notwith 
standing their strength) in large spans of great rise, as on right side of the Fig, oi 

.c ..it of the excessive h ngt 

required for the web members 
and hence it will in such case: 
usually be^found expedient t< 
adopt something analogous t< 
w hat is show n on the left hanc 
of the Fig. Here a truss/, shorte; 
and shallower than that on th« 
right hand,'is substituted for th< 
latter. At its ends provision must 
be made for supporting not only 
itself, but the archstones below 
it. As the pressure of these low*- 
er archstones is comparatively 
small, this may usually be effected 
by resting the end of the framt 
/ upon another and shallower frame o a. This may in large spans be aided by 
either inclined or vertical struts, either single or braced together; or as the trestles 
on p 755. Sometimes one shallow truss like / is sustained upon another truss 
throughout its entire length. The striking-wedges for these various supports may 
be placed at either their tops or their feet, as may be most convenient. 



Art. 7. For flat arches of 10 feet clear span, a mere board o c 

Fig 10, 12 ins deep, by 1.5 ins thick, with another piece c of the same thickness 

on top of it, trimmed to the Curve, and con¬ 
fined to o o by nailing on tw T o cleats of nar 
row board, will answer every purpose, w r ith 
intervals of 18 ins from cen to cen. If the 
upper piece also is as much as 12 ins deep at 
its center, the clear span may be extended 
to 15 ft. 4 

For spans of 10 to 15 ft, and of any 
rise, tw r o thicknesses of plank from 1 to 2 ins 
thick according to span; 8 to 12 ins wide at 
middle of each piece, in lengths as per table, Rem 2, Art 4, well nailed or 
spiked together, according to span, breaking joint as in Fig 6, will answer for 
distances of 2 to 3 ft apart cen to cen. For greater dists apart increase the thick¬ 
ness of the planks proportionally. 

If the centers have to be moved from place to place, to serve 
for other arches, then, to preserve them from injury in handling, their feet should 
be united by nailing on one or both sides of each frame a chord piece of about 1 


























































CENTERS FOR ARCHES. 


717 


.a 


it jach board; and also a vertical piece or pieces of the same size from the center 
f the chord to the top of the frame. 

Even when they are not to he moved, the chord pieces are useful 
•ven in so small spans, inasmuch as they render the striking easier, by not allow- 
J , ig the feet of the ribs to give trouble by spreading outward and pressing against 
“ iie abutments. 

' r For spans of 15 to .TO ft, and for any rise not less than one sixth of the 
pan, the following dimensions, varying with the span, may be used for distances 

* part of 3 ft from cen to cen. 
et ee Fig 11. For the bow b, 

wo thicknesses of 1 to 2 inch 
V; lank from 9 to 12 ins wide 
t the middle; and from 7 to 
11 0 ins at each end, well spiked 
61 agether breaking joint as at B, 

,r 'ig6. For the chord c,two 
i#l hicknesses of plank of same 
11 ize as the bow at its middle; 

* laced on outsides of bow, and 
f| ? ell spiked to its ends. A 

r ertical v, in one piece as 

* bde as a bow plank, and twice 

( '|s thick. Its top is placed under llie bow, and is confined to it by two pieces, o , q, 
jf bow plank twice as long as the bow plank is deep, and spiked to both v and the 
qw. The foot of v passes between the two thicknesses of the chord c, and is 
piked to them. Two oblique tie-struts, s, each of two pieces of bow 
lank, outside of the bow and vertical v; footing against each other; and spiked 
p bow and v. These with v divide the bow into 4 parts. 

. Rem. 1. The above dimensions are suitable to a rise of one sixth. If the 
' ise is one fourth, the thickness only of the planks may be reduced one third 
art; and for a rise of one third or more, we may reduce to one half. 



Rem. 2. If in the larger of these spans the struts s should show any ipcli- 
t iation to bend sideways, nail on some pieces t from frame to frame. Also in the 
m irger ones with rises exceeding one third, insert four double struts s, instead 
te f two; thus dividing the bow into 6 parts, as at left side of Fig. 11. For spans of 
ill 5 to 35 ft, add also two struts like a a, of same size as v. 

in Art. 8. For spans greater than about 30 ft. the writer believes 
a.* hat as a general rule (liable to modifications according to the judgment of the 
il ngineer In charge) the following ideas will lead to safe practice. Namely, to 
dopt a bowstring truss with a simple Warren or triangular web, as at / on ,the 
* eft side of Fig 9. The bow to rest on the chord, and each to be of a single thick- 
‘1 mss. The web members (especially in large spans) to be also of single thickness, 
:e nd placed below the bow, resting on the chords, and well strapped to both, so as 
a o act as either ties or struts. In smaller spans the web members may each be in 
wo thicknesses, one bolted or treenailed to each side of the bow and chord. Other 
nodes will suggest themselves; but we have not space for such details, 
is Or a web of the Howe, or of the Pratt system, as on the right side of Fig 9 may be 
aj ised. But in reference to both of these it may be remarked that the use of 
ong iron rods in centers of large spans is highly objectionable, owing 
o the different rqtes of expansion between iron and w ood. Therefore if these 
u ystems are used, all the members should be of wood. The lattice may be used. 
s Even when the rise of the arch exceeds .25 of the span, it is better not to let 
H.hatof the centers exceed that limit; but adopt the expedient shown at 
b he left side of Fig 9, with a rise of about one sixth of the span. 
lL Rem. 1. To fiix on the number of web triangles in a Warren 
!! ;russ or frame for a center, find the square root of the span, and to it add one 
J |jenth of the span. Divide their sum by 2, and call the quotient n. Divide the 
4 pan by n. If this quotient is a whole number use it; or if the quotient is partly 
lecimal, use the whole number nearest to it, as a distance in feet to be stepped off 
f long the chord; thus dividing the chord into a number of equal parts. All the 
mints thus found on the chord, are the places for the feet of the triangles. 
ll ! \ T ext, from half w T ay between each two of these points, draw vertical lines to the 
11 iow. The points thus found along the bow, are the places of the tops of the 
1 riangles. This rule will be used in connection with the following Table of Areas 
’ >f Bows, as the two are dependent on each other. 

In large arches the timber of the bow should not be wasted by 
e rimming its upper edges to the curve of the arch, but should be left straight; and 
1 eparate pieces so trimmed, like c in Fig. 10, should be spiked on top of them. 













718 


CENTERS FOR ARCHES. 


The transverse area of the how, in square inches, may be taken froE 

the following table; and may in practice be assumed to be uniform throughou 
its entire length ; which in fact it is quite approximately. See Rem 2. 


TA ISLE FOR BOWSTRING CENTERS. 


Table of areas in square inches at the crown of each Bow, of properly 

The frames t 


trussed Bowstring frames for centers of stone or brick arches. T 
be placed 5 feet apart from cen to cen. With these areas, the combined weigh 
of arch, center (of oak), and lagging, will in no case in the table strain the Bo 
at crown of the greatest spans quite 1000 lbs per square inch ; diminishing g 
uallv to 600 or 700 lbs in the smallest spans, which are more liable to casualti 
The depths of the archstones may be taken fully equal to those in our table, 
697. Although centers of moderate span are usually made of white or yello 
pine, spruce, or hemlock, all of which are considerably lighter than oak, we hav 
for safety assumed them to be of oak, in preparing our table. 

For spans of from 10 to 20 feet use the same sizes as for 20 feet. 


Origina 


.5 


.4 


Rise in parts of the Span. 

.35 .3 .25 .2 


.15 


.1 


Span 

in feet. 


Areas of transverse section of 

in square inches. 

Bow, 


20 

14 

17 

19 

21 

24 

29 

38 

59. 

25 

18 

22 

25 

28 

33 

40 

53 

80 1 

30 

23 

28 

32 

37 

43 

51 

71 

103 

35 

28 

34 

40 

45 

54 

64 

87 

125 

40 

34 

41 

48 

55 

65 

77 

106 

150 

45 

40 

49 

57 

65 

76 

92 

126 

175 

50 

47 

57 

66 

76 

89 

107 

146 

203 

55 

53 

64 

75 

87 

102 

121 

166 

233 

60 

60 

73 

85 

99 

115 

135 

187 

263 

65 

68 

81 

95 

110 

129 

151 

209 

294 

70 

75 

90 

105 

122 

143 

168 

233 

325 

•75 

83 

99 

115 

133 

157 

184 

256 

357 

80 

91 

108 

125 

145 

171 

201 

279 

390 

85 

99 

117 

136 

157 

1S5 

218 

302 

423 

90 

108 

127 

147 

169 

199 

235 

325 

457 

. 95 

115 

136 

158 

181 

214 

252 

348 

490 

100 

123 

146 

169 

194 

229 

270 

372 

524 

110 

133 

166 

191 

219 

260 

307 

420 

592 

120 

155 

187 

213 

246 

291 

345 

470 

660 

130 

172 

208 

237 

274 

323 

384 

520 


140 

190 

230 

263 

303 

357 

424 

572 


150 

209 

252 

289 

333 

393 

466 



160 

229 

276 

315 

365 

430 

509 



170 

250 

299 

343 

399 

469 




180 

272 

323 

373 

435 

511 




190 

294 

347 

403 

472 





200 

318 

372 

435 

509 






Rem. 2. The square root of any of these areas gives in inches the side o.< 
a square bow of that area. The distances apart of the triangles which fori , 
the web of the frame, having first been found by Rem 1 (for said Rem and thi 
table are dependent on each other), the above areas for bows 5 ft apart from cei 
to cen, suffice not only to resist the pressure along the bow, but also, as sqium 
beams, to sustain with a safety in no case less than about 5, the load of arch 
stones resting upon them between the adjacent tops of two triangles; and witl 
very trifling deflections. It is therefore unnecessary to deepen the ribs for tha 
purpose; although it may be done (preserving the same area) in case consider 
ations of detail should render it desirable. 

As before suggested, it will generally be best, in spans exceeding 30 or 40 ft, t< 
give the bow a rise not exceeding about one fifth or one sixth of the span ; ant 
to support the frames as at/, Fig 9. 

The size of the chord may be the same as that of the bow; and like i 
uniform from end to end; care however being taken that it be not materially 
weakened by footing the bow upon its ends; or (when too long for single tim¬ 
bers) by the splicing necessary to prevent its being stretched or pulled apart bj 





















CENTERS FOR ARCHES. 


719 


if,he thrust of the bow. When, however, the chord can be placed at, or a little 
jelow the springs of the arch, all danger of this kind may be avoided by simply 
vedging its ends well against the faces of the abutments. 

As to the size of the web members, when a bowstring truss is 
fully loaded on top of the bow, (as is approximately the case with a center 
ind its archstones,) the strains on the web members are quite insignificant, and 
irise chiefly from the weight of the center itself; but while it is being 1 so 
i'll loaded, they are not only greater, but are constantly changing, not only in 
| imount, but also in character—being at one period compressive, and at another 
ensile. 

^ Hence it would be very tedious to calculate the dimensions of the web members. 
a l fortunately the necessity for doing so is in a great measure obviated by the fact 
o hat a center being but a temporary structure, the timber composing it is not ulti- 
4 , nately wasted if a greater quantity of it is used than is absolutely required, 
doreover facility of workmanship is secured by not having to employ timbers 
>f many different sizes. 

Hence the writer will venture to suggest, entirely as a rule of thumb, to give 
5 ?ach web member half the transverse area of the bow, 

aking care to make each of them a tie-strut. 

Rem. 3. As to details of joints, we refer to the Figs on pages 611, 
13; merely suggesting here the use of long and wide iron shoes where timbers 
ire subjected to great pressure sideways. 

Rem. 4. To prevent the thrust of the bow when its rise is small, from split- 
ing off the ends of the chords, the two may be united by many more bolts than 
ire employed in roof trusses, Ac, where only one is generally placed near each end 
>f the chord. But they may when required be inserted at intervals extending to 
nany feet from the ends. They should have strong large washers; and may have 
ibout the same inclination as the shortest web member. 

Another way of securing the same end in smaller spans, is by completely en- 
asing the two sides of the bow and chord, to a distance of a few feet from their 
>nds, in short pieces of board or plank spiked to both of them, and having about 
he same inclination as just suggested for bolts. 

Rem. 5. Build up both sides of the arch at once, in order to strain the cen- 
ers as little as possible. 

Rem. 6. When a bridge consists of more than one arch, and they are to be 
milt one at a time, there must be at least two centers; for a center must not 
>e struck until the contiguous arches on both sides are finished, for fear of over- 
urning the outer unsupported pier. Therefore if there are but two arches, they 
nust be built at once, requiring two centers. 

Rem. 7. Always use supports either vertical or inclined (and pro¬ 
dded with striking-wedges) under the frames, and intermediate of the end sup- 
>orts, when possible; even if they can extend out but a few feet from the abut- 
nents, as at the left side of Fig 9. 

Rem. 8. The weight of large centers and their lagging is greater 
or flat arches than for high ones of the same span; and also approaches nearer 
o that of the supported arch. 

Rem. 9. Thickness of lagging. The following table gives thicknesses 
vhich will not bend more than an eighth of an inch under the weight of any 
jrobable archstones adapted to the respective spans ; and generally not 
o much. 

TABLE OF LAGGING.— Original. 


^ Distance apart 
I, of frames, 


In the clear. 


Feet. 

6 

5 

4 

3 

2 


Span of center in feet. 


10 . 


20 . 


50. 


100 . 


150. 


200 . 


Thickness of close lagging not to bend more than % inch. 


Ins. 
4K 


rk 

I 


Ins. 

i 

i % 


Ins. 

5 

3 % 


2 


Ins. 

5% 

4 

3 

2 


With thicknesses three quarters as great as these, the bending may reach 
a full quarter inch ; which may be allowed in dists apart of 3 or more ft. 

Rem. lO. Centers are framed, or put together, (like iron bridges) on a 
firm, level temporary floor or platform, on which a full-size drawing of a frame is 


















720 


CENTERS FOR ARCHES. 


As each frame is finished, it is removed to its place on the piers oi 

Iphia, 

ressed 
>, and 


first made, 

abuts. 

Art. 9. TIio WissahicUon Bridge of the Reading R R, at Philadelphia 

has five arches of 65 ft span, 23 ft rise, 28 ft wide (archstones 3 ft deep, with dresse 
beds and joints, in cement mortar); with four cutstone piers 9.5 ft thick at top, 
from 35 to 50 ft high. It contains about 15400 cub yds of masonry.* Each center 
consisted of 7 frames or trusses of hemlock timber, of the Bowstring pattern, with 
web-memhers ; and as nearly as may be, of the same span and 
They were placed 4.5 ft apart from center to center; and were 

supported near each end /, Fig 12 
(a transverse section to scale) by 


lattice (p 596) 
rise as the arches. 



hemlock post p, 12 ins square. 


The bow was of two thicknessef 
bb of hemlock plank, 6 ins apart 
clear, in lengths of 6 ft, with their 
upper edges cut to suit the curve 
of the arch. Each piece was 4 ins 
thick, by 13.5 ins deep at its middle 
and 12 ins at its ends. These piece? 
did not break joint; but at each 
joint were four % inch bolts, with 
nuts and washers, uniting them 
with chocks or filling-in pieces 
The bow, bb, footed on top of the ends of the chords f; and the angle formed bj 
their meeting (seen only in a side view) was (for about 2.5 ft horizontal and 5.5 fl 
vertical) filled up solid with vertical pieces, to afford a firmer base for resting th<j 


frame on n ; beyond which it extends (in a side view) about 18 ins. 

Tile fhords/were of two thicknesses of 4X12 hemlock plank, 6 ins apari 
clear, and most of them in two or three lengths; breaking joint, and with two % 
inch bolts, with nuts and washers, at each joint, for bolting them together, and t< 
filling-in pieces. The web member* of each frame were 26 lattices, o, ol j 
3 X 12 inch hemlock,crossing each other about at right angles, at intervals of abou 
3.5 ft from center to center, and passing between the two thicknesses bb of the bow ! 
and//of the chords. A few of the lattices were in two lengths, and the joints wert 
not at the crossings. The lattices were connected at each crossing by two hard woot I 
treenails 9 ins long, and 2 ins diam ; and one such, IS ins long, passed through tin 
intersection of each end of a lattice with a bow or chord. The first lattice foot 
about 4 ft from the end of a chord. They do not extend above the. top of the bow , 
All the spaces between the two thicknesses of bow or chord, where not occupied In 
the ends of lattices, were completely filled by chocks, well spiked. 

Each frame contained about 360 cub ft of timber; and weighed about 11 
tons. They were very flexible laterally until in place, and braced together by • 
transverse horizontal planks spiked to their chords; and by 5 others above them' 
spiked to the lattices. 

Until the keystones were placed, all the joints of the frames continued tight, unde 
the pressure from the arch, and from the unfinished backing to the height of abou 
14 ft above the springing line; but after the keystones were set, all the joints of th 
chords alone opened from .25 to .75 of an inch ; and at the same time the lagging un j 
der the haunches oi the arches became slightly separated from the soffit of the masonry I 

Each center sank but a full inch at the middle, under the pressure fron 
the arch and 14 ft of backing. 

The portion of the bridge above the piers was about two thirds completed befor j 
the centers were struck. 

There was one w'cdgfe w, w, (32.5 ft long, of 12 X 12 inch oak) under eac ' 
end of a center. It was trimmed to form 7 smaller ones w, w, each 4.5 ft long, an 
tapering 7 ins ; one under each end of each frame /. They played between tapere 
blocks a, a, of oak, 2 ft long, 1 ft wide, let 1 inch into the cap c, or into the piece i 
on which last the frames /,/, rested. The sliding surfaces were well lubricated wit! 
tallow when put in place. 

The wedges w ere struck with ease, at one end of a center at a time, by a 
oak log battering-ram 18 ft long, and nearly a ft in diam, susperlded by ropes, an 
swung and guided by 4 men. They generally yielded and moved several inches a 
the second blow with a 3 or 4 ft swing. Although each wedge was loosened entirel 
within 2 or 3 minutes, thus lowering the centers very suddenly, yet on account of th 


* This hridge, finished without accident, in 1882, reflects much credit on the late William Loren 
Esq, Ch. Eng; on Mr. Charles W. Buchholr, Assistant in Charge; and on the skilful aud euerget 


contractors, William & James Nolan, of Heading, Henna. These last most cordially assisted tl 


writer in making observations during the entire progress of the work. 






































CENTERS FOR ARCHES. 


721 


od character of the masonry, not the slightest crack of a mortar joint could after- 
irds be detected in any part of the work. After three days the average sinking of 
e keystones was only .35 of an inch ; the least was %; and the greatest % of an 
a ch. The heads and feet of the posts p compressed the hemlock caps c, and the 
Ils, about % of an inch each, showing that for arches of this size the caps and sills 
id better be of some harder wood, as yellow pine or oak; although probably the 
j mpression was facilitated by the large mortices, 3 by 12 ins, and 6 ins deep. 


m 

eii 

srl 

4ii 

'It 


«il 

E 

i 

'••i 

£ 

lr 

J 

t< 

8 

>al 

)» 

st 

’In 

ot 

II 

b] 

t 

f 


47 




ie 

m 

ft 

118 

T 






722 


RAILROAD CONSTRUCTION 






if 

RAILROADS. 


RAILROAD CONSTRUCTION. 


TABLE OF ACRES REQUIRED per mile, an<l per 100 feei 

lor dillereut widths. 


Width. 

Feet. 

Acres 

per 

Mile. 

Acres 
per 
100 Ft. 

Width. 

Feet. 

Acres 

per 

Mile. 

Acres 

per 

100 Ft. 

Width. 

Feet. 

Acres 

per 

Mile. 

Acres 
per 
100 Ft. 

Width. 

Feet. 

Acres 

per 

Mile. 

Acre 
per 
100 F 

1 

.121 

.002 

26 

3.15 

.06 i 

52 

6.30 

.119 

78 

9.45 

.171 

2 

.242 

.005 

27 

3.27 

.062 

53 

6.42 

.122 

79 

9.58 

.18] 

3 

.364 

.007 

28 

3.39 

.064 

54 

6.55 

.124 

80 

9.70 

.IS- 

4 

.485 

.009 

29 

3.52 

.067 

55 

6.67 

.126 

81 

9.82 

.1S( 

5 

.606 

011 

30 

3.64 

.069 

56 

6 79 

.129 

82 

9.94 

,18f 

6 

.727 

.014 

31 

3.76 

.071 

57 

6.91 

.131 

M 

10. 

.1S{ 

7 

.848 

.016 

32 

3.88 

.073 

% 

7. 

.133 

83 

10.1 

.19( 

8 

.970 

.018 

33 

4.00 

.076 

58 

7.03 

.133 

84 

10.2 

.19 1 

M 

1 . 

.019 

34 

4.12 

.078 

59 

7.15 

.135 

85 

10.3 

•195 

9 

1.09 

.021 

35 

4.24 

.080 

<0 

7.27 

.138 

86 

10.4 

.197 

10 

1.21 

.023 

36 

4.36 

.083 

Cl 

7.39 

.140 

87 

10.5 


n 

1.33 

.025 

37 

4 48 

.0S5 

62 

7.52 

.142 

88 

10.7 

.202 

12 

1,46 

.028 

38 

4.61 

.087 

63 

7.64 

.145 

89 

10.8 

201 

13 

1.58 

.050 

39 

4.73 

.090 

64 

7.76 

.147 

90 

10.9 

.207 

14 

1.70 

032 

40 

4.85 

.092 

65 

7.88 

.149 

% 

11. 

.209 

15 

1.82 

.03 4 

41 

4.97 

.094 

66 

8. 

.151 

91 

11.0 

.209 

16 

1.94 

.037 

M 

5. 

.094 

67 

8.12 

.154 

92 

11.2 

.211 

H 

2. 

.038 

42 

5.09 

.096 

68 

8.24 

.156 

93 

11.3 

.213 

17 

2 06 

.039 

43 

5.21 

.099 

69 

8.36 

.158 

94 

11.4 

.216 

18 

2.18 

.041 

44 

5.33 

.101 

70 

8.48 

.161 

95 

11.5 

.218 

19 

2.30 

.014 

45 

5.45 

.103 

71 

8.61 

.163 

96 

11.6 

.220 

20 

2.42 

.040 

46 

5.58 

.106 

72 

873 

.165 

97 

11.8 

,22i 

21 

2.55 

.018 

47 

5.70 

.108 

73 

8.85 

.168 

98 

11.9 

.221 

22 

2.67 

.051 

48 

5.82 

.110 

74 

8 97 

.170 

99 

12. 

.227 

23 

2.79 

.053 

49 

5.94 

.112 

M 

9. 

.170 

100 

12.1 

.230 

24 

2.91 

.055 

H 

6. 

.114 

75 

9.09 

.172 




% 

3. 

.057 

50 

6 06 

.115 

76 

9.21 

.174 




25 

3.03 

.057 

51 

6.18 

.117 

77 

9.33 

.177 




































RAILROAD CONSTRUCTION 


723 


Table of grade* per mile, and per lOO feet nieasnred hori¬ 
zontally, and corresponding to different angles of incli¬ 
nation. 


;bo a 
<v — 

2 S 

Feet per 
mile. 

Feet pei 
100 ft. 

Deg. 

Min. 

Feet per 
mile. 

Feet pet 
100 ft. 

Deg. 

Min. 

Feet pei 
mile. 

Feet per 
100 ft. 

S? 3 
a 55 

Feet per 
mile. 

Feet per 
100 ft. 

0 1 

1.536 

.0291 

0 45 

69.11 

1.3090 

1 58 

181.3 

3.4341 

3 26 

316.8 

5.9991 

2 

3.072 

.0582 

46 

70.61 

1.3381 

2 0 

184.1 

3.4921 

28 

319.8 

6.0579 

3 

4.608 

.0873 

47 

72.18 

1.3672 

2 

187.5 

3.5506 

30 

322.9 

6.1163 

« 4 

6.141 

.1164 

48 

73.72 

1.3963 

1 

190.6 

3.6087 

32 

326.0 

6.1747 

5 

7.680 

.1455 

49 

75.26 

1.4254 

6 

193.6 

3.6669 

34 

329.1 

6.2330 

6 

9.216 

.1746 

50 

76:80 

1.4545 

8 

196.7 

3.7250 

36 

332.2 

6.2914 

7 

10.75 

.2037 

51 

78.33 

1.4837 

10 

199.8 

3.7833 

38 

335.3 

6.3498 

8 

12.29 

.2328 

52 

79.87 

1.5128 

12 

202.8 

3.8416 

40 

338.4 

6.4083 

* 9 

13.82 

.2619 

53 

81.40 

1.5419 

14 

205.9 

3.8999 

42 

341.4 

6.4664 

« 10 

15.36 

.2909 

51 

82.91 

1.5710 

16 

208.9 

3.9581 

44 

344.5 

6.5246 

I! 11 

16.90 

.3200 

55 

84.47 

1.6000 

18 

212.0 

4.0163 

46 

347.6 

6.5832 

12 

18.43 

.3491 

56 

86.01 

1.6291 

20 

215.1 

4.0746 

48 

350.7 

6.6418 

13 

19.96 

.3782 

57 

87.51 

1.6583 

22 

218.1 

4.1329 

50 

353.8 

6.7004 

11 

21.50 

.4073 

58 

89.08 

1.6873 

24 

221.2 

4.1911 

52 

356.8 

6.7583 

15 

23.04 

.4364 

59 

90.62 

1.7164 

26 

224.3 

4.2494 

54 

359.9 

6.8163 

16 

24.58 

.4655 

1 

92.16 

1.7455 

28 

227.1 

4.3076 

56 

363.0 

6.8751 

17 

26.11 

.4916 

2 

95.23 

1.8038 

30 

230.5 

4.3659 

58 

366.1 

6.9339 

18 

27.64 

.5237 

4 

98.30 

1.8620 

32 

233.5 

4.4242 

4 

369.2 

6.9926 

19 

29.17 

.5528 

6 

101.4 

1.9202 

31 

236.6 

4.4826 

5 

376.9 

7.1381 

i 20 

30.72 

.5818 

8 

104.5 

1.9781 

36 

239.7 

4.5409 

10 

384.6 

7.2842 

21 

32.26 

.6109 

10 

107.5 

2.0366 

38 

242.8 

1.5993 

15 

392.3 

7.4300 

22 

33.80 

.6400 

12 

110.6 

2.0948 

40 

245.9 

4.6576 

20 

400.1 

7.5767 

23 

35.33 

.6691 

14 

113.6 

2.1530 

42 

248.9 

4.7159 

25 

407.8 

7.7231 

21 

36.86 

.6982 

16 

116.7 

2.2112 

41 

252.0 

4.7742 

30 

415.5 

7.8701 

25 

38.40 

.7273 

18 

119.8 

2.2691 

46 

255.1 

4.8325 

35 

423.2 

8.0163 

26 

.39.91 

.7561 

20 

122.9 

2.3277 

48 

258.2 

4.8908 

40 

431.0 

8.1625 

27 

41.47 

.7855 

22 

126.0 

2.3859 

50 

261.3 

4.9492 

45 

438.7 

8.3087 

28 

43.01 

,81 46 

24 

129.1 

2.4441 

52 

264.3 

5.0075 

50 

446.5 

8.4551 

29 

44.51 

.8436 

26 

132.1 

2.5023 

51 

267.4 

5.0658 

55 

454.2 

8.6021 

30 

46.08 

.8727 

28 

135.2 

2.5604 

56 

270.5 

5.1241 

5 

461.9 

8.7489 

31 

47.62 

.9018 

30 

138.3 

2.6186 

58 

273.6 

5.1824 

5 

469.6 

8.8951 

32 

49.16 

.9309 

32 

141.3 

2.6768 

3 

276.7 

5.2407 

10 

477.4 

9.0413 

33 

50.69 

.9600 

34 

141.1 

2.7350 

2 

279.7 

5.2990 

15 

485.1 

9.1875 

31 

52.23 

.9891 

36 

147.1 

2.7932 

4 

282.8 

5.3573 

20 

492.9 

9.3347 

35 

53 76 

1.0182 

38 

150.5 

2.8514 

6 

285.9 

5.4158 

25 

500.6 

9.4819 

36 

55.30 

1.0472 

40 

153.6 

2.9097 

8 

289.0 

5.4742 

30 

508.1 

9.6292 

37 

56 33 

1.0763 

42 

156.6 

2.9679 

10 

292.1 

5.53*26 

35 

516.1 

9.7755 

38 

58.37 

1.1054 

41 

159.7 

3.0262 

12 

295.1 

5.5909 

40 

523.9 

9.9218 

39 

59.90 

1.1345 

46 

162.8 

3.0844 

14 

298.2 

5.6493 

45 

531.6 

10.068 

/ 40 

61.44 

1.1636 

48 

165.9 

3.1427 

16 

301.3 

5.7077 

50 

539.4 

10.215 

41 

62.97 

1.1927 

50 

169.0 

3.2010 

18 

304.1 

5.7660 

55 

547.2 

10.362 

42 

61.51 

1.2218 

52 

172.0 

3.2592 

20 

307.5 

5.8244 

6 

555. 

10.510 

43 

66.04 

1.2509 

51 

175.1 

3.3175 

22 

310.5 

5.8827 




44 

67.57 

1.2800 

56 ! 

178.2 

3.3758 

24 

313.6 

5.9410 





On a turnpike road 1° 38', or about 1 in 35, or 159 It per mile, is the 

•reatest slope that should be given to allow horses to trot down rapidly with safety. In cro sing 
nountains, this is often increased to 3, or even to 5°. It should uever exceed 2>$°, except when abso- 
utely necessary. 


















































724 


RAILROAD CONSTRUCTION 


# 


SLOPES IN FEET PER 100 FT. HORIZONTAL, 


The fractions of minutes are given only to 34 ft in 100. 

A clinometer graduated by the 3d column, and numbered by the first one 

will give at sight the slopes in feet per 100 ft horizontal. No errors. Original. 


Rise in ft 
per 100 
ft hor. 

Length of 
slope per 
100 ft hor. 

Angle of 
slope. 

Rise in ft 
per 100 
ft hor. 

Length of 
slope per 
100 ft hor. 

Angle of 
slope. , 

Rise in ft 
per 100 
ft hor. 

Length of 
slope per 
100 ft hor. 

Angle of 
slope. 


Feet. 

Deg. 

Min. 


Feet. 

Deg. 

Min. 


Feet. 

Deg.- 

Min. 

1 

100.005 

0 

34.4 

35 

105.948 

19 

17 

69 

121.495 

34 

36 

2 

100.020 

1 

8.7 

36 

106.283 

19 

48 

70 

122.066 

35 

0 

3 

100.045 

1 

43.1 

37 

106.626 

20 

18 

71 

122 642 

35 

23 

4 

100.080 

2 

17.5 

38 

106.977 

20 

48 

72 

123.223 

35 

45 

5 

100.125 

2 

51.8 

39 

107.336 

21 

18 

73 

123.810 

36 

8 

6 

100.180 

3 

26.0 

40 

107.703 

21 

48 

74 

124.403 

36 

30 

7 

100.245 

4 

0.3 

41 

108.079 

22 

18 

75 

125.000 

36 

52 

8 

100.319 

4 

34.4 

42 

108.462 

22 

47 

76 

125.603 

37 

14 

9 

100.404 

5 

8.6 

43 

108.853 

23 

16 

77 

126.210 

37 

36 

10 

100.499 

5 

42.6 

44 

105*. 252 

23 

45 

78 

126.823 

37 

57 

11 

100.603 

6 

16.6 

45 

109 659 

24 

14 

79 

127.440 

38 

19 

12 

100 717 

6 

50 6 

46 

110.073 

24 

42 

80 

128 062 

38 

40 

13 

100.841 

7 

24.4 

47 

110.494 

25 

10 

81 

128.690 

39 

1 

14 

100.975 

7 

58.2 

48 

110.923 

25 

38 

82 

129 321 

39 

21 

15 

101.119 

8 

31.9 

49 

111.359 

26 

6 

83 

129.958 

39 

42 

16 

101.272 

9 

5.4 

50 

111 803 

26 

34 

84 

130.599 

40 

2 

17 

101.435 

9 

38.9 

51 

112.254 

27 

1 

85 

131.244 

40 

22 

18 

101.607 

10 

12.2 

52 

112.712 

27 

28 

86 

131.894 

40 

42 

19 

101.789 

10 

45.5 

53 

113.177 

27 

55 

87 

132.548 

41 

1 

20 

101.980 

11 

18.6 

54 

113.649 

28 

22 

88 

133.207 

41 

21 

21 

102.181 

11 

51.6 

55 

114.127 

28 

49 

89 

133,869 

41 

40 

22 

102.391 

12 

24.5 

56 

114.612 

29 

15 

90 

134.536 

41 

59 

23 

102.611 

12 

57.2 

57 

115.104 

29 

41 

91 

135.207 

42 

18 

24 

102.840 

13 

29.8 

58 

115.603 

30 

7 

92 

135.882 

42 

37 

25 

103.078 

14 

2.2 

59 

116.108 

30 

32 

93 

136.561 

42 

55 

26 

103.325 

14 

34.5 

60 

116.619 

30 

58 

94 

137.244 

43 

14 

27 

103.581 

15 

6.6 

61 

117.137 

31 

23 

95 

137.931 

43 

32 

28 

103.846 

15 

38.5 

62 

117.661 

31 

48 

96 

138.622 

43 

50 

29 

104.120 

16 

10 3 

63 

118.191 

32 

13 

97 

139 316 

44 

8 

30 

104.403 

16 

42.0 

64 

118.727 

32 

37 

98 

140.014 

44 

25 

31 

104.695 

17 

13.4 

65 

119.269 

33 

1 

99 

140.716 

44 

43 

32 

104.995 

17 

44.7 

66 

119.817 

33 

25 

100 

141.421 

45 

00 

S3 

105.304 

18 

15.8 

67 

120.370 

33 

49 

101 

142.130 

45 

17 

34 

105.622 

18 

46.7 

68 

120.930 

34 

13 

102 

142.843 

45 

34 


Any ln>r <lisl> is = sloping (list X cosine ang of slope. 

“ sloping; (list is = hor dist - 4 - cosine “ “ “ 

“ vert height is = hor dist X tangent “ “ “ 

or = sloping dist X sine “ “ “ 


u (i 






























GRADES 


725 


able of grades per mile; or per 100 feet measured hori¬ 
zontally. 


Grade 
in ft. 
er mile. 

Grade 
in ft. 

per 100 ft. 

Grade 
in ft. 
per mile. 

Grade 
in ft. 
per 100 ft. 

Grade 
in ft. 
per mile. 

Grade 
in ft. 
per 100 ft. 

Grade 
in ft. 
per mile 

Grade 
in ft. 
per 100 ft 

1 

.01894 

39 

.73864 

77 

1.45833 

115 

2.17803 

2 

.03788 

40 

.75758 

78 

1.47727 

116 

2.19697 

3 

.05682 

41 

.77652 

79 

1.49621 

117 

2.21591 

4 

.07576 

42 

.79545 

80 

1.51515 

118 

2.23485 

5 

.09470 

43 

.81439 

81 

1.53409 

119 

2.25379 

6 

-11364 

44 

.83333 

82 

1.55303 

120 

2.27273 

7 

.13258 

45 

.85227 

83 

1.57197 

121 

2.29167 

8 

.15152 

46 

.87121 

84 

1.59091 

122 

2.31061 

9 

.17045 

47 

.89015 

85 

1.60985 

123 

2.32955 

10 

.18939 

48 

.90909 

86 

1.62879 

124 

2.34848 

11 

.20833 

49 

.92803 

87 

1.64773 

125 

2.36742 

12 

.22727 

50 

.94697 

88 

1.66666 

126 

2.38636 

13 

.24621 

51 

.96591 

89 

1.68561 

127 

2.40530 

14 

.26515 

52 

.98485 

90 

1.70455 

128 

2.42424 

15 

.28409 

53 

1.00379 

91 

1.72348 

129 

2.44318 

16 

.30303 

54 

1.02273 

92 

1.74212 

130 

2.46212 

17 

.32197 

55 

1.04167 

93 

1.76136 

131 

2.48106 

18 

.34091 

56 

1.06061 

94 

1.78030 

132 

2.50000 

19 

.35985 

57 

1.07955 

95 

1.79924 

133 

2.51894 

20 

.37879 

58 

1.09848 

96 

i.81818 

134 

2.53788 

21 

.39773 

59 

1.11742 

97 

1.83712 

135 

2 55682 

22 

.41667 

60 

1.13636 

98 

1.85606 

136 

2.57576 

23 

.43561 

61 

1.15530 

99 

1.87500 

137 

2.59470 

24 

.45455 

62 

1.17424 

100 

1.89394 

138 

2.61364 

25 

.47348 

63 

1.19318 

101 

1.91288 

139 

2.63258 

26 

.49242 

64 

1.21212 

102 

1.931S2 

140 • 

2.65152 

27 

.51136 

65 

1.23106 

103 

1.95076 

141 

2.67045 

28 

.53030 

66 

1.25000 

104 

1.96969 

142 

2.6S939 

29 

.54924 

G7 

1.26894 

105 

1.98864 

143 

2.70833 

30 

.56818 

68 

1.28788 

106 

2.00758 

144 

2.72727 

31 

.58712 

69 

1.30682 

107 

2.02652 

145 

2.74621 

32 

.60606 

70 

1.32576 

108 

2.04545 

146 

2.76515 

33 

.62500 

71 

1.34470 

109 

2.06439 

147 

2.78409 

34 

.64394 

72 

1.36364 

110 

2.08333 

148 

2.80303 

35 

.66288 

73 

1.38258 

111 

2.10227 

149 

2.82197 

36 

.68182 

74 

1.40152 

112 

2.12121 

150 

2.84091 

37 

.70076 

75 

1.42045 

113 

2.14015 

151 

2.85985 

38 

.71970 

76 

1.43939 

114 

2.15909 

152 

2.87879 


If the grade per mile should consist of feet and tenths, add to the grade per 100 feet in the foregoing 
able, that corresponding to the number of tenths taken from the table below ; thus, for a grade of 
3.7 feet per mile, we have .81439 -f- . 01326 “ .82765 feet per 100 feet. 


Ft. per Mile. 

Per 100 Feet. 

Ft. per Mile. 

Per 100 Feet. 

Ft. per Mile. 

Per 100 Feet. 

.06 

.00094 

.4 

.00758 

.7 

.01326 

.1 

.00189 

.45 

.00852 

.75 

.01420 

.15 

.00283 

.6 

.00947 

.8 

.01515 

.2 

.00379 

.66 

.01041 

.85 

.01609 

.25 

.00473 

.6 

.01136 

.9 

.01705 

.3 

.00568 

.65 

.01230 

.95 

.01799 

.36 

.00662 












































726 


RAILROADS, 


Table of Radii, Middle Ordinates, Ac, of Carves. Chord 100 feet 

Contains no error as great as 1 in the last figure. 


Ang. of 
Defl. 

Bad. 

In ft. 

Defl. 
Dist. 
in ft. 

Tang. 
Dist. 
in ft. 

Mid. 

Ord. 

Ang. of 
Defl. 

Rad. 
in ft. 

Defl. 
Dist. 
in ft. 

Tang. 
Dist. 
in ft. 

Mid 

Ord 

O ' 

1 

343775 

.029 

.014 

.004 

O ' 

1 36 

3581 

2.793 

1.396 

.349 

2 

171887 

.058 

.029 

.007 

38 

3508 

2.851 

1.425 

.356 

3 

114592 

.087 

.043 

.011 

40 

3438 

2.909 

1.454 

.364 

4 

85944 

.116 

.058 

.014 

42 

3370 

2.967 

1.483 

.371 

5 

68755 

.145 

.072 

.018 

44 

3306 

3.025 

1.512 

.378 

6 

57296 

.175 

.087 

.022 

46 

3243 

3.084 

1.542 

.385 

7 

49111 

.204 

.102 

.025 

48 

3183 

3.142 

1.571 

.393 

8 

42972 

.233 

.116 

.029 

50 

3125 

3.200 

1.600 

.400 

9 

38197 

.262 

.131 

.033 

52 

3070 

3.257 

1.629 

.407 

10 

34377 

.291 

.145 

.036 

54 

3016 

3.316 

1.658 

.414 

11 

31252 

.320 

.160 

.040 

56 

2964 

3.374 

1.687 

.422 

12 

28648 

.349 

.174 

.043 

58 

2913 

3.433 

1.716 

.429 

13 

26444 

.378 

.189 

.047 

2 

2865 

3.490 

1.745 

.436 

14 

24555 

.407 

.203 

.051 

2 

2818 

3.549 

1.774 

.443 

15 

22918 

.436 

.218 

.054 

4 

2773 

3.606 

1.803 

.451 

16 

21486 

.465 

.232 

.058 

6 

2729 

3.664 

1.832 

.458 

17 

20222 

.494 

.247 

.062 

8 

2686 

3.723 

1.861 

.465 

18 

19098 

.524 

.262 

.065 

10 

2645 

3.781 

1.890 

.473 

19 

18094 

.553 

.276 

.069 

12 

2605 

3.839 

1.919 

.480 

20 

17189 

.582 

.291 

.073 

14 

2566 

3.897 

1.948 

.487 

21 

16370■ 

.611 

.305 

.076 

16 

2528 

3.956 

1.978 

.495 

22 

15626 

.640 

.320 

.080 

18 

2491 

4.014 

2.007 

.502 

23 

14947 

.669 

.334 

.083 

20 

2456 

4.072 

2.036 

.509 

24 

14324 

.698 

.349 

.087 

22 

2421 

4.131 

2.065 

.516 

25 

13751 

.727 

.363 

.091 

24 

2387 

4.189 

2.094 

.523 

26 

13222 

.756 

.378 

.095 

26 

2355 

4.246 

2.123 

.531 

27 

12732 

.785 

.392 

.098 

28 

2323 

4.305 

2.152 

.538 

28 

12278 

.814 

.407 

.102 

30 

2292 

4.363 

2.182 

.545 

29 

11854 

.844 

.422 

.105 

32 

2262 

4.421 

2.210 

.552 

30 

11459 

.873 

.436 

.109 

34 

2232 

4.480 

2.240 

.560 

31 

11090 

.902 

.451 

.113 

36 

2204 

4.537 

2.268 

.567 

32 

10743 

.931 

.465 

.116 

38 

2176 

4.596 

2.298 

.574 

33 

10417 

.960 

.480 

.120 

40 

2149 

4.654 

2.327 

.682 

34 

10111 

.989 

.494 

.123 

42 

2122 

4.713 

2.356 

.589 

35 

9822 

1.018 

.509 

.127 

44 

2096 

4.771 

2.385 

.596 

36 

9549 

1.047 

.523 

.131 

46 

2071 

4.829 ' 

2.414 

.603 

37 

9291 

1.076 

.538 

.134 

48 

2046 

4.888 

2.444 

.611 

38 

9047 

1.105 

.552 

.138 

50 

2022 

4.946 

2.473 

.618 

39 

8815 

1.134 

.567 

.142 

52 

1999 

5.003 

2.501 

.625 

40 

8594 

1.164 

.582 

.145 

54 

1976 

5.061 

2.530 

.632 

41 

8385 

1.193 

.596 

.149 

56 

1953 

5.120 

2.560 

.640 

42 

8185 

1.222 

.611 

.153 

58 

1932 

5.176 

2.588 

.647 

43 

7995 

1.251 

.625 

.156 

3 

1910 

5.235 

2.618 

.654 

44 

7813 

1.280 

.640 

.160 

2 

1889 

5.294 

2.647 

.662 

.669 

45 

7639 

1.309 

.654 

.164 

4 

1869 

5.350 

2.675 

46 

7473 

1.338 

.669 

.167 

6 

1848 

5.411 

2.705 

.676 

47 

7314 

1.867 

.683 

.171 

8 

1829 

5.467 

2.734 

.683 

48 

7162 

1.396 

.698 

.174 

10 

1810 

5.526 

2.763 

.691 

49 

7016 

1.425 

.712 

.178 

12 

1791 

5.583 

2.792 

.698 

50 

6876 

1.454 

.727 

.182 

14 

1772 

5.643 

2.821 

.705 

51 

6741 

1.483 

.741 

.185 

16 

1754 

5.707 

2.850 

.713 

.720 

52 

6611 

1.513 

.757 

.189 

18 

1736 

5 760 

2.880 

53 

6486 

1.542 

.771 

.193 

20 

1719 

5.817 

2.908 

.727 

54 

6366 

1.571 

.786 

.197 

22 

1702 

5.875 

2.937 

.734 

55 

6251 

1.600 

.800 

.200 

24 

1685 

5.935 

2.967 

.742 

56 

6139 

1.629 

.815 

.204 

26 

1669 

5.992 

2.996 

.749 

57 

6031 

1.658 

.829 

.207 

28 

1653 

6.050 

3.025 

.756 

58 

5927 

1.687 

.844 

.211 

30 

1637 

6.108 

3.054 

.764 

59 

5827 

1.716 

.858 

.214 

32 

1622 

6.166 

3.083 

.771 

1 

5730 

1.745 

.872 

.218 

34 

1607 

6.223 

3.112 

.778 

2 

5545 

1.803 

.902 

.225 

36 

1592 

6.281 

3.140 

.785 

4 

5372 

1.862 

.931 

.233 

38 

1577 

6.341 

3.170 

.793 

6 

5209 

1.920 

.960 

.240 

40 

1563 

6.398 

3.199 

.800 

8 

5056 

1.978 

.989 

.247 

42 

1549 

6.456 

3.228 

.807 

10 

4911 

2.036 

1.018 

.255 

44 

1535 

6.515 

3.257 

.814 

12 

4775 

2.094 

1.047 

.262 

46 

1521 

6.575 

3.287 

.822 

14 

4646 

2.152 

1.076 

.269 

48 

1508 

6.631 

3.316 

.829 

16 

4523 

2.211 

1.105 

.276 

50 

1495 

6.689 

3.345 

.836 

18 

4407 

2.269 

1.134 

.284 

52 

1482 

6.748 

3.374 

.843 

20 

4297 

2.327 

1.163 

.291 

54 

1469 

6.807 

3.403 

.851 

22 

4192 

2.385 

1.192 

.298 

56 

1457 

6.863 

3.432 


24 

4093 

2.443 

1.221 

.305 

58 

1445 

6.920 

3.460 

.865 

26 

3997 

2.502 

1.251 

.313 

4 

1433 

6.980 

3.490 

.873 

28 

3907 

2.560 

1.280 

.320 

5 

1403 

7.125 

3.562 

.891 

30 

3820 

2.618 

1.309 

.327 

10 

1375 

7.271 

3.635 

909 

32 

3737 

2.676 

1.338 

.334 

15 

1348 

7.416 

3.708 

.927 

34 

3657 

2.734 

1.367 

.342 

20 

1323 | 

7.561 

3.781 

.945 


































RAILROADS 


727 


*t able of Radii, Middle Ordinates, Ac, of Curves. Chord 100 feet. 

(Continued.) 

The Tangential Angle is always one-half of the Angle of Deflection. 


A 

ag. of 
Deft. 

Rad. 
in ft. 

Defl. 
Dist. 
in ft. 

Tang. 
Dist. 
in ft. 

Mid.* 

Ord. 

Ang. of 
Defl. 

Rad. 
in ft. 

Defl. 
Dist. 
in ft. 

Tang. 
Dist. 
in ft. 

Mid.* 

Ord. 

11! 

a 

3 < 

4 25 

1298 

7.707 

3.854 

.963 

o • 

10 15 

559.7 

17.87 

8.942 

2.238 

M 

30 

1274 

7.852 

3.927 

.982 

30 

546.4 

18.30 

9.160 

2.292 

ill 

35 

1250 

7.997 

3.999 

1.000 

45 

533.8 

18.73 

9.378 

2.347 

hi 

40 

1228 

8.143 

4.072 

1.018 

11 

521.7 

19.17 

9.596 

2.402 

$ 

45 

1207 

8.288 

4.145 

1.036 

15 

510.1 

19.60 

9.814 

2.456 


50 

1186 

8.433 

4.218 

1.054 

30 

499.1 

20.04 

10.03 

2.511 

'OC 

55 

1166 

8.579 

4.290 

1.072 

45 

488.5 

20.47 

10.25 

2.566 

■fli 

5 

1146 

8.724 

4.363 

1.091 

12 

478.3 

20.91 

10.47 

2.620 

•!» 

•a 

5 

1128 

8.869 

4.436 

1.109 

15 

468.6 

21.34 

10.69 

2.675 

10 

1109 

9.014 

4.508 

1.127 

30 

459.3 

21.77 

10.90 

2.730 

■J 

15 

1092 

9.160 

4.581 

1.145 

45 

450.3 

22.21 

11.12 

2.785 

# 

20 

1075 

9.305 

4.654 

1.164 

13 

441.7 

22.64 

11.34 

2.839 

4 

25 

1058 

9.450 

4.727 

1.182 

15 

433.4 

23.07 

11.56 

2.894 

oi 

30 

1042 

9.596 

4.799 

1.200 

30 

425.4 

23.51 

11.77 

2.949 

> 

35 

1027 

9.741 

4.872 

1.218 

45 

417.7 

23.94 

11.99 

3.003 

* 

40 

1012 

9.886 

4.945 

1 236 

14 

410.3 

24.37 

12.21 

3.058 

45 

996.9 

10.03 

5.017 

1.255 

15 

403.1 

24.81 

12.43 

3.113 


50 

982.6 

10.18 

5.090 

1.273 

30 

396.2 

25.24 

12.65 

3.168 

$> 

55 

968.8 

10.32 

5.163 

1.291 

45 

389.5 

25.67 

12.86 

3.223 

!i 

6 

955-4 

10.47 

5.235 

1.309 

15 

383.1 

26.11 

13.08 

3.277 


5 

942.3 

10.61 

5.308 

1.327 

15 

376.8 

26.54 

13.30 

3.332 

il 

10 

929.6 

10.76 

5.381 

1.346 

30 

370.8 

26.97 

13.52 

3.387 

15 

917.2 

10.90 

5.453 

1.364 

45 

364.9 

27.40 

13.73 

3.442 

3 

20 

905.1 

11.05 

5.526 

1.382 

16 

359.3 

27.84 

13.95 

3.496 

il 

25 

893.4 

11.19 

5.599 

1.400 

30 

348.5 

28.70 

14.39 

3.606 

St 

30 

882.0 

11.34 

5.672 

1.418 

17 

338.3 

29.56 

14.82 

3.716 

t 

35 

870.8 

11.48 

5.744 

1.437 

30 

328.7 

30.43 

15.26 

3.825 

oJ 

* 

40 

859.9 

11.63 

5.817 

1.455 

18 

319.6 

31.29 

15.69 

3.935 

45 

849.3 

11.77 

5.890 

1.473 

30 

311.1 

32.15 

16.13 

4.045 

01 

50 

839.0 

11.92 

5.962 

1.491 

19 

302.9 

33.01 

16.56 

4.155 

;< 

55 

828.9 

12.07 

6.035 

1.510 

30 

295.3 

33.87 

17.00 

4.265 

<! 

7 

819.0 

12.21 

6.108 

1.528 

20 

287.9 

34.73 

17.43 

4.375 

3 

5 

809.4 

12.36 

6.180 

1.546 

21 

274.4 

36.44 

18.30 

4.594 

* 

10 

800.0 

12.50 

6.253 

1.564 

22 

262.0 

38.17 

19.17 

4.815 


15 

790.8 

12.65 

6.326 

1.582 

23 

250.8 

39.87 

20.04 

5.035 

11 

20 

781.8 

12.79 

6.398 

1.600 

24 

240.5 

41.58 

20.91 

5.255 

.i 

25 

773.1 

12.94 

6.471 

1.619 

25 

231.0 

43.29 

21.77 

5.476 


30 

764.5 

13.08 

6.544 

1.637 

26 

222.3 

44.98 

22.64 

5.696 


35 

756.1 

13.23 

6.616 

1.655 

27 

214.2 

46.69 

23.51 

5.917 


40 

747.9 

13.37 

6.689 

1.673 

28 

206.7 

48.38 

24.37 

6.139 


45 

739.9 

13.52 

6.762 

1.691 

29 

199.7 

50.08 

25.24 

6.361 


50 

732.0 

13.66 

6.835 

1.710 

30 

193 2 

51.76 

26.11 

6.582 


55 

724.3 

13.81 

6.907 

1.728 

31 

187.1 

53.45 

26.97 

6.805 


8 

716.8 

13.95 

6.980 

1.746 

32 

181.4 

55.13 

27.83 

7.027 

i 

15 

695.1 

14.39 

7.198 

1.801 

33 

176.0 

56.82 

28.70 

7.252 

3 

30 

674.7 

14.82 

7.416 

1.855 

34 

171.0 

58.48 

29.56 

7.473 

1 

45 

655.5 

15.26 

7.634 

1.910 

35 

166.3 

60.13 

30.42 

7.695 


9 

637.3 

15.69 

7.852 

1.965 

36 

161.8 

61.80 

31.29 

7.919 

: 

15 

620.1 

16.13 

8.070 

2.019 

37 

157.6 

63.45 

32.15 

8.142 

i 

30 

603.8 

16.56 

8.288 ] 

2.074 

38 

153.6 

65.10 

33.01 

8.366 

) 

45 

588.4 

17.00 

8.506 

2.128 

39 

149.8 

66.76 

33.87 

8.591 

: 

10 

573.7 

17.43 

8.724 

2.183 

40 

146.2 

68.40 

34.73 

8.816 


For ordinates 5 ft apart, for Chords of 100 ft, see p 730. 

To find tangential and deflection angles for any given rad and 
hord. Divide half the chord by the rad. The quot will be nat sine of the tangl 
ng. Find this tangl ang in the table of nat sines; and mult it by 2 for the def ang. 

To find the def dist for chords 100 ft long. Div 10000 by tbe 
ad in feet. 

To find the def dist for equal chords of any given length. 

>iv chord by rad. Mult, quot by chord. Or div sq of chord by rad. 

To find the tangl dist for equal chords of any given length. 

'irst find the tangl ang as above. Divide it by 2. Find in the table of nat sines 
tte nat sine of the quot. Mult this nat sine by the given chord. Mult prod by 2. 
To find the rad to any given def ang. for equal chords of any 
^ngth. Divide the def ang by 2. Find nat sine of the quotient. Divide half 
he chord by this nat sine. 


* The middle ordinate for a rad of 600 ft or more, (chord 100 ft,) may in 
•actice be taken at one-fourth of the tang dist. Even in 400 ft rad it will be too short only 6 in the 
ird decimal. 




































728 


CIRCULAR CURVES. 


Radii, Arc, of Curves; in metres. Chord, 20 metres — 2 

delta metres. 

The stakes, at the ends of the 2-dekametre chords, should he numbered 2, 4, 6, &c 
meaning 2, 4, 6, &c, dekametres. The tangential angle in the table will then give 
the amount of deflection per unit (dekametre) of measurement. 


0> 

b£) 

a 

CS 

(C 

0) 

Q 

Tangential 

angle. 

Radius. 

Metres. 

Defl dist. 
Metres. 

Tangl dist. 
Metres. 

Mid. ord. 

Metres. 

Defl angle. 

Tangential 

angle. 

Radius. 

Metres. 

Defl dist. 

Metres. 

Tangl dist. 

Metres. 

Mid. ord. 
Metres. 

>° i<y 

0° 5' 

6875.50 

.058 

.029 

.007 

8° 0' 

4° 0' 

143.36 

2.790 

1.396 

.349 

20 

10 

3437.75 

.116 

.058 

.015 

10 

5 

140.44 

2.848 

1.425 

.356 

30 

15 

2291.84 

.175 

.087 

.022 

20 

10 

137.63 

2.906 

1.454 

.364 

40 

20 

1718.88 

.233 

.116 

.029 

30 

15 

134.94 

2.964 

1.483 

.371 

50 

25 

1375.11 

.291 

.145 

.036 

40 

20 

132.35 

3.022 

1.512 

.378 

o o> 

30 

1145.93 

.349 

.175 

.044 

50 

25 

129.85 

3.080 

1.541 

.386 

10 

35 

982.23 

.407 

.204 

.051 

9° 0' 

30 

127.45 

3.138 

1.570 

.393 

20 

40 

859.46 

.465 

.233 

.058 

10 

35 

125.14 

3.196 

1.599 

.400 

30 

45 

763.97 

.524 

.262 

.065 

20 

40 

122.91 

3.254 

1.629 

.407 

40 

50 

687.57 

.582 

.291 

.073 

30 

45 

120.76 

3.312 1.658 

.415 

50 

55 

625.07 

.640 

.320 

.080 

40 

50 

118.68 

3.370 

1.687 

.422 

!° O' 

1° 0' 

572.99 

.698 

.349 

.087 

50 

55 

116.68 

3.428 

1.716 

.429 

10 

5 

528.92 

.756 

.378 

.095 

10° 0' 

5° 0' 

114.74 

3.486 

1.745 

.437 

20 

10 

491.14 

.814 

.407 

.102 

20 

10 

111.05 

3.602 

1.803 

.451 

30 

15 

458.40 

.873 

.436 

.109 

40 

20 

107.58 

3.718 

1.861 

.466 

40 

20 

429.76 

.931 

.465 

.116 

11° 0' 

30 

104.33 

3.834 

1.919 

.4S0 

50 

25 

404.48 

.989 

.494 

.124 

20 

40 

101.28 

3.950 

1.977 

.495 

o o/ 

30 

382.02 

1.047 

.524 

.131 

40 

50 

98.39 

4.065 

2.035 

.509 

10 

35 

361.91 

1.105 

.553 

.138 

12° 0' 

6° 0' 

95.67 

4.181 

2.093 

.524 

20 

40 

343.82 

1.163 

.582 

.145 

20 

10 

93.09 

4.297 

2.152 

.539 

30 

45 

327.46 

1.222 

.611 

.153 

40 

20 

90.65 

4.413 

2.210 

.553 

40 

50 

312.58 

1.280 

.640 

.160 

13° 0' 

30 

88.34 

4.528 

2.268 

.568 

50 

55 

298.99 

1.338 

.669 

.167 

20 

40 

86.14 

4.644 

2.326 

.582 

o 0 ' 

2° O' 

286.54 

1.396 

.698 

.175 

40 

50 

84.05 

4.759 

2.384 

.597 

10 

5 

275.08 

1.454 

.727 

.182 

14° 0' 

7° 0' 

82.06 

4.875 

2.442 

.612 

20 

10 

264.51 

1.512 

.756 

.189 

20 

10 

80.16 4.990 

2.500 

.626 

30 

15 

254.71 

1.570 

.785 

.196 

40 

20 

78.34 5.106 

2.558 

.641 

40 

20 

245.62 

1.629 

.814 

.204 

15° 0' 

30 

76.6b 5.221 

2.616 

.655 

50 

25 

237.16 

1.687 

.844 

.211 

20 

40 

74.96 

5.336 

2.674 

.670 

o (y 

30 

229.26 

1.745 

.873 

.218 

40 

50 

73.37 

5.452 

2.732 

.685 

10 

35 

221.87 

1.803 

.902 

.225 

16° 0' 

8° O' 

71.85 

5.567 

2.790 

.699 

20 

40 

214.94 

1.861 

.931 

.233 

10 

10 

70.40 5.682 

2.848 

.714 

30 

45 

208.43 

1.919 

.960 

.240 

40 

20 

69.00 5.797 

2.906 

.729 

40 

50 

202.30 

1.977 

.989 

.247 

17° 0' 

30 

67.65 

5.912 

2.964 

.743 

50 

55 

196.53 

2.035 

1.018 

255 

20 

40 

66.36 

6 027 

3.022 

.758 

o 0 r 

3° O' 

191.07 

2.093 

1.047 

.262 

40 

50 

65.12 

6.142 

3.080 

.772 

10 

5 

185.91 

2.152 

1.076 

.269 

18° 0' 

9° 0' 

63.92 

6.25713.138 

.787 

20 

10 

181.03 

2.210 

1.105 

.276 

20 

10 

62.77 

6.372 

3.196 

.802 

30 

15 

176.39 

2.268 

1.134 

.284 

40 

20 

61.66 

6.487 

3.254 

.816 

40 

20 

171.98 

2.326 

1.163 

.291 

19° 0' 

30 

60.59 

6.602 

3.312 

.831 

50 

25 

167.79 

2.384 

1.192 

.298 

20 

40 

59.55 

6.717 

3.370 

.846 

o o' 

30 

163.80 

2.442 

1.222 

.306 

40 

50 

58.55 

6.831 

3.428 

.860 

10 

35 

160.00 

2.500 

1.251 

.313 

20° 0' 

10° 0' 

57.69 

6.946 

3.486 

.875 

20 

40 

156.37 

2.558 

1.280 

.320 

21° O' 

30 

54.87 

7.289 

3.660 

.919 

30 

45 

152.90 

2.616 

1.309 

.327 

22° 0' 

11° 0' 

52.41 

7.632 

3.834 

.963 

40 

50 

149.58 

2.674 

1.338 

.335 

23° 0' 

30 

50.16 

7.975 

4.008 

1.007 

50 

55 

146.40 

2.732 

1.367 

.342 

24° 0' 

12° 0' 

48.10 

8.316 

4.181 

1.051 







25° O' 

30 

46.20 

8.658 

4.355 

1.095 


Radius = 


Half the chord 


Deflection dist = 


Sine of tangential angle 

Square of chord 


Half the chord X C08eca,,t of * a "g e » tial 

angle. 

Radius “ «>. chord X 9in<! °l^?* eDt,al 

Tangential dist = Twice the chord X sine of half the tangential angle. 

Middle ord = Radius X (1 — cosine of tangential angle) = Half the chord X 
tangent of half the tangential angle. 

For curves of 60 metres, or greater, radius, the ordinate at 5 metres from 
the end of the 20-metre chord, or midway between the end of the chord and the mid¬ 
dle ordinate, may be taken at three-fourths of the middle ordinate. 
























































TABLE OF LONG CHORDS 


729 


Table of Long Chords. 

:A Lengths of Chord in ft, required to subtend from 1 to 4 stations of 100 ft each, 


W -- 

4 Ai >s- 

of 

Dell. 

1 Sta. 

2 Sta. 

3 Sta. 

4 Sta. 

Ang. 

of 

Defl. 

1 Sta. 

2 Sta. 

3 Sta. 

4 Sta. 

i lo 

100 

200.0 

300.0 

400.0 

% 

100 

199.7 

298.9 

397.5 

1 i A 

100 

200.0 

300.0 

399.9 

6 ° 

100 

199.7 

298.8 

397.3 

% 

100 

200 0 

300.0 

399.9 

M 

100 

199.7 

298.7 

397.0 

% 

100 

200.0 

300.0 

399.8 

\\ 

100 

199.7 

298.6 

396.7 

2 ° 

100 

290.0 

299.9 

399.7 

74 

100 

199.6 

298.5 

396.5 

! % 

100 

200.0 

299.9 

399.6 

7° 

100 

199.6 

298.4 

396.2 

y 

100 

200.0 

299.8 

399.5 

X A 

100 

199.6 

298.3 

396.0 

% 

100 

200.0 

299.8 

399.4 

i| 

100 

199.6 

298.2 

395.7 

3° 

100 

200.0 

299.7 

399.3 


100 

199.6 

298.1 

395.4 


100 

200.0 

299.7 

399.2 

8 ° 

100 

199.6 

298.0 

395.1 

U 

100 

200.0 

299.6 

399.1 

V 

100 

199.5 

297.9 

394.8 

% 

100 

200.0 

299.6 

399.0 

i / 

100 

199.5 

297.8 

394.5 

4° 

100 

199.9 

299.6 

398.9 

74 

100 

199.4 

297.7 

394.3 

% 

100 

199.9 

299.5 

398.7 

9° 

100 

199.4 

297.5 

394.1 

Vo 

100 

199.9 

299.4 

398.5 


100 

199.4 

297.4 

393 7 

% 

100 

199 9 

299.3 

398.3 

1 2 

100 

199.3 

297.3 

393.2 

5° 

100 

199.9 

299.2 

398.0 

74 

100 

199.2 

297.2 

392.8 


100 

199 8 

299.1 

397.8 

10 ° 

100 

199.2 

297.0 

392.4 

Vi 

100 

199.8 

299.0 

397.6 







| Elevation of outer rail in carves theoretically is equal in ins to (square 
)f vel in it per sec X gauge in ins) -=- (Rad of curve in ft X 32.2). Experience 
aas shown that half an inch for each degree of def angle (100 ft chords) does very 
weil for 4 ft 8.5 ins gauge up to 40 miles per hour. At 60 miles use 1 inch per deg. 
In dangerous places this may be increased for safety against high winds. Ap¬ 
proaching the curve raise the outer rail at the rate of 1 inch in about 60 or 80 ft. 

If the ends of the corves are tapered off with very easy “transition ” curves, part 
of the rise may be made on said easier curves instead of on the tangent; or all of 
it if they are long enough. 
































730 


TABLE OF ORDINATES 


Table of Ordinates 5 ft apart. Chord 100 ft. 

For Railroad Curves. 

Ordinates for angles intermediate of those in the table can at once be found by 
simple proportion. 


Distances of the Ordinates from the end of the 100 feet Chord. 


Ang. of 
Defl. 

Mid. 

50 ft. 

45 ft. 

40 ft. 

35 ft. 

30 ft. 

25 ft. 

1 20 ft. 

15 ft. 

10 ft. 

5 ft. 

o * 

4 

.014 

.014 

.014 

.013 

! .012 

.010 

.008 

.008 

.005 

.003 

8 

.029 

.029 

.028 

.026 

.024 

.022 

.018 

.015 

.010 

.005 

12 

.043 

.043 

.041 

.038 

.037 

.033 

.028 

.022 

.015 

.008 

16 

.058 

.058 

.056 

.052 

.049 

.044 

.037 

.030 

.020 

.011 

20 

.073 

.072 

.070 

.066 

.061 

.055 

.047 

.037 

.026 

.014 

24 

.087 

.086 

.083 

.077 

.074 

.066 

.056 

.045 

.031 

.017 

28 

.102 

.101 

.098 

.092 

.086 

.077 

.065 

.052 

.036 

.019 

32 

.116 

.115 

.112 

.106 

.098 

.088 

.075 

.058 

.042 

.022 

36 

.131 

.130 

.126 

.119 

.110 

.099 

.084 

.066 

.047 

.024 

40 

.145 

.144 

.140 

.133 

.123 

.110 

.093 

.074 

.052 

.027 

44 

.160 

.158 

.153 

.145 

.135 

.121 

.103 

.081 

.057 

.030 

48 

.174 

.172 

.167 

.158 

.147 

.132 

.112 

.088 

.062 

.033 

52 

.189 

.187 

.181 

.171 

.159 

.143 

.122 

.095 

.068 

.035 

56 

.204 

.202 

.195 

.185 

.171 

.154 

.131 

.103 

.073 

.038 

1 

.218 

.216 

209 

.198 

.183 

.164 

.140 

111 

.078 

.041 

4 

.233 

.231 

.223 

.211 

.196 

.175 

.150 

.118 

.083 

.043 

8 

.247 

.245 

.237 

.224 

.208 

.186 

.159 

.125 

.088 

.046 

12 

.262 

.260 

.252 

.237 

.220 

.196 

.168 

.133 

.094 

.049 

16 

.276 

.274 

.265 

.251 

.232 

.207 

.177 

.140 

.099 

.052 

20 

.291 

.288 

.279 

.264 

.244 

.218 

.187 

.148 

.104 

.055 

24 

.306 

.303 

.293 

.277 

.256 

.229 

.197 

. 155 

.109 

.057 

28 

.320 

.317 

.307 

.291 

.269 

.240 

.206 

.163 

.114 

.060 

32 

.334 

.331 

.321 

.304 

.281 

.251 

.215 

.171 

.120 

.063 

36 

.349 

.345 

.335 

.317 

.293 

.262 

.224 

.178 

.125 

.06G 

40 

.364 

.360 

.349 

.330 

.305 

.273 

.233 

.185 

.130 

.069 

44 

.378 

.374 

.363 

.343 

.318 

.284 

.242 

.192 

.135 

.072 

48 

.393 

.389 

.377 

.356 

.330 

.295 

.251 

.200 

.141 

.075 

52 

.407 

.403 

.391 

.370 

.342 

.305 

.261 

.208 

.147 

.077 

56 

.422 

.418 

.405 

.383 

.354 

.316 

.270 

.215 

.152 

.080 

2 

.436 

.432 

.419 

.397 

.366 

.327 

.280 

.222 

.157 

.083 

4 

.451 

.446 

.433 

.409 

.379 

.338 

.289 

.230 

.162 

.086 

8 

.465 

.461 

.447 

.425 

.391 

.349 

.298 

.237 

.167 

.088 

12 

.480 

.475 

.461 

.437 

.403 

.360 

.308 

.245 

.173 

.090 

16 

.495 

.490 

.475 

.450 

.415 

.371 

.317 

.252 

.178 

.093 

20 

.509 

.504 

.489 

.463 

.428 

.382 

.326 

.260 

.183 

.096 

24 

.523 

.518 

.503 

.476 

.440 

.393 

.334 

.267 

.188 

.099 

28 

.538 

.533 

.517 

.489 

.452 

.404 

.346 

.275 

.194 

.102 

32 

.552 

.547 

.531 

.503 

.465 

.415 

.355 

.282 

.199 

.104 

36 

.567 

.562 

.545 

.516 

.477 

.425 

.364 

.289 

.204 

.107 

40 

.582 

.576 

.559 

.529 

.489 

.436 

.373 

.297 

.209 

.110 

44 

.596 

.590 

.573 

.542 

.501 

.447 

.382 

.304 

.214 

.113 

48 

.611 

.605 

.587 

.555 

.513 

.458 

.391 

.312 

.219 

.116 

52 

.625 

.619 

.601 

.569 

.526 

.469 

.401 

.319 

.225 

.118 

56 

.640 

.634 

.615 

.582 

.538 

.480 

.410 

.326 

.230 

.121 

3 

.654 

.648 

.629 

.595 

.550 

.491 

.419 

.334 

.235 

.124 

4 

.669 

.662 

.643 

.608 

.562 

.502 

.428 

.341 

.240 

.127 

8 

.683 

.677 

.657 

.621 

.574 

.512 

.438 

.349 

.246 

.130 

12 

.698 

.691 

.671 

.635 

.587 

.523 

.448 

.357 

.251 

.132 

16 

.713 

.705 

.685 

.649 

.599 

.534 

.457 

.364 

.257 

.135 

20 

.727 

.720 

.699 

.662 

.611 

.545 

.466 

.371 

.262 

.138 

24 

.742 

.734 

.713 

.675 

.623 

.556 

.475 

.378 

.267 

.141 

28 

.756 

.749 

.727 

.688 

•685 

.567 

.485 

.386 

.272 

.144 

32 

.771 

-763 

.741 

.702 

.648 

.578 

.494 

.394 

.278 

.146 

36 

.786 

.777 

.755 

.715 

.660 

.589 

.503 

.401 

.283 

.149 

40 

.800 

.792 

.769 

.728 

.673 

.600 

.512 

.408 

.288 

.152 

44 

.814 

.806 

.783 

.741 

.685 

.611 

•521 

.415 

.293 

.155 

48 

.829 

.821 

.797 

.754 

.697 

.621 

.531 

.423 

.298 

.158 

52 

.843 

.835 

.811 

.768 

.709 

.632 

.541 

.431 

.304 

.160 

56 

.858 

.850 

.825 

.781 

.721 

.643 

.550 

.438 

.309 

.163 

4 

.873 

.864 

.839 

.794 

.734 

.655 

.559 

.445 

.314 

.166 

10 

.909 

.900 

.874 

.827 

.764 

.682 

.582 

.464 

.327 

.173 

20 

.945 

.936 

.909 

.860 

.795 

.709 

.606 

.482 

.340 

.179 

30 

.981 

.972 

.944 

.893 

.825 

.736 

.629 

.501 

.354 

.186 

40 

1.017 

1.008 

.979 

.926 

.855 

.764 

.652 

.519 

.367 

.193 

50 

1.054 

1.044 

1.014 

.959 

.886 

.791 

.676 

.538 

.380 

.199 

5 

1.091 

1.080 

1.048 

.993 

.917 

.818 

.699 

.557 

.393 

.207 

10 

1.127 

1.116 

1.083 

1.026 

.947 

.845 

.722 

.576 

.406 

.214 

20 

1.164 

1.152 

1.118 

1.058 

.978 

.872 

.746 

.594 

.419 

.220 

30 

1.200 

1.188 

1.153 

1.092 

1.009 

.900 

.769 

.613 

.432 

.228 


































TABLE OF ORDINATES 


731 


Table of Ordinates 5 ft apart. —(Continued.) 


Distances of the Ordiuates from the end of the 100 feet Chord. 


Ang. of 
Deti. 

Mid. 

50 ft. 

45 ft. 

40 ft. 

35 ft. 

30 ft. 

25 ft. 

20 ft. 

15 ft. 

10 ft. 

5 ft. 

o • 

6 40 

1.236 

1.224 

1.188 

1.124 

1.039 

.927 

.792 

.631 

.445 

.235 

50 

1.273 

1.260 

1.223 

1.157 

1.070 

.954 

.816 

.649 

.458 

.241 

6 

1.309 

1.296 

1.258 

1.191 

1.100 

.982 

.839 

.668 

.472 

.248 

10 

1.345 

1.332 

1.293 

1.224 

1.130 

1.009 

.862 

.686 

.485 

.255 

20 

1.382 

1.368 

1.328 

1.256 

1.161 

1.036 

.886 

.705 

.498 

.262 

30 

1.419 

1.404 

1.362 

1.290 

1.192 

1.064 

.909 

.724 

.511 

.269 

40 

1.455 

1-440 

1.397 

1.323 

1.222 

1.091 

.932 

.742 

.524 

.276 

50 

1.491 

1.476 

1.432 

1.355 

1.253 

1.118 

.956 

.761 

.537 

.283 

7 

1.528 

1.512 

1.467 

1.389 

1.284 

1.146 

.979 

.779 

.551 

.290 

10 

1.564 

1.548 

1.502 

1.422 

1.314 

1.173 

1.002 

.798 

.564 

.297 

20 

1.600 

1.584 

1.537 

1.454 

1.345 

1.200 

1.026 

.816 

.576 

.304 

30 

1.637 

1.620 

1.572 

1.488 

1.375 

1.228 

1.048 

.835 

.590 

.311 

40 

1.073 

1.656 

1.607 

1.521 

1.405 

1.255 

1.071 

.854 

.603 

.318 

50 

1.710 

1.692 

1.641 

1.553 

1.436 

1.282 

1.095 

.872 

.616 

.324 

8 

1.746 

1.728 

1.677 

1.587 

1.467 

1.310 

1.118 

.891 

.629 

.332 

30 

1.855 

1.836 

1.782 

1.687 

1.559 

1.392 

1.188 

.946 

.669 

.353 

9 

1.965 

1.944 

1.886 

1.787 

1.651 

1.474 

1.258 

1.002 

.708 

.373 

30 

2.074 

2.052 

1.991 

1.887 

1.742 

1.556 

1.328 

1.057 

.748 

.394 

10 

2.183 

2.161 

2.096 

1.987 

1.834 

1.637 

1.398 

1.114 

.787 

.415 

30 

2.292 

2.269 

2.201 

2.087 

1 .926 

1.719 

1.468 

1.170 

.827 

.436 

11 

2.401 

2.377 

2.306 

2.186 

2.018 

1.802 

1.538 

1.226 

.866 

.457 

30 

2.511 

2.486 

2.411 

2.286 

2.110 

1.884 

1.609 

1.282 

.906 

.478 

12 

2.620 

2.594 

2.516 

2.386 

2.203 

1.967 

1.680 

1.339 

.946 

.499 

30 

2.730 

2.703 

2.621 

2.485 

2.295 

2.049 

1.750 

1.395 

.985 

.520 

13 

2.839 

2.811 

2.726 

2.585 

2.387 

2.132 

1.820 

1.451 

1.025 

.541 

30 

2.949 

2.920 

2.832 

2.685 

2.479 

2.214 

1.891 

1.507 

1.065 

.562 

14 

3.058 

3.028 

2.937 

2.785 

2.571 

2.297 

1.961 

1.564 

1.105 

.583 

30 

3.168 

3.136 

3.042 

2.884 

2 6<i*4 

2.379 

2.031 

1.620 

1.144 

.604 

15 

3.277 

3.245 

3.147 

2 984 

2.756 

2.462 

2.102 

1.676 

1.184 

.625 

30 

3.387 

3.354 

3.252 

3.084 

2.848 

2.544 

2.172 

1.732 

1.224 

.646 

16 

3.496 

3.462 

3.358 

3.184 

2.941 

2.627 

2.243 

1.789 

1.264 

.667 

17 

3.716 

3.680 

3.569 

3.384 

3.125 

2.792 

2.384 

1.902 

1.344 

.709 

18 

3.935 

3.897 

3.779 

3.584 

3.310 

2.958 

2.525 

2.014 

1.424 

.751 

19 

4.155 

4.115 

3.990 

3.784 

3.495 

3.123 

2.666 

2.127 

1.504 

.793 

20 

4.375 

4.332 

4.201 

3.984 

3.680 

3.288 

2.808 

2.240 

1.583 

.836 

22 

4.815 

4.768 

4.624 

4.386 

4.050 

3.620 

3.093 

2.467 

1.744 

.922 

24 

5.255 

5.204 

5.048 

4.789 

4.423 

3.952 

3.379 

2.695 

1.905 

1.008 

26 

5.697 

5.642 

5.473 

5.192 

4.798 

4.286 

3.665 

2.924 

2.068 

1.094 

28 

6.139 

6.079 

5.898 

5.595 

5.171 

4.622 

3.952 

3.154 

2.232 

1.181 

30 

6.582 

6.517 

6.323 

5.999 

5.544 

4.958 

4.239 

3.385 

2.396 

1.268 

32 

7-027 

6.957 

6.751 

6.406 

5.922 

5.297 

4.530 

3.619 

2.565 

1.356 

34 

7.472 

7.398 

7.179 

6.813 

6.300 

5.637 

4.822 

3.854 

2.733 

1.445 

36 

7.918 

7.841 

7.609 

7.222 

6.679 

5.978 

5.115 

4.090 

2.901 

1.535 

38 i 

8.367 

8.286 

8.041 

7.633 

7.060 

6.320 

5.410 

4.327 

3.069 

1.626 

40 

8.816 

8.731 

8.474 

8.044 

7.442 

6.663 

5.705 

4.565 

3.238 

1.718 


For middle ordinates for bending rails, see p 761 





























732 


LEVEL CUTTINGS. 


To prepare a Table, T, of Level Cutting, for every tJ b °f 
foot of height, or (leptli. For Tables of Level Cuttings, see pp 733 to GO. 
For cost of Earthwork, p 742, &c. 


Let the fig represent the cutting; or, if inverted, 
the filling; in which the horizontal lines are sup- 
posed to be yby foot apart. First calculate the |j, 
area in square feet, of the layer a b c o, adjoining - 
the roadway a b. Then find how many cubic 
yards that area gives in a distance of 100 feet. 
These cubic yards we will call Y; they form the 
first amount to be put into the Table T. 

Next calculate the area in square feet of the triangle a n o. Multiply this area by 4. Find how 
many cubic yards this increased area gives in a distance of 100 feet. Or they will be found ready 
calculated below. We will call them y. This is all the preparation that is needed before 




\ 

1- -IV 


c 


commencing the table. 

Exam.-Let the roadbed a & be 18 feet, and the side-slopes 13-6 to 1. Then for the area of a b c o : 
since the side-slopes are 1% to 1; and s t is .1 foot; c o must be 18.3 feet; aud the mean length of 
abco must be 18.15 feet. Consequents, the area is 18.15 X -1— 1.815 square feet; which, in a 

* 1B1 & 

distance of 100 feet, gives 181.5 cubic feet; which is equal to — =8.7222 cubic yards ; or Y. 


27 


Next, as to the triangle a no: its height a n being .1 foot, and its base no .15 feet; its area 
.1 X .15 .015 

= ’——.0075 square ft. This multiplied by 4, gives .03 square feet; which, in a distance of 
2 a 


2 i 3 

100 feet, gives .03 X 100 3 cubic feet; which is equal to -- = .1111 cubic yard; or y. 


27 


Having thus found Y and y, proceed to make out the table in the manner following, which is so 
plain as to require no explanation. The work should be tested ahout every 5 feet, by calculating the 
area of the full depth arrived at; multiply it by 100, and divide the product by 27 for the cubic yards 
The cubic yards thus found should agree with the table. 


Y... 

.... G.7222 . . 

. Y. 6.722 

.1 

y 

.1111 



y... 

6.8333 
..... .1111 

6.8333 


13.5555 

.2 



6.9444 

6.9444 



1111 



y... 


20 5000 

.3 

y... 

7.0555 
.1111 

7.0555 


27.5555 

.4 


y... 

7.1666 
.... .1111 

7.1666 


34.7222 

.5 



7.2777 

7.2777 




42.0000 

.6 


Table T. 

Height. 

Feet. 

Cub. Yds. 

.1 . 

6.72 Y. 

.2 . 

13.6 

.3 . 

205 

.4 . 

27.6 

.5 . 

34.7 

.6 . 

42.0 

&c. 


The following table contains y. ready calculated for different side-slopes. It plainly 
remains the same for all widths of roadbed. 


Side-slope. 

y 

Side-slope. 

y 

% to 1 . 


to 1 . 


Y, to 1 . 

.0370 

2 to 1. 


y. to l . 

.0556 

2*4' to 1 . 

. 1f5fV7 

1 to 1 . 

.0741 

2*4 to 1 . 

. 1852 

to 1 . 


3 2 to 1 . 


iY to i . 

• •••••.••••••••••a .1111 

4 to 1 . 

• 
















































































RAILROADS 


733 


Table 1. I^evel Cuttings.* 

Roadway 14 feet wide, side-slopes 1% to 1. 


For single-track embankment. 


.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 


5.24 

10.6 

16.1 

21.6 

27.3 

33.1 

39 0 

45.0 

51.2 

57.4 

63.8 

70.2 

76.8 

83.5 

90.3 

97.2 

104.2 

111.3 

118.6 

125 9 

133.4 

141.0 

148.6 

156.4 

164.4 

172.4 

180.5 

188.7 

197.1 

205.6 

214.1 

222.8 

231.6 

240.5 

249.5 

258.7 

267.9 

277.3 

286.7 

296.3 

306.0 

315.8 

325.7 

335.7 

345.8 

356.1 

366.4 

376.9 

387.5 

398.1 

408.9 

419.9 

430.9 

442.0 

453.2 

464.6 

476.1 

487.6 

4993 

511.1 

523.0 

535.0 

547.2 

559.4 

571.8 

584.2 

596.8 

609.5 

622.3 

635 2 

648.2 

661.3 

674.6 

687.9 

701.4 

714.9 

728.6 

742.4 

756.3 

770.3 

784.5 

798.7 

813.1 

827.5 

842.1 

856.8 

871.6 

886.5 

901.5 

916.7 

931.9 

947.3 

962.7 

978.3 

994.0 

1010 

1026 

1042 

1058 

1074 

1090 

1107 

1123 

1140 

1157 

1174 

1191 

1208 

1225 

1243 

1260 

1278 

1295 

1313 

1331 

1349 

1367 

1385 

1404 

1422 

1441 

1459 

1478 

1497 

1516 

1535 

1554 

1574 

1593 

1613 

1633 

1652 

1672 

1692 

1712 

1733 

1753 

1773 

1794 

1815 

1835 

1856 

1877 

1898 

1920 

1941 

1962 

1984 

2006 

2028 

2050 

2072 

2094 

2116 

2138 

2161 

2183 

2206 

2229 

2252 

2275 

2298 

2321 

2344 

2368 

2391 

2415 

2439 

2463 

2487 

2511 

2535 

2559 

2584 

2608 

2633 

2658 

2683 

2708 

2733 

2759 

2784 

2809 

2835 

2861 

2886 

2912 

2938 

2964 

2991 

3017 

3044 

3070 

3097 

3124 

3151 

3178 

3205 

3232 

3259 

3287 

31114 

3342 

3370 

3398 

3426 

3454 

3482 

3510 

3539 

3567 

3596 

3625 

3654 

3683 

3712 

3741 

3771 

3800 

3830 

3859 

3889 

3919 

3949 

3979 

4009 

4040 

4070 

4101 

4132 

4162 

4193 

4224 

4255 

4287 

4318 

4349 

4381 

4413 

4444 

4476 

4508 

4541 

4573 

4605 

4638 

4670 

4703 

4736 

4769 

4802 

4S35 

4868 

4901 

4935 

4968 

5002 

5036 

5070 

5104 

5138 

5172 

5206 

5241 

5275 

5310 

5345 

5380 

5415 

5450 

5485 

5521 

5556 

5592 

5627 

5663 

5699 

5735 

5771 

5807 

5844 

5880 

5917 

5953 

5990 

6027 

6064 

6101 

6139 

6176 

6213 

6251 

6289 

6326 

6364 

6402 

6440 

6479 

6517 

6556 

6594 

6633 

6672 

6711 

6750 

6789 

6828 

6867 

6907 

6916 

6986 

7026 

7066 

7106 

7146 

7186 

7226 

7267 

7307 

7348 

7389 

7430 

7471 

7512 

7553 

7595 

7636 

7678 

7719 

7761 

7803 

7845 

7887 

7929 

7972 

8014 

8057 

8099 

8142 

8185 

8228 

8271 

8315 

8358 

8401 

8445 

8489 

8532 

8576 

8620 

8664 

8709 

8753 

8798 

8842 

8887 

8932 

8976 

9022 

9067 

9112 

9157 

9203 

9248 

9294 

9340 

9386 

9432 

9478 

9524 

9570 

9617 

9663 

9710 

9757 

9804 

9851 

9898 

9945 

9993 

10040 

10088 

10135 

10183 

10231 

10279 

10327 

10375 

10424 

10472 

10521 

10569 

10618 

10667 

10716 

10765 

10815 

10864 

10913 

10963 

11013 

11062 

11112 

11162 

1:1212 

11263 

11313 

11364 

11414 

11465 

11516 

11567 

11618 

11669 

11720 

11771 

11823 

11874 

11926 

11978 

12029 

120S1 

12134 

12186 

12238 

12291 

12343 

12396 

12449 

12502 

12555 

12608 

12661 

12715 

12768 

12822 

12875 

12929 

12983 

13037 

1309 L 

13145 

13200 

13254 

13309 

13363 

13418 

13473 

13528 

13583 

13639 

13694 

13749 

13805 

13861 

13916 

13972 

14028 

14084 

14141 

14197 

14254 

14310 

14367 

14424 

14480 

14537 

14595 

14652 

14709 

14767 

14824 

14882 

14940 

14998 

15056 

15114 

15172 

15230 

15289 

15347 

15406 

15465 

15524 

15583 

15642 

15701 

15761 

15826 

15880 

15939 

15999 

16059 

16119 

16179 

16239 

16300 

16360 

16421 

16481 

16542 

16603 

16664 

16725 

16787 

16848 

16909 

16971 

17033 

17094 

17156 

17218 

17280 

17343 

17405 

17467 

17530 

17593 

17656 

17719 

17782 

17845 

17908 

17971 

18035 

18098 

18162 

18226 

18290 

18354 

18418 

18482 

18546 

18611 

18675 

18740 

18805 

18370 

18935 

19000 

19065 

19131 

19196 

19262 

19327 

19393 

19459 

19525 

19591 

19657 

19724 

19790 

19857 

19923 

19990 

20057 

20124 

20191 

20259 

20326 

20393 

20461 

20529 

20596 

20664 

20732 

20800 

20869 

20937 

21005 

21074 

21143 

21212 

21280 

21349 

21419 

21488 

21557 

21627 

21696 

21766 

21836 

21906 

21976 

22046 

22116 

22186 

22257 

22327 

22398 

22469 

22540 

22611 

22682 

22753 

22825 

22896 

22968 

23039 

23111 

23183 

23255 

23327 

23399 

23472 

23544 

23617 

23689 

23762 















































!i«h 

Ft 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 


RAILROADS 


Table 2. L.evel Cutting's. 

Roadway 24 feet wide, side-slopes 1% to 1. 


For double-track embankment. 


.1 

O 

.3 

.4 

.5 

.6 

.7 

.8 

.9 

Cu. Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

8.94 

18.0 

27.2 

36.4 

45.8 

55.3 

64.9 

74.7 

84.5 

104.5 

114.7 

124.9 

135.3 

145.8 

156.4 

167.2 

178.0 

188.9 

211.2 

222.4 

233.8 

245.3 

256.9 

268.6 

280.5 

292.4 

304.4 

328.9 

341.2 

353.7 

366.3 

379.0 

391.9 

404 8 

417.8 

431.0 

457 8 

471.3 

484.9 

498.6 

512.4 

526.4 

540.4 

554.6 

568.8 

597.8 

612.4 

627.1 

642.0 

656.9 

671.9 

687.1 

702.3 

717.7 

748.9 

764.7 

780.5 

796.4 

812.5 

828.7 

844.9 

861.3 

877.8 

911.2 

928.0 

944.9 

962.0 

979.2 

996.4 

1014 

1031 

1049 

1085 

1102 

1121 

1139 

1157 

1175 

1194 

1212 

1231 

1269 

1288 

1307 

1326 

1346 

1365 

1385 

1405 

1425 

1465 

1485 

1505 

1525 

1546 

1566 

1587 

1608 

1629 

1671 

1692 

1714 

1735 

1757 

1779 

1800 

1822 

1845 

1889 

1911 

1934 

1956 

1979 

2002 

2025 

2048 

2071 

2118 

2141 

2165 

2189 

2213 

2236 

2261 

2285 

2309 

2358 

2382 

2407 

2432 

2457 

2482 

2507 

2532 

2658 

2609 

2635 

2661 

2686 

2713 

2739 

2765 

2791 

2818 

2871 

2898 

2925 

2952 

2979 

3006 

3034 

3061 

3089 

3145 

3172 

3201 

3229 

3257 

3285 

3314 

3342 

3371 

3429 

3458 

3487 

3516 

3546 

3575 

3605 

3635 

3665 

3725 

3755 

3785 

3815 

3846 

3876 

3907 

3938 

3969 

4031 

4062 

4094 

4125 

4157 

4189 

4221 

4252 

4285 

4349 

4381 

4414 

4446 

4479 

4512 

4545 

4578 

4611 

4678 

4711 

4745 

4779 

4813 

4846 

4881 

4915 

4949 

5018 

5052 

5087 

5122 

5157 

5192 

5227 

5262 

5298 

5369 

5405 

5441 

5476 

5513 

5549 

5585 

5621 

56 r >8 

5731 

5768 

5805 

5842 

5879 

5916 

5954 

5991 

6029 

6105 

6142 

6181 

6219 

6257 

6295 

6334 

6372 

6411 

6489 

6528 

6567 

6606 

6646 

6685 

6725 

6765 

6805 

6885 

6925 

6965 

7005 

7046 

7086 

7127 

7168 

7209 

7291 

7332 

7374 

7415 

7457 

7499 

7541 

7582 

7625 

7709 

7751 

7794 

7836 

7879 

7922 

7965 

8008 

8051 

8138 

8181 

8225 

8269 

8313 

8356 

8401 

8445 

8489 

8578 

8622 

8667 

8712 

8757 

8802 

8847 

8892 

8938 

9029 

9075 

9121 

9166 

9212 

9259 

9305 

9351 

9398 

9491 

9538 

9585 

9632 

9679 

9726 

9774 

9821 

9869 

9965 

10012 

10061 

10109 

10157 

10205 

10254 

10302 

10351 

10449 

10498 

10547 

10596 

10646 

10695 

10745 

10795 

10845 

10945 

10995 

11045 

11095 

11146 

11196 

11247 

11298 

11349 

11451 

11502 

11554 

11605 

11657 

11709 

11761 

11812 

11865 

11969 

12021 

12074 

12126 

12179 

12232 

12285 

12338 

12391 

12498 

12551 

12605 

12659 

12713 

12766 

12821 

12875 

12929 

13038 

13092 

13147 

13202 

13257 

13312 

13367 

13422 

13478 

13589 

13645 

13701 

13756 

13813 

13869 

13925 

13981 

14038 

14151 

14208 

14265 

14322 

14379 

14436 

14494 

14551 

14609 

14725 

14782 

14840 

14899 

14957 

15015 

15074 

15132 

15191 

15309 

15368 

15427 

15486 

15546 

15605 

15665 

15725 

15785 

15905 

15965 

16025 

16085 

16146 

16206 

16267 

16128 

16389 

16511 

16572 

16634 

16695 

16757 

16819 

16881 

16942 

17005 

17129 

17191 

17254 

17316 

17379 

17442 

17505 

17568 

17631 

17758 

17821 

17885 

17949 

18013 

18076 

18141 

18205 

18269 

18398 

18462 

18527 

18592 

18657 

18722 

18787 

18852 

18918 

19049 

19115 

19181 

19246 

19313 

19379 

19445 

19511 

19578 

19711 

19778 

19845 

19912 

19979 

20046 

20114 

20181 

20249 

20385 

20452 

20521 

20589 

20657 

20725 

20794 

20862 

20931 

21069 

21138 

21207 

21276 

21346 

21415 

21485 

21555 

21625 

21765 

21835 

21905 

21975 

22046 

22116 

22187 

22258 

22329 

22471 

22542 

22614 

22685 

22757 

22829 

22901 

22972 

23045 

23189 

23261 

23334 

2:1406 

23479 

23552 

23625 

23698 

23771 

23918 

23991 

24065 

24139 

24213 

24286 

24361 

24435 

24509 

24658 

24732 

24807 

24882 

24957 

2503 1 

25107 

25182 

25258 

25409 

25485 

25561 

25636 

25713 

25789 

25865 

25941 

26018 


For continuation to 100 feet, see Table 7. 




































RAILROADS, 


735 


Table 3. Level Cutting's. 

Roadway 18 feet wide, side-slopes 1 to 1. 


For single-track excavation. 

o 


Depth 
in Ft. 

.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 


Cu. Yds. 

Cu. Yds. 

Cu. Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

0 


6.70 

13.5 

20 3 

27.3 

34.3 

41.3 

48.5 

55.7 

6 .0 

1 

70.4 

77.8 

85.3 

92 9 

100.6 

1083 

116.1 

124.0 

132.0 

140.0 

2 

148.1 

156.3 

164.6 

172.9 

181.3 

189.8 

198.4 

207.0 

215.7 

224.5 

3 

233.3 

2423 

251.3 

260.3 

269.5 

278.7 

288.0 

297.4 

306.8 

316.3 

4 

325.9 

335.6 

345.3 

355.1 

3650 

375.0 

385.0 

395.1 

405.3 

415.6 

5 

425 9 

436.3 

446.8 

457.4 

468.0 

478.7 

489.5 

500.3 

511.3 

522.3 

6 

533.3 

544.5 

555.7 

567.0 

578.4 

589.8 

601.3 

612.9 

624.6 

636.3 

7 

648.1 

660.0 

672.0 

684.0 

696.1 

708.3 

720.6 

732.9 

745.3 

757.3 

8 

770.4 

783.0 

795.7 

808.5 

821.3 

834.3 

847.3 

860.3 

873.5 

886.7 

9 

900.0 

913.4 

926.8 

9403 

953.9 

967.6 

981.3 

995.1 

1009 

1023 

10 

1037 

1051 

1065 

1080 

1094 

1108 

1123 

1137 

1152 

1167 

11 

1181 

1196 

1211 

1226 

1241 

1256 

1272 

1287 

1302 

1318 

12 

1333 

1349 

1365 

1380 

1396 

1412 

1428 

1444 

1460 

1476 

13 

1493 

1509 

1525 

1542 

1558 

1575 

1592 

1608 

1625 

1642 

14 

1659 

1676 

1693 

1711 

1728 

1745 

1763 

1780 

1798 

1816 

15 

1833 

1851 

1869 

1887 

1905 

1923 

1941 

1960 

1978 

1996 

16 

2015 

2033 

2052 

2071 

2089 

2108 

2127 

2146 

2165 

2184 

17 

2204 

2223 

2242 

2262 

2281 

2301 

2321 

2340 

2360 

2380 

18 

2400 

2420 

2440 

2460 

2481 

2501 

2521 

2542 

2562 

2583 

19 

2604 

2624 

2645 

2666 

2687 

2708 

2729 

2751 

2772 

2793 

20 

2815 

2836 

2858 

2880 

2901 

2923 

2945 

2967 

29 s9 

3011 

21 

30 53 

3056 

3078 

3100 

3123 

3145 

3168 

3191 

3213 

3236 

22 

3259 

3282 

3305 

3328 

3352 

3375 

3398 

3422 

3445 

3469 

23 

3493 

3516 

3540 

3564 

3588 

3612 

3636 

3660 

3685 

3709 

24 

3733 

3758 

3782 

3807 

3832 

3856 

3881 

3906 

3931 

3956 

25 

3981 

4007 

4032 

4057 

4083 

4108 

4134 

4160 

4185 

4211 

26 

4237 

4263 

4289 

4315 

4341 

4368 

4394 

4420 

4417 

4473 

27 

4500 

4527 

4553 

4580 

4607 

4634 

4661 

4688 

4716 

4743 

28 

4770 

4798 

4825 

4853 

4881 

4908 

4936 

4964 

4992 

5020 

29 

5048 

5076 

5105 

5133 

5161 

5190 

5218 

5247 

5276 

5304 

30 

5333 

5362 

5391 

5420 

5449 

5479 

5508 

5537 

5567 

5596 

31 

5626 

5656 

5685 

5715 

5745 

5775 

5S05 

5835 

5865 

5S96 

32 

5926 

5956 

5987 

6017 

6048 

6079 

6109 

6140 

6171 

6202 

33 

6233 

6264 

6296 

6327 

6358 

6390 

6121 

6453 

6485 

6516 

34 

6548 

6580 

6612 

6644 

6676 

6708 

6741 

6773 

6805 

6S38 

35 

6870 

6903 

6936 

6968 

7001 

7034 

7067 

7100 

7133 

7167 

36 

7200 

7233 

7267 

7300 

7334 

7368 

7401 

7435 

7469 

7503 

37 

7537 

7571 

7605 

7640 

7674 

7708 

7743 

7777 

7812 

7847 

38 

7881 

7916 

7951 

7986 

8021 

8056 

8092 

8127 

8162 

8198 

39 

8233 

8269 

8305 

8340 

8376 

8112 

8448 

8484 

8520 

8556 

40 

8593 

8629 

86G5 

8702 

8738 

8775 

8812 

8848 

8885 

8922 

41 

8959 

8996 

9033 

9071 

9108 

9115 

9183 

9220 

9258 

9296 

42 

9333 

9371 

94)9 

9447 

9485 

9523 

9561 

9600 

9638 

9676 

43 

9715 

9753 

9792 

9831 

9869 

9908 

9917 

99S6 

10025 

10064 

44 

10104 

10143 

10182 

10222 

10261 

10301 

10341 

10380 

10420 

10460 

45 

10500 

10540 

105 SO 

10620 

10661 

10701 

10741 

10782 

10822 

10863 

46 

10904 

10944 

10985 

11026 

11067 

11108 

11149 

11191 

11232 

11273 

47 

11315 

11356 

11398 

11440 

11481 

11523 

11565 

11607 

11649 

11691 

48 

11733 

11776 

11818 

11860 

11903 

11945 

11988 

12031 

12073 

12116 

49 

12159 

12202 

12245 

12288 

12332 

12375 

12418 

12462 

12505 

12549 

50 

12593 

12636 

12680 

127 24 

12768 

12812 

12856 

129!'0 

12945 

12989 

51 

13033 

13078 

13122 

13167 

13112 

13256 

13301 

13346 

13391 

13436 

52 

13481 

13527 

13572 

13617 

13663 

13708 

13754 

13" 00 

13845 

13" 91 

53 

13937 

13983 

14029 

14075 

14121 

14168 

14214 

14260 

14307 

14353 

54 

14400 

14447 

14493 

14540 

14587 

14634 

14681 

14728 

14776 

14823 

55 

14870 

14918 

14965 

15013 

15061 

15108 

15156 

15204 

15252 

15300 

56 

15348 

15396 

15415 

15493 

15541 

15590 

15638 

15687 

15736 

15784 

57 

15833 

15882 

15931 

15980 

16029 

16079 

16128 

17177 

16227 

16276 

58 

16326 

16376 

164'5 

16175 

16525 

16575 

16625 

16675 

16725 

16776 

59 

16826 

1*876 

16927 

16977 

17028 

17079 

17129 

17180 

17231 

17282 

60 1 

17333 

173 v 4 

17436 

17487 

17538 

17590 

17641 

17693 

17745 

17796 


For continuation to 100 feet deep, see Table 7. 




































736 


RAILROADS. 


Table 4. Iievel Cuttings. 

Roadway 18 feet, side-slopes 1% to 1. 


For single-track excavation. 


Depth 
in Ft. 

.0 

.1 

2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 


Cu. Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

0 


6.72 

13.6 

20.5 

27.6 

34.7 

42.0 

49.4 

56.9 

64.5 

1 

72.2 

80.1 

88.0 

96.1 

104.2 

112.5 

120.9 

129.4 

138.0 

146.7 

2 

155.5 

164.5 

173.5 

182.7 

191.9 

201.3 

210.8 

220.4 

230.1 

240.0 

3 

249.9 

260.0 

270.1 

280.4 

290 8 

301.3 

311.9 

322.6 

333.4 

344.5 

4 

355.5 

366.7 

378.0 

389.4 

400.9 

412.5 

424.2 

436 0 

448.0 

460.0 

5 

472.2 

484.5 

496.9 

509.4 

522.0 

534.7 

547.6 

560.5 

573.6 

586.7 

6 

600.0 

613.4 

626.9 

640.5 

654.2 

668.1 

682.0 

696.1 

710.2 

724.5 

7 

738.9 

753.4 

768.0 

782.7 

797.6 

812.5 

827.6 

842.7 

858.0 

873.4 

8 

888.9 

904.5 

920.2 

936.1 

952.0 

968.1 

984.2 

1001 

1017 

1033 

9 

1050 

1067 

1084 

1101 

1118 

1135 

1152 

1169 

1187 

1205 

10 

1222 

1240 

1258 

1276 

1294 

1313 

1331 

1349 

1368 

1387 

11 

1406 

1425 

1444 

1463 

1482 

1501 

1521 

1541 

1560 

1580 

12 

1600 

1620 

1640 

1661 

16S1 

1701 

1722 

1743 

1764 

1785 

13 

1806 

D27 

1848 

1869 

1891 

1913 

1934 

1956 

1978 

2000 

14 

2022 

2045 

2067 

2089 

2112 

2135 

2158 

2181 

2204 

2227 

15 

2250 

2273 

2297 

2321 

2344 

2368 

2392 

2416 

2440 

2465 

16 

2489 

2513 

2538 

2563 

2588 

2613 

2638 

2663 

2688 

2713 

17 

2739 

2765 

2790 

2816 

2842 

2868 

2894 

2921 

2947 

2973 

18 

3O00 

3027 

3054 

3081 

3108 

3135 

3162 

3189 

3217 

3245 

19 

3272 

3300 

3328 

3356 

3384 

3413 

3441 

3469 

3498 

3527 

20 

3556 

3585 

3614 

3643 

3672 

3701 

3731 

3761 

3790 

3820 

21 

3850 

3880 

3910 

3941 

3971 

4001 

4032 

4063 

4094 

4125 

22 

4156 

4187 

4218 

4249 

4281 

4313 

4344 

4376 

4408 

4440 

23 

4472 

4505 

4537 

4569 

4602 

4635 

4668 

4701 

4734 

4767 

24 

4800 

4833 

4867 

4901 

4934 

4968 

5002 

5036 

5070 

5105 

25 

5139 

5173 

5208 

5243 

5278 

5313 

5318 

5383 

5418 

5453 

26 

5489 

5525 

5560 

5596 

5632 

5668 

5704 

5741 

5777 

5813 

27 

5850 

5887 

5924 

5961 

5998 

6035 

6072 

6109 

6147 

6185 

28 

6222 

6260 

6298 

6336 

6374 

6413 

6451 

6489 

6528 

6567 

29 

6606 

6645 

6684 

6723 

6762 

6801 

6841 

6881 

6920 

6960 

30 

7000 

7040 

70S0 

7121 

7161 

7201 

7242 

7283 

7324 

7365 

31 

7406 

7447 

7488 

7529 

7571 

7613 

7654 

7696 

7738 

7780 

32 

7822 

7865 

7907 

7949 

7992 

8035 

8078 

8121 

8164 

8207 

33 

8250 

8293 

8337 

8381 

8424 

8468 

8512 

8556 

8600 

8645 

34 

8689 

8733 

8778 

8823 

8868 

8913 

8958 

9003 

9048 

9093 

35 

9139 

9185 

9230 

9276 

9322 

9368 

9414 

9461 

9507 

9553 

36 

9600 

9647 

9694 

9741 

9788 

9835 

9S82 

9929 

9977 

10025 

37 

10072 

10120 

10168 

10216 

10264 

10313 

10361 

10409 

10458 

10507 

38 

10556 

10605 

10654 

10703 

10752 

10S01 

10851 

10901 

10950 

11000 

39 

11050 

11100 

11150 

11200 

11251 

11301 

11352 

11403 

11454 

11505 

40 

11556 

11607 

11658 

11709 

11761 

11813 

11864 

11916 

11968 

12020 

41 

12072 

12125 

12177 

12229 

12282 

12335 

12388 

12441 

12494 

12547 

42 

12600 

12653 

12707 

12761 

12814 

12868 

12922 

12976 

13030 

13085 

43 

13139 

13193 

13248 

13303 

13358 

13413 

13468 

13523 

13578 

13633 

44 

136S9 

13745 

13800 

13856 

13912 

13968 

14024 

140S1 

14137 

14193 

45 

14250 

14307 

14364 

14421 

14478 

14535 

14592 

14649 

14707 

14765 

46 

14822 

148S0 

14938 

14996 

15054 

15113 

15171 

15229 

15288 

15347 

47 

15406 

15465 

15524 

15583 

15642 

15701 

15761 

15821 

15880 

15940 

48 

16000 

16060 

16120 

16181 

16241 

16301 

16362 

16423 

16484 

16545 

49 

16606 

16667 

16728 

16789 

16851 

16913 

16974 

17036 

17098 

17160 

50 

17222 

17285 

17347 

17409 

17472 

17535 

17598 

17661 

17724 

17787 

51 

17850 

17913 

17977 

18041 

1S104 

18168 

18232 

18296 

18360 

18425 

52 

18489 

18553 

18618 

18683 

18748 

18813 

18878 

18943 

19008 

19073 

53 

19139 

19205 

19270 

19336 

19402 

19468 

19534 

19601 

19667 

19733 

54 

19800 

19867 

19934 

20000 

20068 

20135 

20202 

20269 

20337 

20405 

55 

20472 

20540 

20608 

20676 

20744 

20813 

20881 

20949 

21018 

21087 

56 

21156 

21225 

21294 

21363 

21432 

21501 

21571 

21641 

21710 

21780 

57 

21850 

21920 

21990 

22061 

22131 

22201 

22272 

22343 

22414 

22485 

58 

22556 

22627 

22698 

22769 

22841 

22913 

22984 

23056 

23128 

23200 

59 

23272 

23345 

23417 

23489 

23562 

23635 

23708 

23781 

23854 

23927 

60 

24000 

24073 

24147 

24221 

24294 

24368 

24442 

24516 

24590 

24665 


For continuation to 100 feet deep, see Table 7. 































RAILROADS 


737 


Table 5. I^evel Cuttings. 

Roadway 28 feet wide, side-slopes 1 to 1. 


For double-track excavation. 


Depth 
'n Ft. 

.0 

.1 

,2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 


Cu. Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

0 


10.4 

20.9 

31.4 

42.1 

52.8 

63.6 

74.4 

85.3 

96.3 

1 

107.4 

118.6 

129.8 

141.1 

152.4 

163.9 

175.4 

187.0 

198.7 

210.4 

2 

222.2 

234.1 

246.1 

258.1 

270.2 

282.4 

294.7 

307.0 

319.4 

331.9 

3 

344.4 

357.1 

369.8 

382.6 

395.4 

408.3 

421.3 

434.4 

447 6 

460.8 

: 4 

474.1 

487.4 

500 9 

514.4 

528.0 

541.7 

555.4 

569.2 

583.1 

597.1 

5 

611.1 

625.2 

639.4 

653.7 

668.0 

682.4 

696.9 

711.4 

726.1 

740.8 

6 

755.6 

770.4 

785.4 

800.4 

815.5 

830.6 

845.8 

861.1 

876.5 

891.9 

7 

907.5 

923.0 

938.7 

954.5 

970.3 

986.2 

1002 

1018 

1034 

1050 

8 

1067 

1083 

1099 

1116 

1132 

1149 

1166 

1182 

1199 

1216 

9 

1233 

1250 

1267 

1285 

1302 

1319 

1337 

1354 

1372 

1390 

10 

1407 

1425 • 

1443 

1461 

1479 

1497 

1515 

1534 

1552 

1570 

11 

1589 

1607 

1626 

1645 

1664 

1682 

1701 

1720 

1739 

1759 

12 

1778 

1797 

1816 

1836 

1855 

1875 

1895 

1914 

1934 

1954 

13 

1974 

1994 

2014 

2034 

2055 

2075 

2095 

2116 

2136 

2157 

14 

2178 

2199 

2219 

2240 

2261 

2282 

2304 

2325 

2346 

2367 

15 

2389 

2410 

2432 

2454 

2475 

2497 

2519 

2541 

2563 

2585 

16 

2607 

2630 

2652 

2674 

2697 

2719 

2742 

2765 

2788 

2810 

17 

2833 

2856 

2879 

2903 

2926 

2949 

2972 

2996 

3019 

3043 

18 

3067 

3090 

3114 

3138 

3162 

3186 

3210 

3234 

3259 

3283 

19 

3307 

3332 

3356 

3381 

3406 

3431 

3455 

3480 

3505 

3530 

20 

3556 

3581 

3606 

3531 

3657 

3682 

3708 

3734 

3759 

3785 

21 

3811 

3837 

3863 

3889 

3915 

3942 

3968 

3994 

4021 

4047 

22 

4074 

4101 

4128 

4154 

4181 

4208 

4235 

4263 

4290 

4317 

: 23 

4344 

4372 

4399 

4427 

4455 

4482 

4510 

4538 

4566 

4594 

24 

4622 

4650 

4679 

4707 

4735 

4764 

4792 

4821 

4850 

4879 

i 25 

4907 

4936 

4965 

4994 

5024 

5053 

50S2 

5111 

5141 

5170 

26 

5200 

5230 

5259 

5289 

5319 

5349 

5379 

5409 

5439 

5470 

! 27 

5500 

5530 

5561 

5591 

5622 

56.53 

5684 

5714 

5745 

5776 

2S 

5807 

5839 

5870 

5901 

5932 

5964 

5995 

6027 

6059 

6090 

29 

6122 

6154 

6186 

6218 

6250 

6282 

6315 

6347 

6379 

6412 

30 

6444 

6477 

6510 

6543 

6575 

6608 

6641 

6674 

6708 

6741 

31 

6774 

6807 

6841 

6874 

6908 

6942 

6975 

7009 

7043 

7077 

32 

7111 

7145 

7179 

7214 

7248 

7282 

7317 

7351 

7386 

7421 

f 33 

7456 

7490 

7525 

7560 

7595 

7631 

7666 

7701 

7736 

7772 

34 

7807 

7843 

7879 

7914 

7950 

7986 

8022 

8058 

8094 

8130 

35 

8167 

8203 

8239 

8276 

8312 

8349 

8386 

8423 

8459 

8496 

36 

8533 

8570 

8608 

8645 

8682 

8719 

8757 

8794 

8S32 

8870 

37 

8907 

8945 

8983 

9021 

9059 

9097 

9135 

9174 

9212 

9250 

38 

9289 

9327 

9366 

9405 

9444 

9482 

9521 

9560 

9599 

9639 

39 

9678 

9717 

9756 

9796 

9835 

9875 

9915 

9954 

9994 

10034 

40 

10074 

10114 

10154 

10194 

10235 

10275 

10315 

10356 

10396 

10437 

41 

10178 

10519 

10559 

10600 

10641 

10682 

110724 

10765 

10806 

10847 

42 

10889 

10930 

10972 

11014 

11055 

11097 

11139 

11181 

11223 

11265 

43 

11307 

11350 

11392 

11434 

11477 

11519 

11562 

11605 

11648 

11690 

44 

11733 

11776 

11819 

11863 

11906 

11919 

11992 

12036 

12079 

12123 

45 

12167 

12210 

12254 

12298 

12342 

12386 

12430 

12474 

12519 

12563 

46 

12607 

12652 

12696 

12741 

12786 

12831 

12875 

12920 

12965 

13010 

47 

13056 

13101 

13146 

13191 

13237 

13282 

13328 

13374 

13419 

13465 

48 

13511 

13587 

13603 

13649 

13695 

13742 

13788 

13834 

13881 

13927 

49 

13974 

14021 

14068 

14114 

14161 

14208 

14255 

14303 

14350 

14397 

50 

14444 

14492 

14539 

14587 

14635 

14682 

14730 

14778 

14826 

14874 

51 

14922 

14970 

15019 

15067 

15115 

15164 

15212 

15261 

15310 

15359 

[ 52 

15407 

15456 

15505 

15054 

15604 

15653 

15702 

15751 

15801 

15850 

53 

15900 

15950 

15999 

16049 

16099 

16149 

16199 

16249 

16299 

16350 

54 

16400 

16450 

16501 

16551 

16602 

16653 

16704 

16754 

16805 

16856 

55 

16907 

16959 

17010 

17061 

17112 

17164 

17215 

17267 

17319 

17370 

56 

17422 

17474 

17526 

17578 

17630 

17682 

17735 

17787 

17839 

17892 

57 

17944 

17997 

18050 

18103 

18155 

18208 

18261 

18314 

18368 

18421 

58 

18474 

18527 

18581 

18634 

18688 

18742 

18795 

18849 

18903 

18957 

59 

19011 

19065 

19119 

19174 

19228 

19282 

19337 

19391 

19446 

19501 

60 

19556 

19610 

19665 

19720 

19775 

19831 

19886 

19941 

19996 

20052 


Fat* PAntlnnotinn ta 10A font aoo Tahlfl 7 




































738 


RAILROADS. 


Table 6. Level Cutting's. 

Roadway 28 It wide, side-slopes 1}/, to 1. 
For double-track excavation. 


Depth 
in Ft 

.0 

.1 

.2 

.3 

.4 

.5 

.6 

.7 

.8 

.9 


Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

Cu.Yds. 

0 


10.4 

21.0 

31.6 

42.4 

53.2 

64.2 

75 3 

86.5 

97.9 

1 

109.3 

120.8 

132.5 

144.3 

156.1 

168.1 

180.2 

192.4 

204.8 

217.2 

2 

229.6 

242.3 

255.0 

267.9 

280.9 

294.0 

307.2 

320.5 

334.0 

347.5 

3 

361.2 

374.9 

388.8 

402.8 

416.9 

431.1 

445.4 

459.9 

474.4 

489.1 

4 

503.7 

518.6 

533.6 

548 6 

563.9 

579.3 

594.7 

610.2 

625.8 

641.6 

5 

657.5 

673.4 

689.5 

705.7 

722.1 

738.5 

755 0 

771 7 

788.4 

805.3 

6 

822.2 

839.3 

856.5 

873.8 

891.2 

908.8 

926.4 

944.2 

962.0 

980.0 

7 

998.1 

1016 

1035 

1053 

1072 

1090 

1109 

1128 

1147 

1166 

8 

1185 

1204 

1224 

1243 

1263 

1283 

1303 

1322 

1343 

1363 

9 

1383 

1403 

1424 

1445 

1465 

1486 

1507 

1528 

1549 

1571 

10 

1592 

1614 

1635 

1657 

1679 

1701 

1723 

1745 

1767 

1790 

11 

1812 

1835 

1858 

1881 

1904 

1927 

1950 

1973 

1997 

2020 

12 

2044 

2068 

2092 

2116 

2140 

2164 

2189 

2213 

2238 

2262 

13 

2267 

2312 

2337 

2362 

2387 

2413 

2438 

2464 

2489 

2515 

14 

2541 

2567 

2593 

2619 

2645 

2672 

2698 

2725 

2752 

2779 

15 

2806 

2833 

2860 

2887 

2915 

2942 

2970 

2997 

3025 

3053 

16 

3081 

3109 

3138 

3166 

3195 

3223 

3262 

3281 

3310 

3339 

17 

3368 

3397 

3427 

3456 

3486 

3516 

3646 

3576 

3606 

2636 

18 

3667 

3697 

3728 

3758 

3789 

3820 

3851 

3882 

3913 

3944 

19 

3976 

4007 

4039 

4070 

4102 

4134 

4166 

4198 

4231 

4263 

20 

4296 

4328 

4361 

4394 

4427 

4460 

4493 

4527 

4560 

4594 

21 

4627 

4CC1 

4695 

4729 

4763 

4797 

4832 

4866 

4900 

4935 

22 

4970 

6005 

5040 

6075 

5111 

5146 

5181 

5217 

5253 

5288 

23 

5324 

6360 

5396 

5432 

5469 

5605 

5542 

5578 

5615 

5652 

24 

5689 

5726 

5763 

5800 

6838 

6875 

5913 

5951 

5989 

6027 

25 

6065 

6103 

6141 

6179 

6218 

6257 

6295 

6334 

6373 

6412 

26 

6451 

6491 

6530 

6570 

6609 

6649 

6689 

6729 

6769 

6809 

27 

6850 

6890 

6931 

6971 

7012 

7053 

7094 

7135 

7176 

7217 

28 

7259 

7300 

7342 

7384 

7426 

7468 

7610 

7552 

7594 

7637 

29 

7680 

7722 

7765 

7808 

7851 

7894 

7937 

7981 

8024 

8067 

30 

8111 

8155 

8199 

8243 

8287 

8331 

8375 

8420 

8464 

8509 

31 

8554 

8598 

8643 

8688 

8734 

8779 

8824 

8870 

8915 

8961 

32 

9007 

9053 

9099 

9145 

9191 

9238 

9284 

9331 

9378 

9425 

33 

9472 

9519 

9. 66 

9613 

9661 

9708 

9756 

9804 

9861 

990<* 

34 

9948 

9997 

10045 

10093 

10142 

10190 

10239 

10288 

10337 

10386 

35 

10435 

10484 

10534 

10583 

10633 

10683 

10732 

10782 

10832 

10882 

36 

10933 

109S3 

11034 

11084 

11135 

11186 

11237 

11288 

11339 

11391 

37 

11443 

11494 

11546 

11598 

11649 

11701 

11753 

11806 

111-58 

11910 

38 

11963 

12016 

120 CS 

12121 

12174 

12227 

12281 

12334 

12387 

12441 

39 

12494 

12548 

12602 

12656 

12710 

12764 

12819 

12873 

12928 

12982 

40 

13037 

13092 

13147 

13202 

13257 

13312 

13368 

13423 

13479 

13535 

41 

13591 

13647 

13703 

13759 

13815 

13872 

13928 

13985 

14042 

14099 

42 

14156 

14213 

14270 

14327 

14385 

14442 

14500 

14558 

14615 

14673 

43 

14731 

14790 

14848 

14906 

14965 

15024 

15082 

15141 

15200 

15259 

44 

15318 

15378 

15437 

15497 

15556 

15616 

16676 

15736 

15796 

15856 

45 

15917 

15977 

16038 

16098 

16159 

16220 

16281 

16342 

16403 

16465 

46 

16526 

16587 

16649 

16711 

16773 

16835 

16897 

16959 

17021 

17084 

47 

17146 

17209 

17272 

17335 

17398 

17461 

17524 

17587 

17651 

17714 

48 

17778 

17842 

17905 

17969 

18033 

18098 

18162 

18226 

18291 

18356 

49 

18420 

18485 

18550 

18615 

18680 

18746 

18811 

18877 

18942 

19008 

50 

19074 

19140 

19206 

19272 

19339 

19405 

19472 

19538 

19605 

19672 

51 

19739 

19806 

19873 

19940 

20008 

20075 

20143 

20211 

20279 

20347 

52 

20415 

20483 

20551 

20620 

20688 

20757 

20826 

20894 

20963 

21032 

21730 

53 

21102 

21171 

21211 

21310 

213-0 

21450 

21519 

21589 

21659 

54 

21 S00 

21870 

21941 

22012 

22082 

22153 

22224 

22295 

22366 

22438 

55 

22509 

22581 

22652 

22724 

22796 

22868 

22940 

23012 

23085 

23157 

56 

23230 

23302 

23375 

23448 

23521 

23594 

23667 

23741 

23814 

23888 

57 

23961 

24035 

24109 

24183 

24257 

24331 

24405 

24480 

24554 

24629 

58 

24704 

24779 

24854 

24929 

25004 

25079 

25155 

25230 

25306 

25381 

59 

25457 

25533 

25609 

25686 

25762 

25838 

25915 

25992 

26068 

26145 

60 

26222 

26299 

26376 

26454 

26531 

26609 

26686 

26764 

26842 

26920 


For continuation to 100 feet, see Table 7. 





















































RAILROADS. 


730 


Table 7. Isevel Cuttings. 

Continuation of the six foregoing Tables of Cubic Contents, to 100 feet of height or depth. 


Height 
lor Depth 
, in Feet. 


61 

.5 

62 

.5 

63 

.5 

61 

.5 

63 

.5 

66 

.5 

67 

.5 

68 

.5 

69 

.5 

70 

.5 

71 

.5 

72 

.5 

73 

.5 

74 

.5 

75 

.5 

76 

.5 

77 

.5 

78 

.5 


80 

81 

82 

83 

84 

85 

86 

87 

88 

89 

90 

91 

92 

93 

94 

95 

96 

97 

98 

99 
100 


Table 

1 


Cu. Yds 
23835 
24201 
24570 
24912 
25317 
25694 
26074 
26457 
26843 
27231 
27622 
28016 
28413 
28812 
29215 
29620 
30028 
30138 
30852 
31268 
31687 
32108 
32533 
32960 
33390 
33S23 
31259 
31697 
35139 
35582 
36029 
36t79 
36931 
37386 
37844 
38305 
38768 
39235 
39704 
40650 
41607 
42576 
43555 
44546 
45548 
46561 
47585 
48620 
49667 
50724 
51793 
52872 
53963 
55065 
56178 
57: '.02 
58437 
59583 
60741 


Table 

4 


Cu. Yds, 
26094 
26479 
26S67 
27257 
27650 
28046 
28444 
28816 
29250 
29657 
30067 
30479 
30894 
31313 
31733 
32157 
32583 
33013 
33444 
33879 
34317 
34757 
35200 
35646 
36094 
36546 
37000 
37457 
37917 
38379 
38844 
39313 
39783 
40257 
40733 
41213 
41694 
42179 
42667 
43650 
44644 
45650 
46667 
47694 
48733 
49783 
50844 
51917 
53000 
54094 
55200 
56317 
57414 
58583 
59733 
60894 
62067 
63250 
64414 


Table 

3 


Cu. Yds, 
17848 
IS 108 
18370 
18634 
18900 
19168 
19437 
19708 
19981 
20256 
20533 
20S12 
21093 
21375 
21659 
21945 
22233 
22523 
22S14 
23108 
23404 
23701 
21000 
24301 
24604 
24907 
25214 
25522 
25S32 
26144 
26458 
26774 
27092 
27411 
27733 
28056 
28381 
28708 
29037 
29700 
30370 
31048 
31733 
32426 
33126 
33833 
34548 
35270 
36000 
36737 
37481 
38233 
38993 
39759 
40533 
41315 
42101 
42900 
43704 


Table 

4 


Cu. Yds. 
24739 
25113 
25489 
25868 
26250 
26635 
27022 
27413 
27806 
28201 
28600 
29001 
29406 
29S13 
30222 
30G35 
31050 
31468 
31889 
33313 
32739 
33168 
33600 
34035 
34472 
34913 
35356 
35S01 
36250 
36701 
37156 
37C13 
38072 
38535 
39000 
39168 
39939 
40413 
40889 
41850 
42822 
43806 
44800 
45806 
468 22 
47850 
48889 
49939 
51000 
52072 
53156 
64250 
55356 
56472 
57600 
58739 
69889 
61050 
62222 


Table 


Cu. Yds. 

20107 
20386 
206G7 
20949 
21233 
21519 
21807 
22097 
22389 
22682 
22978 
23275 
23574 
23875 
24178 
21482 
24789 
25097 
25'07 
25719 
26033 
26349 
26GG7 
26986 
27307 
27631 
27956 
28282 
28611 
28942 
29174 
29608 
29944 
30282 
30622 
30964 
31307 
31653 
32000 
32700 
33107 
34122 
34S44 
35574 
36311 
37056 
37807 
38*07 
39333 
40107 
40889 
41678 
42174 
43278 
44089 
44907 
45733 
46567 
47407 


Table 

6 


Cu. Yds. 
26998 
27390 
27785 
28183 
28583 
28986 
29393 
29S01 
30213 
30627 
31044 
31461 
31887 
32312 
32741 
33172 
33605 
34042 
34481 
34924 
35369 
35816 
36267 
36720 
37176 
37635 
38096 
38561 
39028 
39198 
39970 
40416 
40924 
41405 
41889 
42375 
42865 
43357 
43852 
44850 
45859 
46880 
47911 
48954 
60098 
61072 
52148 
53235 
54333 
55413 
5G5G3 
57694 
58837 
59990 
61155 
62331 
63518 
64716 
65926 




























740 


RAILROADS 


Table 8, 

Of Cubic Yards in a 100-foot station of level cutting or filling, to be added to, or su 
tracted from, the quantities in the preceding seven tables, in case the excat 
tious or embankments should be increased or diminished 2 feet in width. 


Cubic Yards iu a length of 100 feet; breadth 2 feet; and of different depths. 


Height or 
Depth 
in Feet. 

Cubic 

Yards. 

'ieight or 
Depth 
in Feet. 

Cubic 

Yards. 

Height or 
Depth 
in Feet. 

Cubic 

Yards. 

Height or 
Depth 
in Feet. 

Cubic 

Yards. 

Height or 
Depth 
in Feet. 

Ctibi 

Yard 

.5 

3.70 

.5 

152 

.5 

300 

.5 

448 

5 

596 

1 

7.41 

21 

156 

41 

304 

61 

452 

81 

600 

.5 

11.1 

.5 

159 

.5 

307 

.5 

456 

.5 

604 

2 

14.8 

22 

163 

42 

311 

62 

459 

82 

607 

.5 

18.5 

.5 

167 

.5 

315 

.5 

463 

.5 

611 

3 

22.2 

23 

170 

43 

319 

63 

467 

83 

615 

.5 

25.0 

.5 

174 

.5 

322 

.5 

470 

.5 

619 

4 

20.6 

24 

178 

44 

326 

64 

474 

84 

622 

.5 

33.3 

.5 

181 

.5 

330 

.5 

478 

.5 

626 

5 

37.0 

25 

185 

45 

333 

65 

4S1 

85 

630 

.5 

40.7 

.5 

189 

.5 

337 

.5 

485 

.5 

633 

6 

44.4 

26 

193 

46 

341 

0G 

489 

86 

637 

.5 

48.1 

.5 

196 

.5 

344 

.5 

493 

.5 

641 

7 

51.9 

27 

200 

47 

348 

67 

496 

87 

644 

.5 

55.6 

.5 

204 

.5 

352 

.5 

500 

.5 

648 

8 

5J.3 

28 

207 

48 

356 

68 

504 

88 

652 

.5 

63.0 

.5 

211 

- .5 

359 

.5 

507 

.5 

656 ; 

9 

66.7 

29 

215 

49 

363 

69 

511 

89 

65S | 

.5 

70.1 

.5 

219 

.5 

367 

.5 

615 

.5 

663 

10 

74 1 

30 

222 

50 

370 

70 

519 

90 

667 

.5 

77.8 

.5 

226 

.5 

374 

.5 

522 

.5 

67C 

11 

81.5 

31 

230 

51 

378 

71 

526 

91 

674 

.5 

85 2 

.5 

233 

.5 

3S1 

.5 

530 

.5 

678 

12 

88.9 

32 

237 

52 

3S5 

72 

633 

92 

681 

.5 

92.6 

.5 

241 

.5 

389 

.5 

537 

.5 

685 

13 

96.3 

33 

244 

53 

393 

73 

541 

93 

682 

.5 

100 

.5 

248 

.5 

398 

.5 

544 

.5 

693 

14 

104 

34 

252 

54 

400 

74 

548 

94 

69* 

.5 

107 

.5 

256 

.5 

404 

.5 

552 

.5 

70C 

15 

111 

oo 

259 

65 

407 

75 

556 

95 

704 

.5 

115 

.5 

263 

.5 

411 

.5 

559 

.5 

707 

16 

119 

36 

267 

56 

415 

76 

563 

96 

711 

.5 

122 

.5 

270 

.5 

419 

.5 

567 

.5 

715 

17 

126 

37 

274 

67 

422 

77 

670 

97 

712 

.5 

130 

.5 

278 

.5 

426 

.5 

574 

.5 

72; 

18 

133 

38 

281 

58 

430 

78 

578 

98 

72( 

.5 

137 

.5 

285 

.5 

433 

.5 

5S1 

.5 

73C | 

19 

141 

39 

289 

'59 

437 

79 

585 

99 

733 

.5 

141 

.5 

293 

.5 

441 

.5 

5S9 

.5 

737 

20 

148 

40 

296 

60 

444 

80 

593 

100 

741 


Remark. The forevroing tables of level cutting's may also 1 
used for widths of roadway greater than those at the heat 
of the tables. Thus, suppose we wish to use Table 1, for a roadbed m n, 16 
wide, instead of <^b, which is only 14 ft, and for which the table was calculated, 
is only necessary first to find the vert dist s a, between these two roadbeds ; and 
add it mentally to each height t s, of the given embkt, when taking out from t 







































RAILROADS. 


741 


ible the numbers of cub yds corresponding to the heights. By this means we obtai* 
511 ie contents of the embkt chop , for any required dist. Next, from these contents 
** ibtract that corresponding to the height s a, for the same dist. The remainder will 
ainly be the embkt m n op. 

In practice it will be sufficiently correct to take sa to the nearest tenth of a 
x>t, which will save trouble in adding it mentally to the heights in the tables. 

. If the roadbed is narrower than the table, as, for instance, if mn be 
, be widrli in the table, but we wish to find the contents for the width cb , then 
l rst find sa, and calculate the cubic yards in 100 feet length of cbmn. Then, 
TJn taking out the cubic yards from the table, first subtract sa mentally from 
ach height ; and to the cubic yards taken out for each 100 feet, opposite this 
educed height, add the cubic yards in 100 feet of cbmn. 

To avoid trouble w r itl» contractors about the measurement of rock 
uts, stipulate in the contract, either that it shall conform with the theoretical 
ross section; or that an extra allowance of say about 2 feet of width of cut 
rill be made, to cover the unavoidable irregularities of the sides. 


Shrinkage of Embankment. Although earth, when first dug, and 
oosely thrown out, swells about \ part, so that a cubic yard in place averages 
bout 1£ or 1.2 cubic yards when dug: or 1 cubic yard dug is equal to f, or to 
3333 of a cubic yard in place; yet when made into embankment it gradually 
ubsides, settles, or shrinks, into a less bulk than it occupied before being dug. 

The following are approximate averages of the shrinkage; or, in other words, 
he earth measured in place in a cut, will, when made into embankment, occupy 
i bulk less than before by about the following proportions: 

Gravel or sand.about 8perct; or 1 in 12^ less. 


Clay 

Loam. “ 

Loose vegetable surface soil. “ 

Puddled clav. “ 


10 per ct; or 1 in 10 less. 
12 per ct ; or 1 in 8^ less. 
15 per ct; or 1 in 6% less. 
25 per ct; or 1 in 4 less. 


The writer thinks, from some trials of his own, that 1 cubic yard of any hard 
ock in place, will make from to 1% cubic yards of embankment; say on an 
iverage 1.7 cubic yards. Or that 1 cubic yard of rock embankment requires 
5882 of a cubic yard in place. He found that a solid cubic yard when broken 
Into fragments, made about as follows (see p 678): 


Cubic 
yards. 

In loose heap. . 1.9 

Carelessly piled. 1.75 

Carefully piled. 1.6 

Rubble, very carelessly scabbled. 1.5 

Rubble, somewhat carefully scabbled. 1.25 


Of which there were 


Solid 

52.6 per cent. 
57 

63 “ 

67 “ 

80 “ 


Voids 

47.4 percent. 
43 “ 

37 “ 

33 

20 “ 


13 

U 


For trestles, see p 755. 

For culverts and stone bridges, see pp 693, &c. 























742 


COST OF EARTHWORK 




COST OF EARTHWORK. 


Art. 1. It is advisable to pay for this kind of work by the cubic yard of excavation only; in¬ 
stead of allowing separate prices for excavation and embankment. By this means we get rid of the 
difficulty of measurements, as well as the controversies and lawsuits which often attend the deter- i 
ruination of the allowance to be made for the settlement or subsidence of the embankments. 

It is, moreover, our opinion that justice to the contractor should lead to the Engslisli j»rao 
tire of imyiiiii the laborers b.v tlie cubic yard, iustead of by the day. 1 
Experience fully proves that when laborers are scarce and wages high, men can scarcely be depended 
upon to do three-fourths of the work which they readily accomplish when wages are low, and when i 
fresh hauds are waiting to be hired in case any are discharged. The contractor is thus placed at the 
mercy of his men. The writer has known the most satisfactory results to attend a system of task- 
w ork, accompanied by liberal premiums for all overwork. By this means the interests of the laborers 
are identified with that of the contractor; and every man takes care that the others shall do their 
fair share of the task. 

Ellwood Morris. C E, of Philadelphia, was, we believe, the first person who properly investigated 
the elements of cost of earthwork, and reduced them to such a form as to enable us to calculate the 
total with a considerable degree of accuracy. He published his results in the Journal of the Franklin 
Institute in 1841. His paper forms the basis on which, with some variations, we shall consider the 
matter; and on which we shall extend it to wheelbarrows, as well as to carts. Throughout this paper 
we speak of a cubic yard considered only as solid in its place, or before it is loosened for removal. It 
is scarcely necessary to add that the various items can of course only be regarded as tolerably close 
approximations, or averages. As before stated, the men do less work when wages are high ; and more 
when they are low. A great deal besides depends on the skill, observation, and energy of the con¬ 
tractor and his superintendents. It is no unusual thing to see two contractors working at the same 
prices, in precisely similar material, where one is making money, and the other losing it, from a want 
of tact in the proper distribution of his forces, keeping his road's in order, having his carts aud bar- 
rows well filled, &c, Ac. Uncommonly long spells of wet weather may seriously affect the cost of exe- - 
cuting earthwork, by making it more difficult to loosen, load, or empty ; besides keeping the roads in 
bad order for hauling. 

The aggregate cost of excavating and removing earth is made up by the following items, namely: 

1st. Loosening the earth ready for the shovellers. 

2d. Loading it by shovels into the carts or barrows. 

3d. Hauling, or wheeling it away, including emptying and returning. 

4th. Spreading it out into successive layers on the embankment. 

5th. Keeping the hauling-road for carts, or the plank gangways for barrows, in good order. 

6th. Wear, sharpening, depreciation, and interest on cost of tools. 

7th. Superintendence, and water-carriers. 

8 th. Profit to the contractor. 

We will consider these items a little in detail, basing our calculations on the assumption that com¬ 
mon labor costs $1 per day. of 10 working hours. The results in our tables must therefore be in¬ 
creased or diminished in about the same proportion as common labor costs more or less than this. 

Art. 2. I.oonoii i ni; the earth ready for the shovellers. This is 

generally done either by ploughs or by picks ; more cheaply by the first. A plough with two horses, 
and two men to manage them, at $1 per day for labor, 75 cents per day for each horse, and 37 cents 
per day for plough, including harness, wear, repairs. Ac. or a total of $3.87, will loosen, of strong 
heavy soils, from 200 to 300 cubic yards a day, at from 1.93 to 1.29 cents per yard; or of ordinary 
loam, from 400 to 600 cubic yards a day, at from .97 to .64 of a cent per yard. Therefore, as an ordi¬ 
nary average, we may assume the actual cost to the contractor for loosening by the plough, as fol¬ 
lows : strong heavy soils, 1.6 cents ; common loam, .8 cent; light sandy soils, .4 cent. Very stiff pure 
clav, or obstinate cemented gravel, may be set down at 2.5 cents; they require three or four horses. 

By the pick, a fair day’s work is about 14 yards of stiff pure clay, or of cemented gravel; 25 yards 
of strong heavy soils; 40 yards of common loam; 60 yards of light sandy soils — all measured in 
place; which, at $1 per day for labor, gives, for stiff clay, 7 cents; heavy soils, 4 cents; loam, 2.5 
cents; light sandy soil, 1.666 cents. Pure sand requires but very little labor for loosening; .5 of a 
cent will cover it. 

Art. S. Shovelling the looseneri earth Into carts. The amount 

shovelled per day depends partly upon the weight of the material, but more upon so proportioning 
the number of pickers and of carts to that of shovellers, ns not to keep the latter waiting for either 
material or carts. In fairlv regulated gangs, the shovellers into carts are not actually engaged in 
shovelling for more than six-tenths of their time, thus being unoccupied but four-tenths of it; while, 
under bad management, they lose considerably more than one-half of it. A shoveller can readily 
load into a cart one-third of a cubic yard measured in place (and which is an average working cart¬ 
load!. of sandy soil, in five minutes ; of loam, in six minutes : and of any of the heavy soils, in seven * 
minutes. This would give, for a day of 10 working hours, 120 loads, or 40 cubic yards of light sandy 
soil; 100 loads, or 33J* cubic yards of loam : or 86 loads, or 28.7 yards of the heavy soils. But from 
these amounts we must deduct, four-tenths for time necessarily lost; thus reducing the actual work¬ 
ing quantities to 24 yards of light sandy soil, 20 yards of loam, 17.2 yards of the heavy soils. When 
the shovellers do less than this, there is some mismanagement. 

Assuming these as fair quantities, then, at $1 per day for labor, the actual cost to the contractor 
for shovelling per cubic vard measured in place, will be, for sandy soils, 4.167 cents; loam, 5 cents; 
heavy soils, clays. Ac, 5.81 cents. 

In practice, the carts are not usually loaded to any less extent with the heavier soils than with the 
lighter ones. Nor. indeed, is there any necessity for so doing, inasmuch as the difference of weight 
of a cart and one third of a cubic yard of the various soils is too slight to need any attention ; espe¬ 
cially when the cart-road is kept in good order, as it will be by any contractor who understands his 




COST OF EARTHWORK. 743 


own interest. Neither is it necessary to modify the load on account of any slight inclinations which 
may occur in the grading of roads. An earth-cart weighs by itself about 34 a ton. 

Art. 4. Hauling- away the earth; (lumping, or emptying* 
and returning to reload. The average speed of horses in hauling is about 2 U niUes 
per hour, or 200 feet per minute; which is equal to 100 feet of trip each wav ; or to 100 feet of lead 
as the distance to which the earth is hauled is technically called.* Besides this, there is a loss of 
about four minutes in every trip, whether long or short, in waitiug to load, dumping turning &c 
Hence, every trip will occupy as many minutes as there are lengths of 100 feet each in the lead - and 
rour minutes besides. Therefore, to fiud the number of trips per day over any given average lead we 
divide the number of minutes in a working day by the sum of -4 added to the number of 100 feet 
lengths contained in the distance to which the earth has to be removed; that is, 

The number (600) o f minutes in a working day _ tlle num ber of trips, or loads 
4 the number of 100- feet lengths in the lead removed per day, per cart. 

And since J4 of a cubic yard measured before being loosened, makes an average cart-load, the num¬ 
ber of loads, divided by 3, will give the number of cubic yards removed per day by each cart; and 
the cubic yards divided into the total expense of a cart per day, will give the cost per cubic yard for 
hauling. J 

Remark. When removing loose rock, which requires more time for loading, say, 

No ' °f (600 > in • working day _ No of loads removed> 

6 -f- No. of 100-feet lengths of lead per day, per cart. 

In leads of ordinary length one driver can attend to 4 carts ; which, at $1 per day, is 25 cents per 
cart. Wheu labor is $1 per day, the expense of a horse is usually about 75 cents; and that of the 
cart, including harness, tar, repairs, &c, 25 ceuts, making the total daily cost per cart $1.25. The 
expense of the horse is the same on Sundays and on rainy days, as when at work ; and this consid¬ 
eration is included in the 75 cents. Some contractors employ a greater number of drivers, who also 
help to load the carts, so that the expense is about the same in either case. 

Example. How many cubic yards of loam, measured in the cut, can be hauled by a horse and cart 
in a day of 10 working hours, (600 minutes,) the lead, or length of haul of earth being 1000 feet, (or 
10 lengths of 100 feet,) and what will be the expense to the contractor for hauling, per cubic yard 
assuming the total cost of cart, horse, and driver, at $1,257 


Here, 


600 minutes 


4-}-10 lengths of 100 feet. 


And 


600 

IT 

125 cents 


43 loads. And 


43 loads 


— 14.3 cubic yards. 


— 8.74 cents per cubic yard. 


14.3 cub yds 

In this manner the 2d and 3d columns of the following tables have been calculated. 


Art. 5. Spreading-, or levelling off the earth into regular 
thin layers OI1 the embankment. A bankman will spread from 50 to lOOcubic 
yards of either common loam, or any of the heavier soils, clays, &c, depending on their dryness. 
This, at $1 per day, is 1 to 2 cents per cubic yard; and we may assume 1 34 cents as a fair average 
for such soils; while l cent will suffice for light sandy soils. 

This expense for spreading is saved when the earth is either dumped over the end of the embank¬ 
ment, or is wasted; still, about )4 cent per yard should be allowed in either case for keeping the 
dumping-places clear and in order. 

Art. 6. Keeping the cart-road in good order for hauling. 

No ruts or puddles should be allowed to remaiu unfilled; rain should at once be led off by shallow 
ditches; and the road be carefully kept in good order; otherwise the labor of the horses, and the wear 
of carts, will be very greatly increased. It is usual to allow so much per cubic yard for road repairs; 

but we suggest so much per cubic yard, per 100 feet of lead ; say -jiy of a cent. 

Art. 7. Wear, sharpening, and depreciation of picks and 

Shovels. Experience shows that about % of a ceut per cubic yard will cover this item. 

Superintendence and water-carriers. These expenses will vary with 
local circumstances; but we agree with Mr. Morris, that \ % cents per cubic yard will, under ordinary 
circumstances, cover both of them. An allowance of about % cent may in justice he added for extra 
trouble in digging the side-ditches; levelling off the bottom of the cut to grade; and general trimming 
up. In very light cuttings this may be increased to J4 cent per cubic yard. 

At 34 cent. all the items in this article amount to 2 cents per cubic yard of cut. 

Art. 8. Profit to the contractor. This may generally be set down at from 6to 
15 per cent, according to the magnitude of the work, the risks incurred, and various incidental cir¬ 
cumstances. Out of this item the contractor generally has to pay clerks, storekeepers, and other 
agents, as well as the expenses of shanties, &c ; although these are in most cases repaid by the profits 
of the stores; and by the rates of boarding and lodging paid to the contractors by the laborers. 

Art. 9. A knowledge of the foregoing items enables ns to 
calcnlate with tolerable accuracy the cost of removing earth. 

For example, let it be required to ascertain the cost per cubic yard of excavating common loam, meas¬ 
ured in place; and of removing it into embankment, with an average haul or lead of 1000 feet; tho 
wages of laborers being $1 per day of 10 working hours; a horse 75 cts a day; and a cart 25 cts. Ono 
driver to four carts. 


-X- When an entire cut is made into an embankment, the mean haul is the dist between centers 
•f gravity of the cut and embkt. 










744 


COST OF EARTHWORK 


Here we have cost of loosening, say by pick, Art 2, per cubic yard, say, 
Loading into carts, Art. 3, “ “ 

Hauling 1000 feet, as calculated previously in example, Art. 4, “ 
Spreading into layers, Art. 5, “ 

Keeping cart-road in repair. Art. 6 , 10 lengths of 100 ft, 

Various items in Art. I, 


Cents. 

2.50 
5.00 
8.74 

1.50 
1.00 
2.00 


Total cost to contractor. 

Add contractor’s profit, say 10 per cent, 


20.74 

2.074 


Total cost per cubic yard to the company, 22.814 

It is easy to construct a table like the following, of costs per cnbicyard, for different lengths of lead. 
Columns 2 and 3 are first obtained by tfie Rule in Article 4; then to each amount in column 3 is added 

the variable quantity of yL of a cent for every 100 feet length of lead, for keeping the road in order; 
and the constant quantity (for any given kind of soil) composed of the prices per cubic yard, for 
loosening, loading, spreading, or wasting, &c, either taken from the preceding articles; or modified 
to suit particular circumstances. In this mauuer the tables have been prepared. 

By Carts. Labor §1 per day, of IO working- Iionrs. 


Length of Lead, or distance to which 
the earth is hauled, in feet. 

Number of cubic yards In place, 
hauled per day by each cart. 

Cost per cubic yard in place, for 
hauling and emptyiug only. 

Common Loam. 

Strong Heavy Soils. 

TOTAL COST PER CUBIC 
YARD, EXCLUSIVE OF 
PROFIT TO CONTRACTOR. 

TOTAL COST PER CUBIC 
YARD, EXCLUSIVE OF 
PROFIT TO CONTRACTOR. 

Picked 

and 

Spread, 

Picked 

and 

Wasted. 

Ploughed 

and 

Spread. 

Ploughed 

and 

Wasted. 

Picked 

and 

Spread. 

Picked 

and 

Wasted. 

Ploughed 

and 

Spread. 

Ploughed 

and 

Wasted. 

Feet. 

Cu.Yds. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

25 

47.0 

2.66 

13.69 

12 44 

11.99 

10.74 

16.00 

14.75 

13.50 

12.25 

50 

44.4 

2.81 

13.86 

12.61 

12.16 

10.91 

16.17 

14.92 

13.67 

12.42 

75 

42.1 

2.97 

14.05 

12.80 

12.35 

11.10 

16.36 

15.11 

13.86 

12.61 

100 

40.0 

3.12 

14.22 

12.97 

12.52 

11.27 

16.53 

15.28 

14.03 

12.78 

150 

36.4 

3 43 

14.58 

13.33 

12.88 

11.63 

16.89 

15 64 

14.39 

13 14 

200 

33.3 

3.75 

14.95 

13.70 

13.25 

12.00 

17.26 

16.01 

14.76 

13.51 

300 

28.6 

4.37 

15.67 

14.42 

13.97 

12.72 

17.98 

16.73 

15.48 

14.23 

400 

250 

5.00 

16.40 

15.15 

14.70 

13.45 

18.71 

17.46 

16.21 

14.96 

500 

22.2 

5.63 

17.13 

15 88 

15.43 

14.18 

19.44 

18.19 

16.94 

15 69 

600 

20.0 

6.25 

17.85 

16.60 

16.15 

14.90 

20.16 

18.91 

17.66 

16.41 

700 

18.2 

6.87 

18.57 

17.32 

16.87 

15.62 

20.88 

19.63 

18.38 

17.13 

800 

16.7 

7.48 

19.28 

18.03 

17.58 

16.33 

21.59 

20.34 

19.09 

17.84 

900 

15.4 

8.12 

19.92 

18.67 

18.22 

16.97 

22.23 

20.98 

19.73 

18.48 

1000 

14.3 

8.74 

20.74 

19.49 

19.04 

17.79 

23.05 

2180 

20.55 

19.30 

1100 

13.3 

9.40 

21.50 

20.25 

19.80 

18.55 

211.81 

22.56 

21.31 

20.06 

1200 

12.5 

10.0 

22.20 

20.95 

20.50 

19 25 

24.51 

23.26 

22.01 

20.76 

1300 

11.8 

10.6 

22.90 

21.65 

21.20 

19.95 

25.21 

23.96 

22.71 

21.46 

1400 

11.1 

11.2 

23.60 

22.35 

21.90 

20.65 

25.91 

24.66 

23.41 

22.16 

1500 

10.5 

11.9 

24.40 

23.15 

23.70 

21.45 

26.71 

25.46 

24.21 

22.96 

1600 

10.0 

12.5 

25.10 

23.85 

23.40 

22.15 

27.41 

26.16 

21.91 

23.66 

1700 

9.52 

13.1 

25.80 

24.55 

24.10 

22.85 

28.11 

26.86 

25 61 

24.36 

1800 

9.09 

13.7 

26.50 

25.25 

24.80 

23.55 

28.81 

27.56 

26.31 

25.06 

1900 

8.70 

14.4 

27.30 

26 05 

25.60 

24.35 

29.61 

28.36 

27.11 

25.86 

2000 

8.33 

15.0 

28.00 

26.75 

26.30 

25.05 

30.31 

29.06 

27.81 

26.56 

2250 

7.54 

16.6 

29.85 

28.60 

28.15 

26.90 

32.16 

30.91 

29.66 

28.41 

2500 

690 

18.1 

31.60 

30.35 

29.90 

28.65 

33.91 

32.66 

31.41 

30.16 

J4 mile 

6.58 

19.0 

32.64 

31.39 

30.94 

29.69 

34.95 

33.70 

32.45 

31.20 

3000 

5.88 

21.2 

35.20 

33.95 

33.50 

32.25 

37.51 

36.26 

35.01 

33.76 

3250 

5.48 

22.8 

37.05 

35.80 

35.35 

34.10 

39.36 

38.11 

36.86 

35.61 

3500 

5.13 

24.3 

38.80 

37.55 

37.10 

35.85 

41.11 

39.86 

38.61 

37.36 

3750 

4.82 

25 9 

40.65 

39.40 

38.95 

37.70 

42.96 

41.71 

40.46 

39.21 

4000 

4.54 

27.5 

42.50 

41.25 

40.80 

39.55 

44.81 

43.56 

42.31 

41.06 

4250 

4.30 

29.1 

44.35 

43.10 

42.65 

41.40 

46.66 

45.41 

44.16 

42.91 

4500 

4.08 

30.6 

46.10 

44.85 

44.40 

43.15 

48.41 

47.16 

45.91 

44.66 

4750 

3.88 

32.2 

47.95 

46.70 

46.25 

45.00 

50.26 

49.01 

47.76 

46.51 

5000 

3.70 

33.8 

49.80 

48.55 

48.10 

46.85 

52.11 

50 86 

49.61 

48.36 

1 mile 

3 52 

35.5 

51.78 

50.53 

50.08 

48.83 

54.09 

52.84 

51.59 

50.34 

w- 

2.86 

43.8 

61.40 

6 *). 15 

59.70 

58.45 

63.71 

62.46 

61.21 

59.96 

134 m. 

2.40 

52.1 

71.02 

69.77 

69.32 

68.07 

73.83 

72.08 

70.83 

69.58 

194 

2.07 

60.4 

80.64 

79.39 

78.94 

77.69 

82.95 

81.70 

80.45 

79.20 

2 m. 

1.82 

68.7 

90.26 

89.01 

88.56 

87.31 

92.57 

91.32 

90.07 

88.82 

















































COST OF EARTHWORK 


745 


By Carts. Labor $1 per day, of 10 working- hours. 


Length of Lead, or distance to which 
the earth is hauled, in feet. 

Number of cubic yards in place, 
hauled per day by each cart. 

Cost per cubic yard in place, for 
hauling and emptying only. 

Pure stiff Clay, or cemented 
Gravel. 

Light Sandy Soils. 

TOTAL COST PER CUBIC 
YARD, EXCLUSIVE OF 
PROFIT TO CONTRACTOR. 

TOTAL COST PER CUBIC 

YARD, EXCLUSIVE OF 

PROFIT TO CONTRACTOR. 

Picked ‘ 

and 

Spread. 

Picked 

and 

Wasted. 

Ploughed 

and 

Spread. 

Ploughed 

and 

Wasted. 

Picked 

and 

Spread. 

Picked 

and 

Wasted. 

Ploughed 

and 

Spread. 

Ploughed 

and 

Wasted. 

Feet. 

Cu.Yds. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

25 

47.0 

2.66 

19.00 

17.75 

14.50 

13.25 

11.52 

10.77 

10.25 

9.50 

50 

44.4 

2.81 

19.17 

17.92 

14.67 

13.42 

11.69 

10.94 

10.42 

9.67 

75 

42.1 

2.97 

19.36 

18.11 

14.86 

13.61 

11.88 

11.13 

10.61 

9.86 

100 

40.0 

3.12 

19.53 

18.28 

15.03 

13.78 

12.05 

11.30 

10.78 

10.03 

150 

36.4 

3.43 

19.89 

18.64 

15.39 

14.14 

12.41 

11.66 

11.14 

10.39 

200 

33.3 

3 75 

20.26 

19.01 

15.76 

14.51 

12.78 

12.03 

11.51 

10.76 

300 

28.6 

4.37 

20.98 

19.73 

15.48 

15.23 

13.50 

12.75 

12.23 

11.48 

400 

25.0 

5.00 

21.71 

20.46 

17.21 

15.96 

14.23 

13.48 

12.46 

12.21 

500 

22.2 

5.63 

22.44 

21.19 

17.94 

16.69 

14.96 

14.21 

13.69 

12.94 

600 

20.0 

6.25 

23,16 

21.91 

18.66 

17.41 

15.68 

14.93 

14.41 

13.66 

700 

18.2 

6.87 

23.88 

22.63 

19.38 

18.13 

16.40 

15.65 

15.13 

14.38 

800 

16.7 

7.48 

24.59 

23.34 

20.09 

18.84 

17.11 

16.36 

15.84 

15 09 

900 

15.4 

8.12 

25.23 

23.98 

20.73 

19.48 

17.75 

17.00 

16.48 

15.73 

1000 

14.3 

8.74 

26.05 

24 80 

21.55 

20.30 

18.57 

17.82 

17.30 

16.55 

1100 

13.3 

9.40 

26.81 

25.56 

22.31 

21.06 

19.33 

18.58 

18.06 

17.31 

1200 

12.5 

10.0 

27.51 

26.26 

23.01 

21.76 

20.03 

19.28 

18.76 

18.01 

1300 

11.8 

10.6 

28.21 

26.96 

23.71 

22.46 

20.73 

19.98 

19.46 

18.71 

1400 

11.1 

11.2 

28.91 

27.66 

24.41 

23.16 

21.43 

20.68 

20.16 

19.41 

1500 

10.5 

11.9 

29.71 

28.46 

25.21 

23.96 

22.23 

21.48 

20.96 

20.21 

1600 

10.0 

12.5 

30.41 

29.16 

25.91 

24.66 

22.93 

22.18 

21.66 

20.91 

1700 

9.52 

13.1 

31.11 

29.86 

26.61 

25.36 

23.63 

22.88 

22.36 

21.61 

1800 

9.09 

13.7 

31.81 

30.56 

27 31 

26.06 

24.33 

23.58 

23.06 

22.31 

1900 

8.70 

14.4 

32.61 

31.36 

28.11 

26.86 

25.13 

24.38 

23.86 

23.11 

2000 

8.33 

15.0 

33.31 

32.06 

28.81 

27. 56 

25.83 

25.08 

24.56 

23.81 

2250 

7.54 

16.6 

35.16 

33.91 

30.66 

29.41 

27.68 

26.93 

26 41 

25.66 

2500 

6.90 

18.1 

36.91 

35.66 

32.41 

31.16 

29.43 

28.68 

28.16 

27.41 

14 mile 

6.58 

19.0 

37.95 

36.70 

33.45 

32.20 

30.47 

29.72 

29.20 

28.45 

*3000 

5.88 

21.2 

40.51 

39.26 

36.01 

34.76 

33.03 

32.28 

31.76 

31.01 

3250 

5.48 

22.8 

42.36 

41.11 

37.86 

36.61 

34.88 

34.13 

33.61 

32.86 

3500 

5.13 

24.3 

44.11 

42.86 

39.61 

38.36 

36.63 

35.88 

35.36 

34.61 

3750 

4.82 

25.9 

45.96 

44.71 

41.46 

40.21 

38.48 

37.73 

37.21 

36.46 

4000 

4.54 

27.5 

47.81 

46.56 

43.31 

42.06 

40.33 

39.58 

39.06 

38.31 

4250 

4.30 

29.1 

49.66 

48.41 

45.16 

43.91 

42.18 

41.45 

40.93 

4018 

4500 

4.08 

30.6 

51.41 

50.16 

46.91 

45.66 

43.93 

43.18 

42.66 

41.91 

4750 

3.88 

32.2 

53.26 

52.01 

48.76 

47.51 

45.78 

45.03 

44 51 

43.76 

5000 

3.70 

33.8 

55.11 

53.86 

50.61 

49.36 

47.63 

46.88 

46.36 

45.61 

1 mile 

3.52 

35.5 

57.09 

55 84 

52.59 

51.34 

49 61 

48.86 

48.34 

47.59 

1J4 m. 

2.86 

43.8 

66.91 

65.46 

62.21 

60.96 

59.23 

58.48 

57.96 

57.21 

1)4 m. 

2.4(7 

52.1 

76.33 

75.08 

71.83 

70.58 

68.85 

68.10 

67.58 

66.83 

Wk m. 

2.07 

60.4 

85.95 

84.70 

81.45 

80.20 

78.47 

77.72 

77.20 

76.45 

2 * m. 

1.82 

68.7 

95.57 

94.32 

91.07 

89.82 

88.09 

87.34 

66.82 

86.07 


Art. 10. By wheelbarrows. The cost by barrows may be estimated in the same 
ianner as by carts. See Articles 1, &c. Men in wheeling move at about the same average rate as 
-rses do in hauling, that is, 2% miles an hour, or 200 feet per minute, or 1 minute per every 100 feet 
mgth of lead. The time occupied in loading, emptying, &c (when, as is usual, the wheeler loads his 
wn barrow,! is about 1.25 minutes, without regard to length of lead; besides which, the time lost in 
•ccasional short rests, in adjusting the wheeling plank, and in other incidental causes, amounts to 
bout JL. part of his whole time; so that we must in practice consider him as actually working but 
hours out of his 10 working ones. Therefore 

The number of minutes in a working day X .9 _ the number of trips or of loads 

T^5 + the number of 100-/ee< lengths of lead ~ removed per day per barrow. 

. See Remark, next page. m , , , 

The number of loads divided by 14 will give the number of cub yards, since a cub yard, measured 
place averages about 14 loads. And the cost of a wheeler and barrow per day. (say $1 per man. 
id 5 cents per barrow.) dit ided by the number of cub yards, will give the oost per yard for loading 
/heeling, and emptying. 
































746 


COST OF EARTHWORK 


Ex. How many cubic yards of common loam, measured in place, will one man load, wh 
and empty, per day of 10 working hours, (or (iOO minutes;) the lead, or distance to which the eart 
removed being 1000 feet, (or 10 lengths of 100 feet;) and what will be the expense per yard, suppo> 
the laborer and barrow to cost $1.05 per day ? 

600 minutes X .0 510 

Here, — ~ —— , ~ 48 trips, or loads per day. 

1.2o -f- 10 lengths ’' 


48 


And yy = 3.43 cub yds per day. 


11.28 
And 


105 cent 8 


= 30.6 cents 


3.43 cub yds 

per cub yard for loading, wheeling away, emptying, and returning. This would be increased aln 
inappreciably by the cost of the shovel, which, in the following tables, however, is included in 
cost of tools. 

Rom. For rock, which requires more time for loading, say 4 

No of minute s in a working d ay X 0 _ No of loads removed 

per day, per barrow. 


1.6 -J- No of 100- feet lengths of lead 

Aft. 11. The following tables are calculated as in the case of carts, by first finding columt 
and 3 by means of the Rule in Art 10,and then adding to each sum in column 3, the variable quam 
of .1 of a cent per cubic yard per 100 feet of lead for keeping the wheeling-planks in order; aud 
prices of loosening, spreading, superintendence, water-carrying, Ac, per cubic yard, as given in 
preceding Articles 2 to 7. 


By Wheelbarrows. Labor $1 per <lay, of 10 working- lioui 


Length of Lead, or distance to which 
the earth is wheeled, in feet. 

Number of cubic yards in place, 
loaded, and wheeled per day; 
each barrow. 

* ---- - ■ 

Cost per cubic yard in place, for 
loading, wheeling, and emptying. 

Common Loam. 

Strong, Heavy Soils. 

TOTAL COST PER CUBIC 
YARD, EXCLUSIVE OF 

PROFIT TO CONTRACTOR. 

TOTAL COST PER CUBIC 

YARD, EXCLUSIVE 01 

PROFIT TO CONTRACTOR 

Picked 

and 

Spread. 

Picked 

and 

Wasted. 

Ploughed 

and 

Spread. 

Ploughed 

and 

Wasted. 

Picked 

and 

Spread. 

Picked 

and 

Wasted. 

Ploughed 

and 

Spread. 

72 

s \ 

tserr t 
3 C k 

Feet. 

Cu.Yds. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

25 

25.7 

4.09 

10.12 

8.87 

8.42 

7.17 

11.62 

10.37 

9.12 

7.87 

50 

22.1 

4.75 

10.80 

9.55 

9.10 

7.85 

12.30 

11.05 

9.80 

8.55 

75 

19.3 

5.44 

11.52 

10.27 

9.82 

8.57 

13.02 

11.77 

10.52 

9.27 

100 

17.1 

6.14 

12.24 

10.99 

10.54 

9.29 

13.74 

12.49 

11.24 

9.99 

150 

14.0 

7.50 

13.65 

12.40 

11.95 

10.70 

15.15 

13.90 

12.65 

11.40 

200 

11.9 

8.82 

15.02 

13.77 

13.32 

12.07 

16.52 

15.27 

14.02 

12.77 

250 

10.3 

10.2 

16.45 

15.20 

14.75 

13.50 

17.95 

16.70 

15 45 

14.20 

300 

9.07 

11.6 

17.90 

16.65 

16.20 

14.95 

19.40 

18.15 

16.90 

15 65 

350 

8.14 

12.9 

19.25 

18.00 

17.55 

16.30 

20.75 

19.50 

18 25 

17.00 

400 

7.36 

14.3 

20.70 

19.45 

19.00 

17.75 

22.20 

20.95 

19.70 

18.45 

450 

6.71 

15.6 

22.05 

20.80 

20.35 

19.10 

23.55 

22.30 

21.05 

19.80 

500 

6.17 

17.0 

23.50 

22.25 

21.80 

20.55 

25.00 

23.75 

22.50 

21.25 

600 

5.32 

19.7 

26.30 

25.05 

24.60 

23.35 

27.80 

26.55 

25.30 

24.05 

700 

4.67 

22.5 

29.20 

27.95 

27.50 

26.25 

30.70 

29.45 

28.20 

76.95 

800 

4.17 

25.2 

32.00 

30.75 

30.30 

29.05 

33.50 

32 25 

31.00 

29.75 

900 

3.76 

27.9 

34.80 

33,55 

33.10 

31.85 

36.30 

35.05 

33.80 

32.55 

1000 

3.43 

30.6 

37.60 

36.35 

35.90 

34.65 

39.10 

37.85 

36.60 

35.35 

1200 

2.91 

36.1 

43.30 

42.05 

41.60 

40.35 

44.80 

43.55 

42.30 

41.05 

1400 

2.53 

41.5 

48.90 

47.65 

47.20 

45.95 

50.40 

49.15 

47.90 

46.65 

1600 

2.24 

46.9 

54.50 

53.45 

52.80 

51.55 

56.00 

54.75 

53.50 

52.25 

1800 

2.00 

52.5 

60.30 

59.05 

58.60 

57.35 

61.80 

60.55 

59.30 

58.05 

2000 

1.81 

58.0 

66.00 

64.75 

64.30 

63.05 

67.50 

66.25 

65.00 

63.75 

2200 

1.66 

63.3 

71.50 

70.25 

69.80 

68.55 

73.00 

71.75 

70.50 

69.25 

2400 

1.53 

68.6 

77.00 

75.75 

75.30 

74.05 

78.50 

77.25 

76.00 

74.75 

*4 mile. 

1.39 

75.5 

84.14 

82.89 

82.44 

81.19 

85.64 

84.39 

83.14 

81.89 








































COST OF EARTHWORK. 747 


uiSy Wheelbarrows. 

»i 


I,abor Si per day, of lO working- hours. 


Length of Lead, or distance to 
which the earth is wheeled. 

Number of cubic yards in place, 
loaded, aud wheeled, per day; 
each barrow. 

| Cost per cubic yard in place, for 
loading, wheeling, and emptying. 

Pure Stiff Clay, or Ce¬ 
mented Gravel. 

Light Sandy Soils. 

TOTAL COST PER CUBIC 
Y ART), EXCLUSIVE OF 
PROFIT TO CONTRACTOR. 

TOTAL COST PER CUBIC 
Y-ARO, EXCLUSIVE OF 
PROFIT TO CONTRACTOR. 

Picked 

and 

Spread. 

Picked 

aud 

W asted. 

Ploughed 

and 

Spread. 

Ploughed 

' and 

Wasted. 

Picked 

and 

Spread. 

Picked 

aud 

Wasted. 

Ploughed 

and 

Spread. 

Ploughed 

aud 

Wasted. 

Feet. 

Cu.Yds. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

25 

25.7 

4.09 

14.62 

13.37 

10.12 

8.87 

8.79 

8.04 

7.52 

6.77 

50 

22.1 

4.75 

15.30 

14 05 

10.80 

9.55 

9.47 

8.72 

8.20 

7.45 

75 

19.3 

5 44 

18.02 

14.77 

11.52 

10.27 

10.19 

9.44 

8.92 

8.17 

too 

17.1 

6.14 

16.74 

15.49 

12.24 

10.99 

10.91 

10.16 

9.64 

8.89 

150 

14 0 

7.50 

18.15 

16.90 

13.65 

12.40 

12.32 

11.57 

11.05 

10.30 

200 

11.9 

8.82 

79.52 

18.27 

15.02 

13.77 

13.69 

12.94 

12.42 

11.67 

250 

10.3 

10.2 

20.95 

19.70 

16.45 

15.20 

15.12 

14.37 

13.85 

13.10 

300 

9.07 

11.6 

22.40 

21.15 

17.90 

16.65 

16.57 

15.82 

15.30 

14.55 

350 

8.14 

12.9 

23.75 

22.50 

19.25 

18.00 

17.92 

17.17 

16.65 

15.90 

400 

7.36 

14.3 

25.20 

23.95 

*>0 70 

19.45 

19.37 

18.62 

18.10 

17.35 

450 

6.71 

15.6 

26.55 

25.30 

22.05 

20.80 

20.72 

19.97 

19.45 

18.70 

500 

6.17 

17.0 

28.00 

26.75 

23.50 

22.25 

22.17 

21.42 

20.90 

20.15 

600 

5.32 

19.7 

30.80 

29.55 

26.30 

25.05 

24.97 

24.22 

23.70 

22.95 

700 

4.67 

22.5 

33.70 

32.45 

29.20 

27.95 

27.87 

27.12 

26.60 

25.85 

800 

4.17 

25.2 

36.50 

35.25 

32.00 

30.75 

30.67 

29.92 

29.40 

28.65 

900 

. 3,76 

27.9 

39.30 

38.05 

34.80 

33.55 

33.47 

32.72 

32.20 

31.45 

1000 

3.43 

30.6 

42.10 

40.85 

37.60 

36.35 

36.27 

35.52 

35.00 

34.25 

1200 

2.91 

36.1 

47.80 

46.55 

43.30 

42 05 

41.97 

41.22 

40.70 

39.90 

1400 

2.53 

41.5 

53.40 

52.15 

48.90 

47.65 

47 57 

46.82 

46.30 

45.55 

1600 

2.24 

46.9 

59.00 

57.75 

54.50 

53.25 

53.17 

52.42 

51.90 

51.15 

1800 

2.00 

52.5 

64 80 

63.55 

60.30 

59.05 

58.97 

58.22 

57.70 

56.95 

2000 

1.81 

58.0 

70.50 

69.25 

66.00 

64.75 

64.67 

63.92 

63.40 

62.65 

2200 

1.66 

63.3 

76.00 

74.75 

71.50 

70.25 

70 17 

69.42 

68.90 

68.15 

2400 

1.53 

68.6 

81.50 

80.25 

77.00 

75.75 

75.67 

74.92 

74.40 

73.65 

mile. 

1.39 

75.5 

88.64 

87.39 

84.14 

82.89 

82.81 

82.06 

81.54 

80.79 


Art. 12. By wheeled scrapers and drag scrapers. The body 
f the wheeled scraper is a box of smooth sheet-steel about '6% ft square by 15 ins 
sen, containing about % cubic yard of earth when “even full.” The box is open 
i front (in some machines it is closed by an “ end gate ” when full), and can be raised 
nd lowered, and revolved on a horizontal axis. To fill the box, it is lowered into, 
nd held down in, the earth, while the team draws the machine forward. When full, 

; is raised to about a foot above ground ; and, on reaching the dump, is unloaded by 
eing overturned on its axis. All the movements of the box are made by means of 
evers, and without stopping the team, which thus travels constantly. The wheels 
ave broad tires, to prevent them from cutting into the ground. 

In the drag scraper the box, owing to the greater resistance to traction, is made 
mch smaller. It contains about .15 to .25 cubic yard in place, and is always open in 
ront. The operation of the drag scraper is similar to that of the wheeled scraper, 
xcept that the box, when filled, rests upon the ground and is dragged over it by the 
earn. 

Each scraper (“wheeled” or “drag”) requires the constant use of a team of two 
orses with a driver. Besides, a number of men, depending on the shortness of the 
>ad and the number of scrapers, are required in the pit and at the dump to load the 
drapers (by holding the box down into the earth) and unload them (by tipping the 
ox). Except in sand, or in very soft soil, it is economical to use a plow before 
craping. 

The severest work for the team is the filling of the box ; and this occurs oftenest 
here the lead is shortest. Hence smaller scrapers are used on short than on long 
auls. We base our calculations on the following loads: 


For drag scrapers (used only on short hauls)...2 cubic yard 

For wheeled scrapers 

lead less than 100 feet.33 “ 

“ 100 to 300 feet.4 “ 


“ 400 to 500 feet.5 “ 

“ over 500 feet...6 “ 



















































748 


COST OF EARTHWORK. 


The daily expense per scraper, for driver’s wages and the use of a 2-horse team, is 
about $150. For leads of 400 feet and over, we add 50 cts per day for use of “ snatch - 
team ” to help load the larger scrapers then used. One snatch team generally serves 
a number of scrapers. 

Owing to the fact that the teams are constantly in motion without rest, they travel 
somewhat more slowly than with carts. We take 150 ft per minute (or 75 ft of lend, 
per minute j as an average. 

In loading and unloading, the teams not only go out of their way in order to turn 
around, but travel more slowly than when simply hauling. To cover this we make 
an .addition of 25 ft to each length of lead, whether long or short, for wheeled scrap¬ 
ers; and 15 feet for drag scrapers. 

We add 1 cent per cubic yard for the cost of loading and dumping the scrapers; and 
estimate the approximate cost of the other items as follows: 

Repairs of cart-road ct per cub yd in place for each 100 ft of lead 

Light Soils Heavy Soils 

Loosening cts per cub yd in place cts per cub yd in place 

by i>ick. * . 5. 

by shovel.;. * . 2 . 

Spreading. 1. 1.5 

Superintendence, wear and tear etc. 1. 1. 

We repeat that our figures are to be regarded merely as tolerable approximations, 
and subject to great variations according to skill of contractor and superintendent, ] 
strength of teams, character of material moved, state of weather etc etc. 

No. of trips per day No. (600) of mins in a working day 
per wheeled scraper ^- 0 0 f 75 f t i en gths in (lead -f 25 ft) 


No. of trips per day _ No. (600) of mins in a working day 
per drug scraper No. of 75 ft lengths in (lead + 15 ft) 


No. of cub yds in place moved _ No. of trips per 

per day by each scraper — day per scraper 


X 


No. of cub yds in place, 
per scraper per trip 


Cost per cub yd in place, 
for loading, hauling, 
dumping and returning 


Daily expense of one scraper ^ j f or ] oa ,jj n g 
No. of cub yds in place, moved and dumping 
per day by each scraper 


Total cost per Cost per cub yd .1 ct per cub yd 

cubic yard in in place, for in place for each 

place exclusive — loading, haul- -f- 100 ft of lead, -(- 
of contractor’s ing, dumping, for repairs of 

profit and returning road 


Cost, per cub yd in place, 
of loosening, spreading 
or wasting, and super¬ 
intendence &c. 


By Wheeled Scrapers. Labor $1 per day of 10 working hours. 


Length of lead, or dist 
to which earth is 
hauled 

Quantity in place, 
hauled per day by 
each scraper 

Cost per cub yd in 
place, for loading, 1 
hauling, dumping, 
and returning. 

Total cost, 

per cubic yard in place, exclusive of contractor's profit. 

Light Roils 

Heavy Soils 

Spread 

Wasted 

Picke 

Spread 

d and 

Wasted 

Plowt 

Spread 

id and 

Wasted 

Feet, 

cub vds 

cts 

cts 

cts 

cts 

cts 

cts 

cts 

50 

200 

2.8 

4.9 

3.9 

10 

8.5 

7.3 

5.8 

100 

140 

3.4 

5.5 

4.5 

11 

9.5 

8. 

6.5 

150 

105 

4.3 

6.5 

5.5 

12 

11 

9 

7.5 

200 

80 

5.4 

7.6 

6.6 

13 

12 

10 

8.5 

300 

56 

7.3 

9.6 

8.6 

15 

14 

12 

11 

400 

50 

8.5 

11 

10 

16 

15 

13 

12 

600 

43 

10 

13 

12 

18 

17 

15 

14 

800 

33 

13 

16 

15 

21 

20 

18 

17 

1000 

27 

16 

19 

18 

25 

24 

22 

21 


* Light soils can generally be advantageously loosened by the scrapers themselves in the act of 
loading. 























































COST OF EARTHWORK. 


749 


By Brag Scrapers. Labor $1 per day of 10 working hours. 


er. •<— 

•3 a 

«Tb 

O -3 

• s i. I 

Total cost 

per cubic yard in place, exclusive of contractor's profit. 

t. 2 


►."S s 








'O 

o 3 SC 







•e ® 

ei 

o 

2-3 
~ £ a. 

— Jj 

jd 'C a 

3 ’5 

Light Soils 


Heavy Soils 


° -3 T5 

o 

3? .B # 







J3 & ® 

c =•§ 

— <V -JZ ~ 

• Sat 



Picked and 

Plowed and 

a O a 

rt 2 ei 
d -a <y 




Spread 




J 

O’ 


Spread 

Wasted 

Wasted 

Spread 

Wasted 

Feet. 

cub yds 

cts 

cts 

cts 

cts 

cts 

cts 

cts 

loss') 








rlian y 

220 

2 .G 

4.6 

3.6 

10 

8.5 

7 

5.5 

40 j 









50 

140 

3.5 

5.5 

4.5 

11 

9.5 

8 

6.5 

75 

100 

4.5 

6.6 

5.6 

12 

11 

9 

8 

100 

80 

5.4 

7.5 

6.5 

13 

12 

10 

9 

150 

54 

7.5 

9.6 

8.6 

15 

14 

12 

11 

200 

42 

9.3 

12 . 

11 

17 

16 

14 

13 


Both wheeled and drag scrapers are made by Western Wheel Scraper Co, Mount 

I Pleasant, Iowa; by Kilbourne & Jacobs Mfg Co, Columbus, Ohio; by Fay Manufactur¬ 
ing Co, Elyria, Ohio, and others. A medium-sized wheeled scraper, weighing 450 lbs, 
and carrying .4 cubic yard, costs about from $50 to $70. A drag scraper weighs about 
100 ft>8, and costs about $14. 


Art. 13. By cars and locomotive, on level track. We have based our 
calculations upon the following assumptions: Trains of 10 cars, each car containing 1 % 
cubic yards of earth measured in place. Average speed of trains, including starting 
: and stopping, but not standing, 10 miles per hour, — 5 miles of lead per hour. Labor 
$1 per day of 10 working hours. Loosening, loading (by shovelers), spreading, wear 
; &c of tools, superintendence, &c, the same as with carts, Arts 2, 3, 5, and 7. Loss of 
time in each trip for loading, unloading, &c, 9 minutes, = .15 hour. Therefore 

Number of trips per | __ The number (10) of hours in a working day 
day, per train j .15 -f- the number of 5-mile lengths in the lead 

Number of cubic") Number of Number (10) Number (1.5) of cubic 

yards in place, per > = trips per day X of cars in a X yards in place in each 

day per train j per train train car 


i Cost per cubic yard, in place, 
i for hauling, dumping, and 
returning 


} 


One day’s train expenses -f 1 day’s cost of track 
Number of cubic yards in place per day per train 


One day’s train expenses: 

Cost of 10 cars @ $100. $1000 

“ locomotive. 3000 

-$4000 

One day’s interest at 6 per cent, on cost of train. $0.67 

Wages of engine driver (who fires his own engine). 2.00 

“ foreman at dump. 2.00 

“ 3 men at dump at $1. 3.00 

Fuel. 2.00 

Water. 1.00 

Repairs of locomotive and cars. 2.33 


Total daily expense of one train.$13.00 


Tile daily expense of track, for interest and repairs, may be taken at 
$3 for each mile, or fraction of a mile, of lead. 

Therefore, 













































750 


COST OF EARTHWORK. 


Cost per cubic yard in place.] $13 + ($3 for each mile of lead) 

for hauling, dumping, and 1- " Number of Number (10) Number (1.5) of 

returning j trips per day X of ears in a X cubic yards in 

per train train each car 


Total cost per cubic 
yard in place, ex¬ 
clusive of contrac¬ 
tor's profit 


Cost per cub yd in Cost per cubic yard, in place, for 
_ place lor hauling, . loosening, loading, spreading or j 
~ dumping, and re- ' wasting, and superintendence, &c. 
turning (Arts 2, 3, 5, and 7.) 


By Cars and Locomotive. Labor $1 per day of 10 working hours. 


Length of lead, or dis¬ 
tance to which the 
earth is hauled. 

Numberofcubicyards, 
in place, hauled per 
day by each train. 

1 Cos t per cubic y ard, i n 
place, for hauling, 
dumping, and re¬ 
turning. 

Total cost per cubic yard, in place, exclusive of contractor’s profit. 

Light Sandy Soils. 

Strong Heavy Soils. 

Picked and 
Spread. 

Picked and 
Wasted. 

Ploughed and 
Spread. 

Ploughed and 

Wasted. 

Picked and 

Spread. 

Picked and 

Wasted. 

Ploughed and 

Spread. 

Ploughed and 

Wasted. 

Miles. 

Cu yds. 

Cts. 

Cts. 

Cts. 

Cts. 

Cts. 

Ct 8 . 

Cts. 

Cts. 

Cts. 

1 

4350 

.4 

9.7 

8.4 

8.4 

7.2 

13.7 

12.4 

11.3 

10 . 

2 

2700 

.7 

10 . 

8.8 

8.8 

7.5 

14. 

12.8 

11.6 

10.4 

3 

1950 

1.1 

10.4 

9.2 

9.2 

7.9 

14.5 

13.3 

12.1 

10.9 

4 

1500 

1.7 

11 . 

9.7 

9.7 

8.5 

15. 

13.7 

12.6 

11.3 

5 

1200 

2.3 

116 

10.4 

10.4 

9.1 

15.6 

14.4 

13.2 

12 . 

6 

1050 

3. 

12.3 

11 . 

11 . 

9.8 

16.3 

15. 

13.9 

12.6 

7 

900 

3.8 

13.1 

11.8 

11.8 

10.6 

17.1 

15.8 

14.7 

13.4 

8 

750 

4.9 

14.2 

13. 

13. 

11.7 

18.2 

17. 

15.8 

14.6 

10 

600 

7.2 

16.5 

15.2 

15.2 

14. 

20.5 

19.2 

18. 

16.8 


Where large amounts of work are to be done, the steam excavator, land 
dredge or steam shovel generally economizes time and money. Where the 
depth of cutting is less than 10 ft, so much time is lost in moving from place to place 
that the excavators do not work to advantage. In still soils, cuttings may be made 
about from 17 to 20 ft deep without changing the level of the machine. For greater 
depths in such soils the work is done in two levels, since the bucket or dipper cannot 
reach so high. But in sand and loose gravel, much deeper cuts may be made from a 
single level. 

The excavator resembles a dredging machine in its appearance and operation. A 
large plate-steel bucket, like a dredging bucket, with a flat hinged bottom, and pro¬ 
vided with steel cutting teeth, is forced into and dragged through the earth by 
steam power. It dumps its load, by means of the hinged bottom, either into cars 
for transportation, or upon the waste bank, as desired. 

Each machine is mounted on a car of standard gauge, which can be coupled in an 
ordinary freight train. The car is made of w r ood or iron, as desired, and is provided 
with a locomotive attachment, by which it can be moved from point to point as the 
work proceeds. The machines can be used as wrecking or derrick ears. 
Each machine has a water tank, holding from 300 to 550 gallons, for the supply of 
its boiler. 

Before beginning to excavate, the end of the car nearest the work is lifted from 
the track by hydraulic or screw jacks, upon which it rests while working. 

In stiff soils the excavator leaves the sides of the cut nearly vertical; and the de¬ 
sired slope is afterwards given by pick and shovel. When the soil is hard or much 
frozen, it may be loosened by blasting in advance of the excavator. 

Steam excavators are made by Osgood Dredge Co, Albany NY; by John 
Souther & Co, (the “Otis” excavator) Boston Mass; by Vulcan Iron Works, Toledo 
0; by Industrial Works, Bay City Mich; and by Pound Manufacturing Co., Lock- 
port N. Y. 

The Osgood is made in two sizes. In No 1 the car is 34 ft X 10 ft, and its floor is 4 
ft above the rails. It has a four-wheeled truck near each end. The dipper holds 2 
cubic yards, struck measure. The machine weighs, complete, about 40 tons, and 
costs about $7500 on track at works (Albany, N Y). In the No 2 machine, the car is 
28 ft X 10 ft; floor 5 ft above the rails. It has two pairs of wheels, 16 ft apart from 
center to center of axles. The dipper holds V/ 2 cubic yards, struck measure. The 
machine weighs, complete, about 28 tons, and costs about $6000. 


I 

t 

I 

f 

3 






4 


» 


A. 

or 


5 

'ii 

i:!( 

thj 

Mj 

fee 

, ear 

occ 


in. 

feel 

AS I 
her 

a 

pin 

































COST OF EARTHWORK. 


751 


' 


The excavator has to he moved forward (as the work advances) abt 8 ft at a time, 
s regularly made, it can dig at a distance of 17 ft, horizontally, from the center of 
a le car in any direction, and can dump 12 ft above the track. In sand or gravel it 
kes out, while actually digging, 3 dipperfuls (= 4)^ to 6 cub yards in the dipper, 
= 3.75 to 5 cubic yards in place) per minute; in stiff clay, 2 dipperfuls per minute 
= 3 to 4 cub yards in the dipper, --- 2.5 to 3.33 cubic yards in place). An average 
iy’s work (10 hours) for a “ No 1 ” machine, including time lost in moving the ma- 
line, &c, is about 500 cubic yards in “ hard-pan,” and from 1200 to 1500 in sand and 
ravel. This allows for the usual and generally unavoidable delays in having cars 
iady for the excavator. 

The excavators carry about 80 to 90 lbs of steam. They burn from 100 to 150 lbs 
good hard or soft coal per hour; and require one engineer, one fireman, one 
•anesman, and 5 to 10 pitmen, including a boss. The pitmen are laborers, who 
tend to the jacks, lay track for the excavator and for the dump cars, assist in 
oving the latter, bring or pump water, &c, &c. 

After reaching the site of the work, about 30 minutes are required for getting the 
Kcavator into working condition; and an equal length of time, after completion 
the work, in getting it ready for transportation. 

The following figures are taken from the records of work done by a No 1 machine, 
om May to Nov, 1883. The material was hard clay with pockets of sand. The 
cpeuses per day of 12 working hours, at $1.50 per such day for labor, were 

Water (a very high allowance). $ 5.00 

Coal, 1% tons bituminous. 10.00 

Wages of engineer. 4.00 

“ “ fireman . 1.50 

“ “ cranesman or dipper-tender. 2.50 

“ “ pit boss. 3.00 

“ “ 8 pitmen at $1.50. 12.00 

Oil, waste, repairs, &c (estimated). 5.00 

Interest on cost ($7500) of machine. 1.25 


$44.25 


Reduced to our standard of $1 for labor per day of 10 working hours, this would 
e say $30.00 per day. Reduced to the same standard, and allowing for the greater 
roportional loss of time in stopping at evening and starting in the morning: the 
verage daily quantity excavated, measured in place, was, in shallow cutting, 530 
ubic yards; in deep cutting, 1200 cubic yards; average of whole operation, 800 
ubic yards. This would make the cost, per cubic yard measured in place, for 
>o8ening and loading into cars, 5.67 cts, 2.5 cts, and 3.75 cts respectively; while the 
oat by ploughing and shoveling, in strong heavy soils, by Arts 2 aud 3, is 7.4 cts; and 
ypicking and shoveling, say 10 cts. 


Art. 14. Removing: rock excavation by wheelbarrows. 

0 cubic yard of hard rock, in place , or before being blasted, will weigh about 1.8 tons, if sandstone 
d r conglomerate. (150 lbs per cubic foot:) or 2 tons if good compact granite, gneiss, limestone, or 
( larble, (168 lbs per cubic foot.) So that, near enough for practice in the case before us, we may as- 
e lime the weight of any of them to be about 1.9 tons, or 4256 lbs per cubic yard, in place; or lf>8 lbs 
'• ;r cubic foot. 

f Now, a solid cubic yard, when broken up by blasting for removal by wheelbarrows 
r carts, will occupy a space of about 1.8, or 1(1 cubic yards ; whereas average earth, when loosened, 
11 wells to but about 1.2, or li of its original bulk in place; although, after being made into embank- 
lent, it eventually shrinks into less than its original bulk. In estimating for earth, it is assumed 
bat -rV cubic yard, in place, is a fair load for a wheelbarrow. Such a cubic yard will weigh on an 

2430 

verage 2430 lbs, or 1.09 tons; therefore, - ' ■ — 174 lbs, is the weight of a barrow-load, of 2.31 cubic 

it ^ , 

pet of loose earth. Assuming that a barrow of loose rock should weigh about the same as one of 

4256 

■ /artb. we mav take it at J T of a cubic yard; which gives —— = 177 lbs per load of loose rock, 
- 2 4 24 


ccupving 2 cubic feet of space. 

I In the following table, columns 2 and 3 are prepared on the same principle as for earth, as directed 
j a Article 10. Column 4 is made up by adding to each amount in column 3, .2 of a cent for each 100 
| ;et length of lead, for keeping the wheeling-planks in order: and 45 cents per cubic yard, in place, 
s the actual cost for loosening, including tools, drilling, powder, &c; as well as moderate drainage, 
nd every ordinary contingency not embraced in column 3. Contractor’s profits, of course, are not 
ere included. 

Ample experience shows that when labor is at $1 per day. the foregoing 45 cents per cubic yard, in 
lace, is a sufficiently liberal allowance for loosening hard rock under all ordinary circumstances, 
n practice it will generallv range between 30 and 60 cents : depending on the position of the strata, 
ardness, toughness, water, and other considerations. Soft shales, and other allied rocks, may fre- 
uently be loosened by pick and plough, as low as 15 to 20 cents; while, on the other hand, shallow 
littlngs of very lough rock, with an unfavorable position of strata, especially in the bottoms of ex- 
avations, may cost Si. or even considerably more. These, however, are exceptional cases, of com- 
aratively rare occurrence. The quarrying of average hard rock requires about J4 to % lb of powder 
er cubic yard, in place; but the nature of the rock, the position of the strata, &c, may increase il 

















752 


COST OF EARTHWORK 


to M lb, or more. Soft rock frequently requires more powder than hard. A pood churn-driller will j, 
drill 8 to 10 feet in depth, of holes about 2)4 feet deep, and 2 inches diameter, per day, iu average 
hard rock, at from 12 to lb cents per foot. Drillers receive higher wages than common laborers. 


Hard Rook, by Wheelbarrows. 

Labor $1 per day, of 10 working hours. 




? e 

100 


lie 

or 


Length of 
Lead, or dis¬ 
tance to 
which the 
rock is 
wheeled. 

Number of 
cubic yards, 
in place, 
wheeled per 
day by each 
barrow. 

Cost per 
cubic yard, 
in place, 
for loading, 
wheeling, 
and 

emptying. 

Total cost 
per cubic 
yard, in 
place, ex¬ 
clusive of 
profit to 
contractor. 

Length of 
Lead, or dis¬ 
tance to 
which the 
rock is 
wheeled. 

Number of 
cubic yards, 
in place, 
wheeled per 
day by each 
barrow. 

Cost per 
cubic yard, 
in place, 
for loading, 
wheeling, 
and 

emptying. 

Total cost 
per cubic 
yard, in 
place, ex¬ 
clusive of 
profit to 
contractor 

Feet. 

Cubic Yds. 

Cents. 

Cents. 

Feet. 

Cubic Yds. 

Cents. 

Cents. 

25 

12.2 

8.64 

53.7 

600 

2.96 

35.5 

81.7 

50 

10.7 

9.81 

54.9 

700 

2.62 

40.1 

86.5 

75 

9.58 

11.0 

56.2 

800 

2.34 

44.8 

91.4 1 

100 

8.66 

12.1 

57.3 

900 

2.12 

49.5 

96.3 

150 

7.26 

14.5 

59.8 

1000 

1.94 

51.1 

101.1 

200 

6.25 

16.8 

62.2 

1200 

1.65 

63.6 

115.0 

250 

5.49 

19.1 

64.6 

1400 

1.44 

72.9 

120.7 

300 

4.89 

21 5 

67.1 

1600 

1.28 

82.2 

130.4 

350 

4.41 

23.8 

69.5 

1800 

1.15 

91.5 

140.1 1 

400 

4.02 

26.1 

71.9 

2000 

1.04 

100.8 

149.8 

450 

3.69 

28.5 

71 4 

2200 

.953 

110.2 

159.6 

500 

3.41 

30 8 

76.8 

2400 

.879 

119.5 

169.3 


mi 


Art. 15. Removing' rook excavation by carts. A cart-load of 

rock may be taken at i of a cubic yard, in place. This will weigh, on an average, 851 lbs; or but 41 
lbs more than a cart ioad of average soil. Since the cart itself will weigh about )4 a ton, the total 
loads are very nearly equal iu both cases. Columns 2 and 3 of the following table are prepared on the 
same principle as for earth, as directedin Art. 4. Column 4 is made up by adding to each amount in 
column 3, the following items : For blasting, (and for everything except those in column 3 ; loading, 
and repairs of cart-road,) 45 cents per cubic yard, in place; for loading, 8 cents, per cubic yard, in 
place; and for repairs of road, .2, or i of a cent for each 100-feet length of lead. Contractor's profit 
not included. 5 


Hard Rock, by Carts. 

Labor $1 per day, of 10 working hours. 


Length of 
Lead, or dis¬ 
tance to 
which the 
rock is 
hauled. 

Number of 
cubic yards, 
in place, 
hauled per 
day, by each 
cart. 

Cost per 
cubic yard, 
in place, 
for hauling, 
and 

emptying. 

Total cost 
per cubic 
yard, in 
place, ex¬ 
clusive of 
profit to 
contractor. 

Length of 
Lead, or dis 
tancc to 
which the 
rock is 
hauled. 

Number of 
cubic yards, 
in place, 
hauled per 
day, by each 
cart. 

Cost per 
cubic yard, 
in place, for 
hauling, 
and 

emptying. 

Total cost 
per cubfo 
yard, in 
place, ex¬ 
clusive of 
profit to 
contractor 

Feet. 

Cubic Yds. 

Cents. 

Cents. 

Feet. 

Cubic Yds. 

Cents. 

Cents. 

25 

19.2 

6.51 

59.6 

1800 

5.00 

25 0 

81.6 

50 

18.5 

6.77 

69.9 

1900 

4.80 

26.0 

82 8 

75 

17.8 

7.03 

60.2 

2000 

4.62 

27.1 

84.1 

100 

17.1 

7.29 

60.5 

2250 

4.21 

29.7 

87.2 

150 

16.0 

7.81 

61.1 

2500 

3.87 

32.3 

90.3 

200 

15.0 

8.33 

61.7 

J 4 mile 

3.70 

33.7 

92.0 

300 

13.3 

9.37 

63.0 

3000 

3.33 

37.5 

96.5 

400 

12.0 

10.4 

64.2 

3250 

3.12 

40.1 

99.6 

600 

10.9 

11.5 

65.5 

3500 

2.92 

42.8 

102 8 

600 

10 0 

12.5 

66.7 

3750 

2.76 

45.3 

105.8 

700 

9.23 

13.6 

68.0 

4000 

2.61 

47.9 

108.9 

800 

8.57 

14.6 

69.2 

4250 

2.47 

50.6 

112.1 

900 

8.00 

15.6 

70.4 

4500 

2.35 

53.2 

115.2 

1000 

7.50 

16.7 

71.7 

4750 

2.24 

55.8 

118.3 

1100 

7.06 

17.7 

72.9 

5000 

2.14 

58.4 

121.4 

1200 

6.67 

18.7 

74.1 

1 mile 

2.04 

61.2 

124.8 

1300 

6.32 

19.8 

75.4 

i y* “ 

1 67 

75.0 

141.2 

1400 

6.09 

20.8 

76.6 

IH “ 

1.41 

88.8 

157.6 

1500 

5.71 

21.9 

77.9 

IK “ 

1.22 

102.5 

174.0 

1600 

5.45 

22.9 

79.1 

2 “ 

1.08 

116.3 

190.4 

1700 

5.22 

24.0 

80.4 

2 H “ 

.962 

130.0 

206.8 


“ Loose rock” will cost about 30 cts per yd less; and even solid rock will 

average about 10 cts less than the tables. 















































COST OF EARTHWORK. 753 

Art. 16. Removing rock excavation by cars and locomo- 
ive, on level track. Our calculations are based upon the following assumptions: 
rains of 10 cars, each car containing 1 cubic yard of rock measured in place. Aver¬ 
se speed of trains, including starting and stopping, but not standing, 10 miles per 
our = 5 miles of lead per hour. Labor $1 per day of 10 working hours. Loosening, 
"> cts per cubic yard in place. Loading, 8 cts per cubic yard in place. Cost of track, 
>r interest and repairs, S3 per day per mile of lead. The calculations are the same, in 
rinciple, as those in Art. 13. 

Hard Rock, by Cars and locomotive. 

Labor $1 per day of 10 working hours. 


length of lead, or distance to which the rock 

is hauled...miles 

lumber of cubic yards, in place, hauled per 

day by each train.. 

lost, per cubic yard in place, for hauling, 

dumping, and returning.cents 

!otal cost, per cubic yard in place, exclusive 
of contractor’s profit.^.cents 


1 

3 

5 

7 

10 

2900 

1300 

800 

600 

400 

.6 

1.7 

3.5 

5.7 

10.8 

53.6 

54.7 

56.5 

58.7 

63.8 


49 
















754 


TUNNELS. 


TUNNELS. 


i" 

tp 

hi 

in 


Tunnels for railroads should, if possible, be straight, espe¬ 
cially when there is but a single track ; inasmuch as collisions or other accidentspli 
in a tunnel would be peculiarly disastrous. A tunnel will rarely be expedient !! 
before the depth of cutting exceeds 60 feet. Firm rock of moderate hardnessli 
and of a durable nature, is the most favorable material for a tunnel r 
especially if free from springs, and lying in horizontal strata. In soft rock, 01 
in shales (even if hard and firm at first), or in earth, a lining of hard brick oiw 
masonry in cement, is necessary. A tunnel should have a grade or incli* n 
nation in one direction, for ease of future drainage and ventilation. Ncto 


special arrangement is essential for ventilation either during construction 
or after, if the length does not exceed about. 1000 feet; but beyond that, gen¬ 
erally during construction either shafts are resorted to, or means provided foi an 
forcing air into the tunnel through pipes from its ends. But after the work Isjfoi 
finished, except under peculiar circumstances, nothing of the kind is necessary, er: 
Shafts often draw air down wards; and frequently, even when aided by a steep, 
uniform grade, do not secure ventilation. The Mont Cenis tunnel under the* 
Alps, completed in 1871, is 7% miles long, and has no shafts, although it. gradesjup 
up from each end, which is the most unfavorable of all conditions for ventila-lbl; 
tion without shafts. It was made so for facilitating drainage. Its ventilation si 
is maintained by air forced in from the ends. The Hoosac tunnel, Mass, 4^ ih 
miles long, has shafts : one of them 1030 feet deep; but they were for expediting ik 
the work. Kliafts generally cost from 1V£ to 3 times as much per cubic 
yard as the main tunnel, owing to the greater difficulty of excavating and re-I s 
moving the material, and getting rid of the water, all of which must be done da 


by hoisting. When through earth, they must be lined as well as the tunnel 
and the lining must, usually be an under-pinning process. Or the lining may 
first be built over the intended shaft, and then sunk by undermining it grad¬ 
ually; see page 6.50. Their sectional area commonly varies from about 40 to 
100 square feet. They have the great advantage of expediting the work by in-'H 
creasing the number of points at which it can be carried on ; but if placed toe 
close together, their cost more than compensates for this. The air in some 
tunnels, while being constructed, is much more foul than in others; so that 
after the work has been commenced, shafts with forced air may be expedient 
where they were not anticipated. In excavating the tunnel itself, a heading 
or passage-way, 5 or 8 feet high, and 3 to 12 feet wide, is driven and maintained 
a short distance (10 to 100 feet, or more, according to the firmness of the*ma- 
terial) in advance of the main work. In rock, the heading is just below the 
top of the tunnel, so that the men can convenientlv driil holes'in its floor for 
blasting; hut in earth, the heading is driven along the bottom of the tunnel 
that being the most convenient for enlarging the aperture to the full tunnel 
size, by undermining the earth, and letting it fall. In earth, the top and sides 
of the heading, as well as of the tunnel, must be carefully prevented from 
caving in before the lining is built.; and this is done bv mean's of rows of verti¬ 
cal rough timber props, and horizontal caps or overhead pieces, between which 
and the earth rough boards are placed to form temporary supporting sides and 
ceiling to the excavation. The props and caps are placed first ; and the boards 
are then driven in between them and the earthen sides of the excavation, 
These are gradually removed as the lining is carried forward. Tlie lining-, 
when of brick, is usually from 2 to 3 bricks thick (17 to 26 inches) at bottom, 
and from 1>£ to 2% bricks thick at top; and when of rough rubble in cement., 
about half again as thick. It is important that the bricks or stone should hr 
of excellent hard quality, and laid in good cement.. The bricks should b< 
moulded to the shape of the jirch. As the lining is finished in short lengths, 
and before the centers are removed, any cavities or voids between it and 
the earth should be carefully and compactly filled up. Even in rock, if much 
fissured, or if not of durable character, as common shale, lining is necessary, 
The cross-section of a single-track railroad tunnel, in the clear of every¬ 
thing, and for cars of 11 feet extreme width, should not be less than about 15 
feet wide, by 18 feet high ; nor a double-track one, less than 27 feet wide bv 2-1 
feet high; unless in the last case the material is firm rock, in which a high arch 
is not necessary for lining. The root may then be much flatter, so that a height! 
ot 20 feet may answer. With cars of 10 feet extreme width, the width of the* 
tunnel may be reduced to 25 feet; or with 9 feet cars, to 23 feet. Many have 
been made 22 feet.. The Mont Cenis is 26 feet, wide, by 25 high. The rate of 
daily progress from each face of a tunnel varies from 18 inches to 9 feet of 
length per 24 hours, with three relays of workmen. On the Mont Cenis the ex- 






TRESTLES. 


755 


retries were about 4 to 9 feet daily for a whole year, froin each face. Drills 
/orked by compressed air were employed in the headings, which were 12 feet 
dde bv 8 feet high. Ordinarily, from \ l /£ to 3 feet may be taken as averages, 
'be difference of rate of progress between a single and a double track tunnel 
! j not so great as might be supposed ; inasmuch as a larger force can be etn- 
! doved on the wider one. If the tunnel is in earth, the construction of the 
1 iniug about makes up for the slower excavation of one in rock. In rock, with 
j abor at $1 per day, tlie cost will usually vary with the character of the rock, 

1 rom $2 to $5 per cubic yard for the main tunnel; and from $3 to $10 for the 
leading; while shafts will average about 50 percent, more than heading. The 
; ost of a single-track tunnel, when common labor is $1 per day, will generally 
I range between $30 and $75 per foot of length. Tunnel work, however, is liable 
o serious contingencies which cannot be foreseen. Since the sides and roof are 
ough as blasted, the width and height should each be estimated to the con- 
ractor as about 18 inches or 2 feet greater than the established clear ones. At 
1 my rate, the mode of measurement should be clearly stated in the specifications 
1 or the work. When a tunnel is made with a uniform grade, the work gen- 
1 (rally progresses in a more satisfactory manner from the lower end, because 
he descent favors the drainage of the spring water that is usually met with; 
vhereas, at the upper end, it must be removed by pumps or by bailing. The 
ipper end has, however, the advantage of sooner getting rid of the smoke in 
blasting. Before commencing a tunnel, or even deciding upon one, trial 
shafts should be sunk to ascertain the nature of the material. In long ones, 
he greatest care and accuracy are necessary for preserving the line of direc¬ 
tion, so that the work from both ends shall meet properly at the center. 

In the heading of the Vosburg (Pa) tunnel of the Lehigh Valley R R, built 
il884, cioss-section 7*4 feet X 26 feet, the average progress per working 
lay of 24 hours with two shifts of 12 hours each, was as follows; by hand 
Irilling 2.8 feet and 2.4 feet respectively from each end ; by machine drills 
two rival drills in competition) 5.6 feet and 7.8 feet. The material was hard 
*ray sandstone. For the whole tunnel the rate was about 2 feet per day. 

For further information respecting tunnels, the reader is referred to Mr. 
H. S. Drinker’s very full treatise on the subject, published by the Messrs Wiley. 

For Stone bridges and culverts, see pp 693, &c. 

For Trusses, see pp 547, Ac. 


TEESTLES. 



x 


Figs 1, 2, 3, 5, 6, 7, are elevations of trestles; taken across the track or 
roadway. We may consider Fig 1 as adapted to a height of about 10 to 20 ft; tigs 2 


























































756 


TRESTLES. 


and 3, to heights from 20 to 30 ft; Fig 5, from 30 to 40 ft; Fig f>, from 40 to 60 ft, nt 


rough approximations merely. A single framework, such as that shown m eacli of 


tii ese six figures, is called a “bent.'’ These bents of course admit of many modifi-!™ 
cations. They are usually supported by bases of masonry, as in the figures. These el 
preserve the lower timbers from contact with the earth, which would hasten theii 
decay. It is advisable to make these bases high enougli to prevent injury from cattle," 
orpassing vehicles, Ac. Up to heights of about 40 or 50 ft, a single row of posts or up- 1 
rights,o, a, a, Figs 1 to 9, as shown at ee. under Figs 1 and 6, will answer. But as the 1 , 1 
height becomes greater, more posts should be introduced, as shown at x x under Fig 1 ' 
5; or two entire rows of them ; or three rows, as under Fig 7 ; and as also in Fig 8. ■" 
which is an end view of Fig 7. Figs 7 and 8 bear much resemblance to the trestles'* 1 
190 ft high, with masonry bases 30 ft high (3. Seymour, C. E.), which carried the! ' 
Erie Rway (now the N Y, Lake Erie & West’n R It) over the (ilenesee River at 1 
Portage, N Y. There each bent had 21 posts 14 ins square, at its base; and 15 3 


posts of 12 X 12, at its top. 
pairs, embracing the posts. 


The other timbers were 6 X 12; many of them were in 1 


.to 


This single-track viaduct was begun July 1, 1851, and 


completed Aug. 14,1852. It contained 1,602,000 ft (B M) of timber, and 108,862 lbs 


if: 


of iron. In the foundations were 9200 cub yds of masonry. The entire cost was 
about $140,000. It was burned down in 1875, and was replaced, in less than 3 mosJ ee 
with a single-track viatlurt of wrought-iron trest les, (described below)?™ 
containing, in all, 1,340,000 lbs of iron, and 130,600 ft (B M) of timber; and costing' 51 
complete, above the masonry, about $95,000. Frequently the posts of trestles are in" 
pairs; and the other timbers pass between ; all bolted together. 

In Fig 4, the posts «, a, a, are end views of three trestles or bents, such as Fig 3; M( 
andtt are diag braces extending from trestle to trestle; the two outer ones inclining' 5 ' 
in one direction; and the central one crossing them. These may be placed either" 
intermediate of the posts, as in Fig 3; with the heads of the two outer ones confined'. 1 
to the cap c c of one trestle; and their feet to the sill y y of the next one; or they*' 5 ' 
may all be spiked or bolted to the posts themselves, as in Fig 4. The last is the best, ‘ 
as it serves also directly to stiffen the posts: as do also the braces o o, n n, Fig 2. 
Such bracing is too frequently omitted. During the passage of trains, the backward 
pressure of the steam, exerted through the driving wheels against the track, pro¬ 
duces a serious strain lengthwise of the road, and tending to upset the trestles; and 
the sudden application of brakes to a moving train, produces a similar strain in the 
opposite direction. These strains become more dangerous as the ht increases. Hence i 
the need for such braces. Usually the outer posts may loan 1.5 to 2.5 ins to a ft. 

The posts should not be less than about 12 ins square, except in quite low trestles; 
and even then not less than about 10 X 10. The diag bracing may generally be about 
as wide as the posts; and half as thick. The dist apart of the bents, when the road¬ 
way is supported by simple longitudinal beams, should not exceed 10 or 12 ft, for 
railroads. But if these beams receive support from braces beneath, like ss, Fig 8: or 
from iron truss rods, as at Fig 52, page 514, the dist may be extended to 15 or 20 or 
more ft. But when the trestles become very high, and contain a great deal of tim¬ 


ber, Jt becomes cheaper to place them farther apart, say 30 to 60 ft; and to carry 
the railway upon regular framed trusses, as at u u, Figs 7 and 8; as in a 


_ _ o _.. _ , —.bridge with; 

stone piers. In the Genesee viaduct, the trestles were 50 ft apart, center to center. 4 
When such a trestle as Fig 8 becomes very narrow in proportion to its height, we 
may add to its stability by introducing beams w, extending from trestle to trestle; ! 
and still further by inserting diag braces v r, as in the old Genesee viaduct. 

Fi{£H 02, p 613, -‘Trusses,” will show how the timbers may be joined. In de¬ 
signing trestles, (as in wooden bridges,) it is advisable, as far as practicable, to arrauge tbe pieces 
so that any one may be removed if it becomes decayed; and another put iu its place. Os curves, 
additional strength should be given on the convex side ; as suggested by the dotted lines in Fig 5. 
<)u very high trestles especially, (as well as on bridges,) wheel-guards, g g, Fig 10, either inside or 
outside of the rails, should never be omitted, as is commonly done. 

In marshy ground , piles may be driven to support the trestles; or may be left so 
far above ground, as themselves to constitute the posts. Such trestles may often be 
used advantageously, even when to be afterward filled in by embkt. They then sus¬ 
tain the rails at their proper level until the embkt has reached its final settlement. 


on 


They are generally used to avoid the expense of embkt; especially when earth 
ly bo obtained from a great dist. Even when earth and timber are equally c 


can 

vement, they will rarely much exceed about half the cost of embkt; even when but 
about 30 ft high ; but owing to their liability to decay, they should be resorted to 
only iu case of necessity; or as a temporary expedient. 

Iron tre*tle«. At the Crumlin double-track iron viaduct,in England,1500 
ft long, (spans 150 ft.) they are about 180 ft high ; 60 by 27 ft at bale; 30 by 18 at top ; ' 
each composed of 14 cast-iron posts, arranged as a long hexagon ; each post being r 1 
formed of 17-t‘t lengths of iron pipes, 1 ft outer diam, by 1 inch thick. At each 17-ft ' 
length, the pipes are firmly connected by hor iron pieces; and between these diff 'l 1 

Ctei u in i I i II » l.»ii/«ikw< ..4* 4 s. s I V/ 1 / 2 1 . .. It J m • < . _ . _ 


stages is diag bracing of 4 X 4 X Vi inch rolled T iron, arranged as in Fi 
The viaduct contains about 3,000,000 lbs of wrought-irou uud nearly as much 


ly as much cast-iron 









TRESTLES. 


757 


20 ft. 


ol | The new Portage Tiaduct, referred to above, consists of iron Pratt 

J -usses, resting upon six towers. Fig 11 is a longitudiual view, and Fig 12 a trans* 
arse view, of one of the tallest towers, 203 ft 8 
is high from top of masonry to track-rail, and 
It jeighing 285,000 lbs. Each tower consists of 2 
ip ients B B, Fig 11. Each bent has two wrought- 
li, on columns CC, Fig 12, inclining toward each 
j, ther with a batter of 1 in 8. They are 20 ft 
! part at top, and, in the tallest tower, 69 ft 8 ins 
@ etween centers, at base. The two bents of a 
>wer are 50 ft apart, and are connected with 
hi *ch other by hor longitudinal struts S, F’ig 11, 
li ad diag tie-rods R. The spans between 
ij owers vary from 50 to 118 ft. 
u The columns are put together in lengths 
j f 25 ft. F'igs 13, 14, and 15 show the upper end 
a f one of these lengths. Fig 13 is an elevation, 
j 3en from between the feet of a bent, Fig 14 a 
, ross section, and Fig 15 a side view. Each col 
, t composed of three plates, P P P, and 4 angle¬ 
's ars 4 X 4 X inch, as shown. In the tallest 
jjywers, the 2 opposite side plates are 15 ins wide, 


nd from % to % inch thick; their thickness 
lcreasing with their dist from the top of the 
ower. The third, or back, plate is 17 ins X M 
icli throughout. The fourth side, L, has only 




jrsoftTI 
1 1 / 

nU 



1 , 

i /a 

j 



K 69.8.cen’sx 


Fifir.ll 


Fig.12 


Fig.13 


.a zig-zag lacing, Z Z Z, Fig 13, of 
at bars, so that the interior of the col is accessible for painting. 

At tile upper end U of each 25-ft length, are riveted two small iron plates 
pp, forming a tenon. The foot of the next length fits over this 
tenon, and is confined to it by a turned iron pin, 1% ins diam, 
passing through carefully bored holes. To this pin, the longitudinal 
diag rods, R, Fig 11,1ins diam, are attached. The longitud¬ 
inal hor struts, S, Fig 11, are light latticed girders of uniform 
width (1 ft) and depth (2 ft). They abut against the sides of the 
columns, and are bolted to lugs of angle iron, riveted to them. 
They are connected with the corresponding transverse struts, S, Fig 
12. by hor diag angle-bars fastened to each strut 10 ft from its end. 
The transverse struts are of different de¬ 
signs, depending upon their lengths. At their ends 
they are held by pins passing through holes o, Fig 
15, in the side plates of the columns. These pins 
also hold the 1^ inch diag rods, R, Fig 12. Each 
of the lowest three transverse struts is in two 
lengths, and is supported by an intermed iate 
vert post, I, Fig 12. shown in cross section by 
Fig 16; and each of the lowest two is connected 
with the corresponding strut in the other bent of 
the same tower by a longitudinal strut and hor 
diag rods. 

The cols rest upon east-iron pedestals. 

Each pedestal is tenoned into the foot of its col. 

The pedestals on the north side of the bridge are 
doweled to cast-iron plates built into the masonry. 

Those on the south side are on rollers, rolling at 
right angles to the axis of the bridge. The two 
pedestals of each bent are connected with each 
other, and held in position, by eye-bars, which 
prevent them from spreading; and by stints, 
which prevent them from coming together when 
the diags are screwed up. V V, Fig 12, are an¬ 
chor rods, bolted to the masonry, and attached to 
the columns at such a lit as not to interfere with 
the hor expansion of the tower. 

Cast-iron caps are tenoned and bolted into the tops of the cols. On each cap 
est the ends of the upper chords of two trusses, as shown in Fig 11. 

One end of each long-span truss is bolted to the cap of a col. The other end rests 
ipon rollers on the cap of the next col, but is connected with the fixed end of the 
djoining truss by iron loops which pass over the end pins of both trusses, and allow 
uly sufficient motion for contraction and expansion. The short spans over the 
owers are bolted to the caps at both ends,but those between the towers are arranged 




55 


V 


f 

p 

k 

—cr 


4 ins 


P 
Fig. 14 


v 

I 

I 

l 

i • 

W 

0 


i 


Fig. 16 



































































758 


TRESTLES. 


r 


like the long spans. Where the ends of a long, and of a short, span rest upon the 
same cap, they are placed respectively 3 and 6 ins from its center, so that their cen¬ 
ter of pressure coincides with the center of cross section of the col. 

The towers are made strong enough for a double-track road ; and the trusses are 
so arranged that they can be placed closer together, and additional trusses placed 
alongside of them ; so that the track can he doubled at any time. 

The bridge is so designed that the greatest compressive strain per sq 
inch that can come upon a col, under a uouble-track bridge and load, is 6(300 lbs; 
and the greatest tensile strain per sq inch on a diag, 15,000 lbs. The 
greatest weight on the foot of any one col will be 357,500 lbs, or 155 lbs per sq inch 
of the 4 ft sq pedestal-stone. 

The lowest section of a tower was first erected by means of a wooden framework 
on a flooring resting on the stone piers. A gin-pole 55 ft high was then lashed to 
each col, and by means of these poles the flooring and framework were raised to the 
tops of the columns, and used in placing the second section in position on top of the 
first. This process was repeated until the tower attained its full height. One of the 
tallest towers was erected in 11 days. 

The Kinzua Viaduct, on a branch of the N Y, Lake Erie & Western R It 

in Penna, is single track, 2052 ft long, and its greatest height is 285 ft from the top 
of the masonry piers to the track-rails. 

There are 21 spans of 61 ft each, placed between 20 towers; and 20 spans of 38% 
ft each, over the towers. The towers are similar, in general arrangement, to those ii> 
of the Portage viaduct, above described. The bents are 10 ft wide at top, and 103 ft 
at the feet of the tallest towers, which are 278 ft high. The 2 bents of a tower are 
38% ft apart. The legs are of Phoenix Iron Co’s 4-segment columns, pattern C, p 
449,7^ ins inner diam, and from i to in thick. They are made up of lengths of j 
about 33 ft, which are held in place, one on top of another, by plain cylindrical 
wrought-iron tubes about 14 ins long, and of such diam that they just fit inside of 
the cols. The abutting ends of two column-lengths slide over such a tube, and meet h, 
at the middle of its length. The tube is held in place by 4 bolts, which pass through L 
the column from side to side, two ot them being at right angles to the other two. |ei 
These bolts hold in place the longitudinal and transverse lior braces, most of which 
are lattice-girders, spaced about 30 ft apart vertically. 

The cols rest upon smooth iron plates, which allow a sliding movement of 1 inch 
transversely, and .38 inch lengthwise, of the roadway. Each plate is bolted, by two 
1%-inch bolts, from 9 to 12 ft long, to a square, pyramidal masonry pier, from 10 to lC 
13 ft deep, 8 ft square at bottom and 4 ft square at top. 

The cols have cast-iron caps, to which the ends of the lower chords of the 38% ft . 
spans are firmly bolted, and on which those of the 61-ft spans rest. The latter are rt 
bolted to the shorter spans through oval bolt-holes, which allow .17 inch play, longi¬ 
tudinally, for expansion and contraction. 

The greatest compression on a col is 8000 lbs per sq in. The ult load, 
by U S Govt experiments at Watertown Arsenal, is 35,000 lbs per sq in. 
The greatest tension on the diags is 15,000 lbs per sq in. 

The heaviest column weighs about 5000 lbs. The entire structure, 
3,500,000 llis. It cost $275,000, and was built by a gang averaging 125 
men, aided by 2 steam-hoisters and a traveling crane. Clarke, Reeves 
& Co, Phila, builders: now (1886) I’hoenix Bridge Co. 

Mode of erect ion. At each of the four corners of the site for a f a ; 
tower a mast 60 ft long was set up. These were guyed With ropes.‘and 
by means of them the 4 columns about 33 ft long, forming thd lowest 
story of a tower, were erected and braced together. The masts were then 11 
raised about 30 ft, and clamped one to each of the columns, and used for 1 
raising the second story into position on top of the first, and so on,except 




I 


Kill 



h 


for the top story, which was bolted together on the ground, in two pieces, ril 


and hoisted into position by the travelling crane. 

Verrugas Viaduct, near Lima, Peru, 575 ft long, 252 ft high, car¬ 
rying the Oroya R R (single track) over the A gnu de Verrugas. Built by 


S P 


Fiff 


the late Baltimore Bridge Co, Chas. II. Latrobe, Engr, in less than four 
• 17 months in 1872. It coutains 1,325,000 lbs of iron, and cost $165,000. 


There are four Fink truss spans, three of 100 ft, and one of 125 ft. They 
rest on 3 tow'ers. Each tower consists of three parallel bents. 25 ft apart, and each 
bent has four columns, arranged as in Fig 17. The bents are 15 ft, w'ide at top and 
the outer columns of each beut batter 1 in 12. The columns are all of the Phoenix 
segment pattern. The two outer ones of each bent have six segments each, diam 
11% ins, area of cross section, 20 sq ins. The inner ones have four segments each, 
diam 8 ins, area 13 sq ins. The columns are put together in lengths of 23% ft, 
which are joined by cast-iron couplings. The hor braces extending from col to col 
are about 25 ft apart vert. 










RAILROAD CONSTRUCTION. 


759 


15 ALL AST. 

Table of eubic yards of ballast per mile of road. 

•t Side-slope of the ballast 1 to 1. Width in clear between 2 tracks 6 ft. The ties 
d 'd rails may be laid first, for carrying the ballast along the line; then raised a 
w ft of length at a time, and the ballast placed under them. Deduct for ties, 
„ below. 


Denth 

r in 

las. 


Top width, 

Single Track. 

Top width, 

Double Track. 

10 Ft. 

11 Ft. 

12 Ft. 

21 Ft. 

22 Ft. 

23 Ft. 


Cub. Y. 

Cub. Y. 

Cub. Y. 

Cub. Y. 

Cub. Y. 

Cub. Y. 

12 

2152 

2347 

2543 

4303 

4499 

4695 

18 

3374 

3667 

3960 

6600 

6S94 

7188 

24 

4694 

5085 

5 47 4 

8996 

9388 

9780 

30 

6111 

6600 

7087 

11490 

11980 

12470 


A lliatl Call breali 3 to 4 cub yds per day, of hard quarried stone, to a size suitable 
j >r ballast; say Averaging cubes of 3 insouauedge. Where other ballast cauuot be had, hard-burnt 
! av is a good substitute. The slag from iron furuaces is excellent. The ties decay more rapidly 
j lieu gravel or sand is used instead ofbrokeu stoue, because these do cot drain off the rain, but keep 
, lie ties damp longer. For stone crushers, see p 680. 

TIES. 


In the United States the life of a tie is about as follows: 


liestnut, 

Years. 

5 to 12 

Average, 

Years. 

7 

White Oak. 

Years. 

6 to 12 

Average, 

Years. 

7 

eda r, 

6 to 15 

9 

Spruce Pine, 

4 to 7 

5 

lemlock, 

3 to 8 

5 




As shown by table, Art 3,p 815, the annual expense for renewal of ties, 
l the U. S. alone, is about ten millions of dollars. 


The Penna R R, in 1883, used 575,000 ties in construction on its main line 
nd branches (= 2552 miles of single track) and 779,000 ties in repairs. 

It will often, especially in the case of the softer and more perisLcble woods, be 
rue economy to preserve ties by the injection of creosote. See p 425. Creosote 
reserves the spikes. 

The writer believes that most of the fault usually ascribed to cross-ties, as well as 
p rail-joints, is in reality due to imperfect drainage of the roadbed. Hence, he does 
ot agree with those who advocate very long ties; but considers that with good 
allast, on a well-drained roadbed, 8% ft is as good as more; and that 8% ft, bv 9 
as, by 7 ins; and 2% ft apart from center to center, is sufficient for the heaviest 
Iraffic. On many important roads they are but 8 ft; and on some only 7% ft long; 
rack 4 ft 8%. On narrow-gauge roads the ties are generally from 6 to 7 ft long, 
■'he actual cost of cutting down the trees, lopping off the branches, and hewing 
he ties ready for hauling away to be laid, is about 6 to 9 cts per tie, at $1.75 per 
ay per hewer. 

The narrow bases of rails resting immediately on the cross-ties, without chairs, 
requently produce in time such an amount of crushing in the ties as to injure them 
naterially even before decay begins. Burnetised ties rust the spikes away rapidly. 
Jreosoted ones preserve them. 

Cross-ties Of 8)4 ft, by 9 ins, by 7 ins, contain 3.719 cub ft each ; and if placed 2>^ ft apart 
rom center to center, there will be 2112 of them per mile, amounting to 291 cub yds. Therefore, if 
hey are completely embedded in the ballast, they will diminish its quantity by that amount. At 2 ft 
| .part there will be 2640 of them ; occupying 364 cub yds ; and at 3 ft apart, 1760 of them ; 243 cubyds. 

Cubic feet contained in cross-ties of different sizes. 


Dimensions. 


Dimensions. 


Dimensions. 


Ft. Ins. 

Ins. 

Cub. Ft. 

Ft. 

Ins. 

Ins. 

Cub. Ft. 

Ft. 

Ins. 

Ins. 

Cub. Ft. 

8 hv 8 bv 6 

2.667 

8« by 8 by 6 

2.833 

9 by 8 by 6 

3.J00 

H " 9 

6 

3.000 

8U 

9 

6 

3.188 

9 

9 


3.3i5 

8 9 

7 

3.500 

8t<j 

9 

7 

3.719 

9 

9 

7 

3.938 

8 10 

6 

3.333 

8% 

10 

6 

3.542 

9 

10 

6 

3.750 

8 10 

7 

3.889 

8'4 

10 

7 

4.132 

9 

10 

7 

4.375 

8 10 

8 

4.444 

8Vt 

10 

8 

4.722 

9 

10 

8 

5.000 

8 12 

8 

5.333 

8)4 

12 

8 

5.667 

9 

12 

8 

6.000 


Bessemer steel ties, furnished by International Railway Tie Co., office 210 
■WiOiinetnn St.. Boston Mass., and laid iu the Boston <te Maine it. It. near Boston in July 1885, under 















































760 


RAILS. 


RAILS. 

Every sq inch of sectional area of rail, corresponds to 10 lbs per yard of a single 
rail ; t or to 15.7143 tons per mile of single-track road. Consequently, 

Wt in tons per mile 

Wt in lbs per vd of rail, of single-track _ Area of rail 

W ° F 15A143 in S( 1 ins - 

Thus, a rail of 100 tons per mile of single track, will have a section of 6.364 sq 
ins ; and will weigh 63.64 ft>s per yd of single rail. Add for turnouts, sidings, road- 
crossings, and a trifle for waste in cutting. When the ties are in place, and the rails 
distributed in piles at short intervals, a gang of 6 men can lay % a mile of rails per 
day, of single track; or after the ballast is in place, a gang of 15 men will lay about 
one mile of complete single-track superstructure per week. 

Steel rails last from 9 to 25 years; average 15 years. 

In the U. S. steel rails weigh, on 4 ft 8}^ in and 5 ft gauges, usually from 
55 to 70 lbs per yard. 56 and 60 are common. Sometimes as light as 50, and as 
heavy as 76 to 80. 3 ft gauge, 30 to 40, sometimes 50. 2 ft, 25. The usual length 
of rails is 30 ft. They have been made much longer, even up to 60 ft, but such lengths 
have not come into use to any extent. They of course have fewer joints, but the 
great space necessary between the rail-ends at the joints, to allow for expansion in 
hot weather, is a serious objection. This might be obviated by beveled joints,p 764. 
For sections of rails, to scale, see pp 764 and 765. 

Annual production of rails in the United States, in tons of 2240 lbs. 



1872 

1885 

Bessemer steel. 


959,470 

Open-hearth steel. 


1,250 

Iron. 


13,118 


892,857 

973,838 


Steel rails usually contain from .3 to .5 of one per cent of carbon. 

The Penna R R listed. in 1883, on its main line and branches (= 2552 miles 
of single track), 6600 tons of steel rails in construction of new lines, and 14,300 tons 
in repairs. 

Price of Steel rails, at mill, in 1886, about $35 per ton of 2240 lbs. 








RAILROADS, 


761 


'able of Middle Ordinates, to be used for the bending of rails of different 
lengths, so as to form portions of curves of different radii. Ordinates for lengths 
or radii intermediate of those in the table, may be found by simple proportion. 


Def. 

Ang. 

Radius. 




LENGTHS 

OF RAILS. 





30 

28 

26 

24 

22 

20 

48 

16 

14 

12 

10 

8 

6 

Deg. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet. 

Feet 

.5 

11460. 

.010 

.008 

.006 

.005 

.004 

.004 

.003 

.002 

.002 

.001 

.001 

.000 

.000 

1. 

5730. 

.020 

.016 

.013 

.011 

.009 

.008 

.006 

.005 

.004 

.003 

002 

.001 

.001 

1.5 

3820. 

.029 

.026 

.021 

.018 

.016 

.013 

.010 

.008 

.006 

.004 

.003 

.002 

.001 

2. 

2865. 

.038 

.034 

.029 

.025 

.021 

.017 

.014 

.011 

.008 

.006 

.004 

.003 

.001 

2.5 

2292. 

.049 

.043 

.037 

.031 

.027 

.022 

.018 

.014 

.010 

.007 

.005 

.003 

.002 

3. 

1910. 

.058 

.051 

.044 

.037 

.031 

.026 

.022 

.017 

.012 

.009 

.006 

.004 

.002 

3.5 

1637. 

.070 

.061 

.052 

.043 

.037 

.031 

.025 

.020 

.015 

.011 

.008 

.005 

.003 

4. 

1433. 

.079 

.069 

.060 

.050 

.042 

.035 

.029 

.023 

.018 

.013 

.009 

.006 

.003 

4.5 

1274. 

.088 

.077 

.067 

.056 

.047 

.039 

.032 

026 

.020 

.015 

.010 

.007 

.1X14 

5. 

1146. 

.099 

.086 

.074 

.063 

.053 

.044 

.035 

.029 

.022 

.016 

.011 

.007 

.004 

5.5 

1042. 

.108 

.094 

.082 

.070 

.059 

.048 

.039 

.032 

.024 

.018 

.012 

.1X18 

.004 

6. 

955.4 

.117 

.102 

.088 

.076 

.064 

.052 

.042 

.034 

.026 

.019 

.013 

.008 

.005 

6.5 

882. 

.128 

.112 

.097 

.082 

.069 

.057 

.046 

.037 

.028 

.021 

.014 

.009 

.005 

7. 

819. 

.137 

.120 

.104 

.088 

.074 

.061 

.049 

.039 

.030 

.022 

.015 

010 

.005 

7.5 

764.5 

.146 

.127 

■ 111 

.094 

.079 

.065 

.053 

.042 

.032 

.024 

.016 

.010 

.006 

8. 

716.8 

.158 

.137 

.119 

.100 

.085 

.070 

.056 

.045 

.034 

.025 

.017 

.011 

.006 

8.5 

674.6 

.166 

.145 

.126 

.106 

.090 

.074 

.060 

.048 

.036 

.027 

.018 

.012 

.007 

9. 

637.3 

.175 

.153 

.133 

.112 

.095 

.078 

.063 

.050 

.038 

.029 

.019 

.012 

.007 

9.5 

603.8 

.187 

.163 

.141 

.119 

.101 

.083 

.067 

.054 

.042 

.031 

.021 

.013 

.008 

10 

573.7 

.196 

.171 

.148 

.125 

.106 

.087 

.071 

.057 

.045 

032 

.022 

.014 

.008 

11 

521.7 

.216 

.188 

.163 

.139 

.117 

.096 

.078 

.063 

.049 

.036 

.024 

.016 

.009 

12 

478.3 

.236 

.206 

.179 

.151 

.128 

.105 

.085 

.069 

.053 

.039 

.026 

.017 

.010 

13 

441.7 

.254 

.222 

.192 

.163 

.138 

.113 

.092 

.075 

.057 

.012 

.028 

.019 

.010 

14 

410.3 

.275 

.239 

.207 

.175 

.148 

.122 

.099 

.080 

.061 

.045 

.030 

.020 

.011 

15 

383.1 

.295 

.257 

.223 

.188 

.159 

.131 

.106 

.085 

.065 

.049 

.033 

.021 

.012 

16 

359.3 

.313 

.273 

.236 

.200 

.170 

.139 

.113 

.091 

.070 

.052 

.035 

.023 

.013 

17 

338.3 

.333 

.290 

.252 

.213 

.180 

.148 

.120 

.096 

.074 

.055 

.037 

.024 

.014 

18 

319.6 

.351 

.306 

.265 

.225 

.190 

.156 

.127 

.102 

.078 

.058 

.039 

.025 

.014 

19 

302.9 

.371 

.324 

.280 

.238 

.201 

.165 

.134 

.108 

.082 

.061 

.041 

.027 

.015 

20 

287.9 

.392 

.341 

.296 

.250 

.212 

.174 

.141 

.114 

.087 

.066 

.044 

.028 

.016 

21 

274.4 

.410 

.357 

.309 

.262 

.222 

.182 

.148 

.120 

.091 

.069 

.046 

.030 

.017 

22 

262. 

.430 

.375 

.325 

.275 

.233 

.191 

.155 

.126 

.096 

.072 

.048 

.031 

.018 

23 

250.8 

.450 

.390 

.338 

.287 

.243 

.199 

.162 

.131 

.100 

.075 

.050 

033 

.019 

24 

240.5 

.469 

.408 

.354 

.299 

.253 

.208 

.169 

.137 

.104 

.078 

.052 

.034 

.019 

*25 

231. 

.486 

.424 

.367 

.311 

.263 

.216 

.176 

.142 

.108 

.081 

.054 

.035 

.020 

26 

222.3 

.506 

.441 

.382 

.323 

.274 

.225 

.183 

.148 

.112 

.084 

.056 

.037 

.021 

27 

214.2 

.524 

.457 

.396 

.335 

.284 

.233 

.190 

.153 

.116 

.087 

.058 

.038 

022 

28 

206 .1 

.545 

.475 

.411 

.348 

.294 

.242 

.197 

.158 

.120 

.090 

.060 

.039 

.022 

29 

199.7 

.564 

.491 

.424 

.361 

.303 

.250 

.203 

.163 

.124 

.093 

.062 

.041 

.023 


T 


For ordinates for center line of road, see pp 726 to 731 








































762 


RAILROAD SPIKES 



WEIGHT OF RAIEICOAD SPIKES. 

Tl»e Iiook-lioadcd spikes t. commonly used for confining rails to 
the cross-ties, vary within the limits of the following table; the lightest ones 
for light rails on short local branches; and the heaviest ones for heavy rails 
on first-class roads. The spikes are sold in kegs usually of 150 ft>s. For the 
weight of spikes of larger dimensions, we may near enough take that of a 
square bar of the same length. What is saved at the point, suffices for the 
addition at the head. Price, Phila, 1886, from 2 cts per lb for j 9 b inch thick ; 
to 3 for %. 


Size in 

ins. 

N o. per keg 

Length. 

Side. 

of 150 tbs. 

* l A X 

fs 

k 

526 

41/ 

2 X 

400 

5 

X 

% 

705 

5 

X 

is 

k 

488 

5 

X 

390 

5 

X 

* 

295 

5 

X 

% 

257 


No. per lb. 

Size in 
Length.j 

ins. 

Side. 

3.5 

2.66 

X 

f>A X 

A 

4.7 


X 

3.25 

6 

X 

A 

2.6 

6 

X 

fs 

k 

1.97 

1.71 

6 

X 


No. per keg 
of 150 lbs. 

No. per lb. 

350 

2.33 

289 

1.93 

' 218 

1.46 

310 

2.07 

262 

1.75 

196 

1.30 


A size in very common use, is 5% X Tb ’> which weighs about x / z lb per 
spike. A mile of single-track road, with 2640 cross-ties, 2 feet apart from center 
to center; and with rails of the ordinary length of 30 feet, or 15 ties to a rail; 
will have 352 rail-joints per mile; and with 4 spikes to each tie, will require lo560 
spikes, weighing about 5500 lbs. 

But au allowance must be made for rail guards at road-crossings, which we may assume to be 30 ft 
wide, or the length of a rail. A guard will usually cousist of 4 extra rails for protecting the track- 
rails, and spiked to the 15 ties by which said track-rails are sustained. Consequently, such a crossing 
requires 15 X 8 = 120 spikes. For turnouts, sidiugs, loss, <fcc, we may roughly average 700* spikes 
more per mile ; thus making in all (if we assume one road-crossing per mile) 10560-f-120 -)- 700 — lisao 
spikes per mile; or say 6u00 lbs, or -40 kegs of 150 lbs. 

Adhesion of spikes. Professor W. R. Johnson found that a plain spike 
.375, or % inch square, driven 3% ins into seasoned Jersey yellow pine, or unseasoned 1 
chestnut, required about 2000 lbs force to extract it; from seasoned white oak,about 
4000; and from well-seasoned locust, about 6000 lbs. Bevan found that a 6-pennv 
nail, driven one inch, required the following forces to extract it: Seasoued beecli 
667 lbs; oak. 507 ; elm. 327 ; pine, 187. 

Very careful experiments in Hanover. Germany, 

by Enginekk Funk, give from 2465 to 3940 lbs, (mean of many experiments, about 3000 
lbs,) as the force necessary to extract a plain }4 inch square iron spike,6 ins long, wedge- 
poiuted for one inch, (twice the thickuess of the spike,) aud driveu 4% ins iuto white or 
yellow pine. When driven 5 ins, the force reqd was about yfy part greater. Similar 
spikes, inch square, 7 ins long, driveu 6 ius deep, reqd from 3700 to 6745 lbs to ex¬ 
tract them from piue; the mean of the results being 4873 lbs. In all cases about twice as 
much force teas reqd to extract them from oak. The spikes were all driven across the 
grain of the wood. Experience shows that when driveu with the gra n, spikes or nails 
do not hold with much more than half as much force. 

Jagged spikes, or twisted ones (like an auger, j or those which were either swelled or diminished 
near the middle of their length, all proved interior to plain square ones. When the length of the 
wedge point was increased to 4 times the thickuess of the spike, the resistance to drawing out was r 
trifle less. But see “ Jag-spike ” in Glossary. 

When the length of the spike is fixed, there is probably no better shape than the plain square cross 
section, with a wedge poiut twice as long as the width of the spike, as per this fig. 



* This allows that turnouts and sidings amount to about 1 mile of extra track on 15 miles of road. 





































RAIL-JOINTS 


763 


t 

'# 

k 

li 

i 


- 




RAIL-JOINTS. 

er 

•' Art. 1. A track, being weakest at the joints between the rails, where they 
j' 'e deprived ot' their vertical strength, has of course a greater tendency to bend at 
j# lose points; and this bending produces an irregularity in the movement of the 
lain, which is detrimental to both rolling-stock and track. Moreover, that end of a 
11 til upon which a loaded wheel is moving, bends more than the adjacent unloaded 
id of the next rail; so that when the wheel arrives at said second rail, it imparts to 
s end a severe blow, which injures it. Thus, the ends of the rails are exposed to 
r more injury than its other portions. Numerous devices have been resorted to for 
lengthening the joints of the rails, with a view of preventing this bending entirely; 
•. at least, of causing the two adjacent rail-ends to bend equally, and together: so 
i to avoid the blows alluded to. None of these joint-fastenings, known as chairs, 
sh-plates, wooden blocks, &c, have proved entirely satisfactory. 

Much of the deficiency ascribed to the fastenings, is, however, really due to want of stability in the 

i o-is-ties at the joints; and more attention must be directed to this latter consideration, before an 
Bcient fastening can be obtained. Observation shows that when the joint-ties are very firmly 
aided, almost any of the ordinary fastenings will, (if the joint is placed between two ties, instead 
resting upon a tie,)* answer very well; whereas, when the cross-ties are so insecurely bedded as to 
ay up and down for half an inch or more under the driving-wheels of the engines, the strongest and 
ost effective fastenings soon become comparatively inoperative. All the parts of the best of them 
ill in that case become gradually loosened, warped, bent, or broken. 

Experience has established the superiority of suspended joints over supported 
lies. Long fastenings, perhaps, possess but little superiority over short ones, where 
he track is not kept in good repair; for the great bearing of the former, although 
aiparting increased firmness on a good track, becomes converted into a powerful 
average, by which it accelerates its own destruction, in a bad one. An element in 
he injury of joints, is the omission of proper fastenings at the center of the rails. 
Jach rail should be so firmly attached to the cross ties at and near its center, as to 
ompel the contraction and expansion to take place equally from that point, toward 
ach end. It would probably be somewhat difficult to accomplish this perfectly, 
'he attempts hitherto made have failed. 


* In the first case the joint is called a suspi'ton-u one • i:i the last a supported one. 









764 


RAIL-JOINTS. 


Under the extremes of temperature in the United States, bar Iron expand 
or contracts about 1 part in 916; or 1 inch in 76% feet; consequently, a ra 
30 ft long will vary T 7 S inch; and one 20 ft long fully % inch. 

Beside this, the rails are very liable to move or creej 
bodily in the direction of the heaviest trade, especially when tli 
grade descends in the same direction; and by this process also the joint-fastening 
are exposed to additional strain and derangement.* 

All rails appear to become elongated very slightly at their ends by use; and th 
renders a full allowance for contraction and expansion the more necessary. 

Art. 2. “Even ” Joints and •• broken” Joints. If the road-bed i 
of a yielding character, it is advisable to place the joints (on the two rails of the sam 
track) opposite to each other (‘"even joints”); because “staggere 1 ' ” or “broken 
joints are more liable to give the cars a sideways swaying motion. But if the roa( 
bed is solid, and in good condition, the broken joints are preferable, because, wit 
them, the jar of passing from rail to rail is less severe than where both wheels mak 
that passage at the same time. 

Art. 2. Beveled, or mitred Joints. To lessen this jar, Mr. Sayre sug 
gests cutting the rails so that the vertical plane forming the rail end shall make a | 
angle of 45° to 60° with the longitudinal vert plane of the web of the rail, instead o I 
the usual right angle. This would permit the use of longer rails than are now lai< ! 
as the great space (% inch or more) between the ends of such long rails in col [ 
weather, would not be so serious an objection when the ends were thus cut obliquely I 
This method of cutting the rails has been tried, with good results, but lias not yt j 
come into general use. It is claimed that a comparatively inexpensive change in tli 
arrangement of the saws at the rolling mill, would permit the rails to be cut wit 
ends at any angle, as readily as with square ends, and without further increase i 
the cost of sawing. 

Art. 4. Fisli-plates, Fig 1, and Angle-plates, Figs 2, 3, 4, and 5, hav 

nearly supplanted all other forms of joint on the principal 
railroads of the U. S. Although the rails are almost uni¬ 
versally of Bessemer, or similar, steel, the joint-plates are 
generally made of iron. They are rolled in long bars, and 
cut off in any desired length, generally about 2 ft; and 
are bolted together, and to the rails, by4 bolts, 2 in each 
rail-end. 

Art. 5. Tlie fish-plate Joint was one of the 

earliest suggested. It was introduced upon the Newcastle 
and Freuchtown It R, in Delaware, by Robt. H. Barr, in 
1843. The weight of a complete fish-plate joint, includ¬ 
ing bolts and nuts, is about 20 lbs. 

Fig 1, one-fifth of real size, represents a fish-plate joint 
made by Cambria Iron Co, office 218 S 4th St, Phila,with a steel rail weighin 
50 lbs per yard, by the same Co. The thickness of plate at /is T 9 6 inch. The plate 
weigh 414 lbs per lineal foot of a single plate. 

Art. 0. The principal advantage of the angle-plates is that the spike! 
which are driven through slots in their flanges to confine them to the cross-tie! 
serve also to counteract the tendency of the rails to “creep.”* (See Art 1.) Wher 
fish -plates are used, tlie flauges of tlie rails have to be slotted for this purpose. 


rTTTTTTl 

INCHES 

Fig.2. 

Art. 7. Fig 2 (one-fifth of real size) shows the standard angle-plate joint (1884 
of the Pliila A Reading R R. The bars are 2 ft long, and weigh 16 lbs each 

* On the St Louis bridge (steel arches) and its eastern approach (plate girders on iron column; 
the rails creep, in the direction of the traffic, about a foot per day , both up «nd down a grade of £ 
it per mile, and with such force that although various fastenings were used, in order to prevent tfc 
creeping, none proved effectual. The track is now adjusted daily to accommodate the creeping. 




























RAIL-JOINTS. 


765 



8 . The wheel-tread and 76 lb steel rail, shown (one-fifth of real size) in 

Fig 3, are those designed by Robt. If. Sayre, 
C. E., and used by the Lehigh Valley R R, 
under very heavy traffic. Tile angle-plate 
joint was designed by Mr. Jolin Frit* *, 
Supt Bethlehem Iron Co, Bethlehem, Pa, and Mr. 
Sayre, and has been in use on the I,eliig-it 
Valley R R for the past 12 years. 

These forms of wheel-tread, rail, and splice, are 
the result of careful study, and each detail has 
been modified from time to time as experience 
dictated, until now they are probably the most per¬ 
fect in this country. Mr. Sayre places the stems 
of the two plates much farther apart than usual, 
thus giving the joint greater lateral strength; at 
the same time adding to its vertical strength by 
the support given to the lower side of the rail¬ 
head by the upper enlargement c; while the lower 
one a secures a full bearing on the flange of the 
rail. The joint, for 76 lb rail, complete, 2 ft long, 
with 4 bolts % inch diam, weighs 40 to 48 lbs, 
depending upon the thickness of the angle plate. 
The drilled bolt-holes in the stem of the rail, arel 
inch diam, to allow the rails to contract and expand. 
Figs 4 and 5 (one-fiftli of actual size) show an angle-plate joint 

made by Eambria Iron Vo, Johns¬ 
town, Pa, office 218 S 4th St, Phila, and fur¬ 
nished with their patent uut-lock, 
which consists of a small piece, or “key,” 
p, of Bessemer steel, semi-circular in cross- 
section at one end, and tapered to a hori¬ 
zontal edge at the other. After the nut has 
been screwed to its place, the key is driven 
close up to it, and then the pointed end of 
the key is bent up (as shown in Fig 5) by a 

special tool with a lever attached. The key 
is prevented from falling out sideways by 
the edge of the longitudinal groove, Fig 4, in 
the angle-plate, into which it fits. This nut- 
lock resembles that used with the old form 
f Fisher-joint, see B Figs 15. See nut-lock washers, p 408. 

Arf. 10. Both fish- and angle-plates are apt to crack vertically about the mid- 
le of their length, or opposite to the joint in the rail. To obviate this, the 
' 8amson-bar ” (which is made either of fish or of angle form) is rolled about 
alf inch thicker at the middle than at its ends. The thickened portion is about 8 
as long, extending say 4 ins each way from the joint, but the upper edge of the bar, 
! re jpon which the head of the rail bears, and in fish-bars the lower edge also, are made 
f this increased thickness throughout their length. These joints are (1886) largely 
sed on the Western railroads of the U. S. Their cost is about the same as that of 
rdinary fish-and angle-joints. They are made by Morris Sellers & Co, office No. 6 
tshlaud Block, Chicago, Ill. 

Art. 11. Fish- and angle-plates, of all the patterns shown, and others, are 

*o!lcd to suit different sizes and shapes of rails. The bolt 
leads are usually round, and the shoulders of the bolts, immediately under 
he heads, are therefore made of oval cross-section, fitting into corresponding oval 
mles in the fish- or angle-plate. The bolt is thus prevented from turning when the 
int is screwed on, and afterwards. Many devices have been tried, with a view to 
ireventiug the nuts from wearing- loose (see lock-nut washers, 

,408). The Vulcanized Fibre Co, Wilmington, Del, furnish a vulcanized 
washer, which is intended to act as an elastic cushion, deadening shocks and 
ibrations. They are said to become hard, and lose their elasticity, in time. 

The plates are frequently rolled,as in Figs 17, with a longitudinal groove, 
is wide as the head or nut of the bolt, and about % inch deep, running their entire 
engtli. This groove receives either the head of the bolt, which in such cases is 
nade square or oblong and inserted first, and the nut afterwards screwed on; or 
Ise the nut is first placed in the groove, and the holt afterwards screwed into it. 
['liia i s intended to prevent the unscrewing of the nut, but cannot be relied upon to 
lo so. 

' It is well to have the slots in the flanges of rails or of angle-bars so spaced that 


HT 

Fig-,4. 


—3- 

INCHES 


Fig-.5. 






































766 


RAIL-JOINTS. 


tlie two spikes of a joint, driven into the same cross-tie 
shall not be directly opposite to each other, but “staggered,” so as t 
diminish the danger of splitting the tie. 

Joints are frequently laid with one tish- and one angle-plate. 

In 1886, angle-and fish-plates cost about 2 ets per ft; bolts and nuts, 3 cts pe 
ft. The cost of a complete angle-joint for 70-ft rail, is from 80 cts to $1.20; of 
fish-joint, 60 to 80 cts. 

Art. 12. It will be noticed that both fish- and angle-plates act by placing 
support under the head of the rail. The Fisher bridge-joint. Figs 6 to 8 
made by Mr. Clark Fisher, Trenton, N J, applies the support under the base of th 

rail. . . 

The principal feature of this joint is a flanged beam, Fig 6, about 6 ms wide aiv 
22 ins long, which extends across, and is spiked to, the two joint-ties, as in Fig \ 
The holes for the spikes are placed so that the two spikes in the same tie are no 
opposite to each other; and the flanges F F also are staggered, so as not to interfer 
with the driving of the spikes. The joint-ties T T are placed 7 inches apart in th 
clear. The beam has an upward camber of about one-eiglith of an inch. The tw 


1 

r 

ii 

ii 

* 

i 




x i u • 



rail-ends, forming the joint, rest upon the beam, and meet at the middle of itf 
length. They are held down to it by a single U-slmped bolt B, of 1 inch diam 
with a nut on each leg. These nuts bear directly upon the horizontal upper sides 
of the “fore-locks” L L, one of which is shown separately in F'ig 9. The fore-Iockt 
are rolled to fit accurately to the rail-flanges. The legs of the U-bolt pass first 
through the circular holes h /», in the beam. Fig 6; next through rounded notches 
cut in the corners of the rail-flanges; then through the holes in the fore-locks; anc 
lastly through the nuts. Between the U-bolt and the bottom of the beam is place' 
a small piece s,of spring steel,slightly cambered downward, and having two semi-cii 
cular notches for the legs of the U-bolt, which hold i^in place. This is intended tokeej 
the joint elastic, to take up any loose space produced by the wear of the surfaces it 
contact, to render less abrupt the strains on the bolt, and, by keeping the threads 
of the nut pressed against those of the bolt, to prevent the nuts from becoming 
loose. The joints are shipped from the factory complete, and with all the parts 
bolted together; the nuts being screwed down to within about two threads of theit 
final places, so that the ends of the rail-flanges can be easily slid into place unde) 
the fore-locks. 

As an additional precaution against creeping of the rails, the rail-flanges may b< 
slotted near their ends, as in cases where fish-plates are used, and spikes driver 
through these slots. For such cases the beams are punched, at the mill, with font 
additional square holes a little further from the edges of the beam than the others 
Unlike the angle- and fish-plate joints, the Fisher may be used with any section of 
T-rail; and the head of the rail may be made stronger by being rolled pear-shaped 
which is inadmissible with fish- and angle-joints, because these require a nearlj 
















































RAIL-JOINTS. 


767 


.||, rizontal bearing on the under side of the head. The “ Fisher” requires no drill- 
■ Mr or punching of the stem of the rail. It costs about 25 per cent more than a 
r -01 angle-joint for the same rail. Its weight, complete, for 65-ft> rail, is about 

JWr. Fisher makes also an extra strong joint with three U-bolts 
It heavy curves and tor places liable to wash-outs. It is intended to support the 
t lit, even if the ballast is removed from under the joint-ties. Either of the Fisher 
njlnts can be made of any desired weight. The “Fisher” is largely used on some 
tifl the puncipal eastern roads, and with very satisfactory results. Figs 15 n 768 
l>w an old form of this joint. 6 ’ v ' 

Jm?TW rail-joint, Figs 10,11, and 12,invented 

Mr. I bos. II. tiibbon, 0 E, Albany, N Y, is a supported one. Two inches in length 



Fig. 12. 

the head, at each rail-end, have to be cut off, as in Fig 10; but no further cutting, 
d no punching or drilling, of the rail, is required. Over the ends of two rails, thus 
t, and placed together in their final position on the ties, the Bessemer steel “sad- 
>casting,” Fig 11, is placed. The top of this casting is shaped so as to correspond 
th the head of the rail; and its length, 4 ins, just occupies the space cutaway 
[>m the ends of the two adjoining rail-heads. The feet of the casting fit into notches 
t in the joint-tie at the sides of the rail. The ends of the rails are then raised 
glitly, and the base-plate. Fig 12, of iron or steel, is slipped through the slots in 
e side of the saddle-casting, then under the base of the rail, and lastly through a 
rresponding slot in the opposite leg of the saddle-casting. Spikes are then driven 
to the tie through the four holes in the base-plate, and the joint is complete. The 
ddle-casting and base-plate weigh, together, about 24 fbs, and cost (1886) $1.00. 
Art. 14. The following are some forms of rail-joint that have fallen into disuse, 
have been proposed but not adopted. They illustrate early practice, and may bo 
;eful as hints. 

Fig 13 was an early 
form of supported 
w rongli t-i roil 
cliair. It was about 
7 ins square, % thick, 
and weighed lOlhs. Fig 
14 is a later form, still 

furnished, to some ex- ^ ^ . 

tent, by the Tredegar 1 lg.14. 

Co, of Richmond, Va. 



iliinuiimiuiummimniitilmiiiimimmumiunt 

Fig. 13. 



Figs 15 show one of the earliest forms of the Fisher joint. It 

as at one time largely used on the Lehigh Valley R R. (For the present form of 
iis joint, see p 766.) It was a suspended joint; and, instead of the long supporting 
im, or bridge-plate, of the present pattern, it had a plate c, 6 ins square. It had 
U-bolts, each of 1 inch diam, and the fore-locks 11 were 6 ins long, and had two 
ales each. The lower side of the thread on the bolt was made horizontal as at j. 
he thread in the nut was somewhat as at l. In screwing on the nut, these two 





















768 


RAIL-JOINTS. 


threads were forced into conformity, thus employing the principle of the prese 
Harvey lock-nut, p 408. A nut-lock B, somewhat similar to the Cambria pate 
lock, p 765, was used as an additional precaution. 


< 4 ' > 



Figs 16 show a remarkable suspended fastening, ealled the Ring! 

joint; highly approved of at one time on the Camden & Amboy road (now Peni 
It R, United Railroads of New Jersey Division) on which it was employed for uiai 
years, under a heavy traffic; to the almost entire exclusion of others, except f< 
experimental comparison. 



This fastening consisted of a simple welded triangular ring tea, (in the end view 
or m in the side view,) % an inch thick, and ins wide. This ring passed throng 
a slot v r\ (see middle fig,) 4 ins long, cut into the adjacent rail-ends. Two cast-iro 
wedges w tv, 6 to 8 ins long, of a shape to fit the ring and the rail, were inserted b< 
tween them ; and a thinner one s s, of plate iron, below the rail. The first were cm 
around a cylindrical rod of rolled iron, (i v of the end view, * i of the side view,) aboi 
% inch diam; and a little longer than the wedges; for increasing their strengt’ 
and for preventing them from falling out from the chair in case they should brea 
which they sometimes did. The rail-ends sometimes split, as shown .at l and k. 

The joint was suspended betweeu two cross-ties, 1 ft apart in the clear. 










































































RAIL-JOINTS. 


769 


* Figs 17 represent the combined suspended joint-fastening introduced upon the 
hilada <fc Reading railroad by J. Dill toil Steele, C E. Some of them are still 
1 use (1884). 



The joint is suspended between 2 cross-ties, placed 1 ft apart in the clear. B is a 
lock (known as Trimble’s splice) of oak, 3 by 3 ins, and 3 ft long, dressed on one 
de to fit the outside of the rail; and c is a rolled fish, 17 ins long, (shown more in 
etail at F,) placed on the inside of the rail. This fish and the oak block are bolted 
„ igether, through the rail, by two %-inch screw-bolts a a, 13 inches apart. Under 
D, ne rail is a rolled chair d , 2 ft 8 ins long, 5 ins wide, and y 2 inch thick; turned up 
aD ( inch along each of its two sides, and fastened to the 2 wooden cross-ties by 4 hook- 
d eaded spikes, ]/ 2 inch square, by ins long. The heads of the two screw-bolts are 
lade somewhat oblong, (about % inch by 1%,) for fitting into the groove nn, seen 
long one side of the fish ; so as to prevent the tendency of the bolts to revolve under 
le action of the trains, and thus unscrew the nut at the other end. The nuts, how¬ 
ler, unscrew themselves, notwithstanding this precaution. The strain on the screw- 
olts is great, both vert and hor; and it becomes greater as the wooden blocks in 
me lose (as thev do)their close fit to the sides of the rails. The blocks then cease 
> act in perfect unison with the other parts of the fastening, in sustaining passing 
.ads; and when the track is not kept in good order, the various parts may plainly 
a seen to yield and move in various directions, independently of each other. The 
olt-uuts then loosen; and the fish pieces, long chairs,and long bolts, become bent; 
id sometimes split, or break entirely. The long wooden blocks B crush and decay 
KUiest near where their tops are in contact with the rail. They, however, have an 
.rerage life of 6 to 8 years, upon roads kept in tolerable order. 



Fie-18 is a ioint for U-rail. It was of rolled iron, 6 ft long, and rested on 3 ties, 
.was riveted loosely to one flange of the rail, as shown on the right-hand side. It 

F P ?gl9 -any years since by Alex. W. Itae, € E, 

Fb^Twas^Isoone of the numerous joint-fastenings suggested at an early day; 
^uiie.he fcre.oin* it never 

S,'C VT«h. loo. Of the ciinir. This would din,in- 

h t T e difficulty of sliding the chairs on or off of the rail; and would thus make i t 
tsy to employ larger ones; besides insuring a firm bearing lor the base of the lail 

pon the fastening. 


50 











































770 


TURNOUTS, 


TUKNOUTS, 


Art. 1. To enable an engine and train to pass from one track, A B, Fig 1, to 
another, A D, a turnout is introduced. This consists essentially of a 



switch, q mp s, a froj?,/, and two fixed ^uaril- rails, g and g'. If a switch 
is made to serve for two turnouts, A D and A D'. Fig 2, one on each side of the main 
track, A B, it is called a three-throw switch. 



Art. 2. When a train approaches a switch in the direction of either arrow, Fie 
1; or so that it passes the frog before reaching the switch, it is said to “trail* 
the switch. When it approaches in the opposite direction, passing the switch before 
reaching the frog, it is said to 44 face” the switch. Fig 3 represents a portion of 
a double-track road in which the trains keep to the right, as shown by the arrows. 
In this fig, V and W are “ trailing ” switches; and X and Y are “ facing ” switches. 
In order to leave the main track by a trailing switch, a train must move in a direc¬ 
tion contrary to the proper one on said track. 

Art. 3. misplaced switches. A moving train, facing any switch, must 
plainly go as the switch is set, whether right or wrong. If wrong, serious accident 
may result. For instance, the train may run upon, and over the end of, a short 
trestle siding, or may collide with a train standing or moving upon the turnout 
Safety switches, such as the Lorenz, Arts 13, Ac, and Wharton, Arts 18, Ac 
are so arranged that trains trailing them can pass them safely, even if the switch 
is misplaced. But in the case of the plain stub-sw r itch. Art 4, when mis 
placed, a trailing train will leave the rails at b and r, ore and u, Fig4, and rut 
upon the ties. » 

Stub-switches are frequently provided with 44 safety-castings ” of iron 
bolted to their sides, and reaching front their toes m and s, Fig 4,several feet towart 
p and q. These, in case of misplacement of the switch, receive the flanges of tin 
wheels of a trailing train, and guide the w heels safely on to the switch-rails q m am 
ps. The “Tyler” switch is arranged in this way. 


























TURNOUTS. 


771 


Ar*. 4. The common blnnt*en<lo(l or s(i!b>switcli consists 
*»s«*n tially of two rails, q m and p s, Figs 1 and 4. The ends, q and p , of these 
ils, where they are fixed in line with the main track, form the •* heel ” of the 
;yitch. Their other ends, m and s, form the 4 * toe,” and are free to move from d 




On main lines of road, the switch-rails are usually from 18 to 26 feet long from 
id to toe. Formerly their heels ,q and p , were fixed by being confined in the same 
lairs which held the adjoining ends of the main-line rails ; or by being connected 
ith said rails by short fish-plates. See p 764. In either case they remained prac- 
mlly straight, even when set for the turnout. Now, however, they are generally 
ade long enough to extend 4 to 6 feet back from their heels, toward A ; and this 
Iditional length is spiked unyieldingly to 
e ties; so that, when set for the turn- 
it, the switch-rails bend so as to form (at 
ast approximately) a part of the turn- 
it curve. Their toes, rn and *, in any 
se, rest and slide upon iron 44 liea<l« 
lates,?* P P, Fig 4, shown in detail 
Fig 5. These head plates also receive 
e ends, be and ru, of the main-track 
id turnout rails. In three-throw switches, 
e head-plates must of course be longer, 
give room for the three rail-ends side 
' side. 

Frequently a plain strip of iron, about 3 
s wide by half inch thick, is fastened to 
e upper surface of each tie, under the 
.se of the rail, for the latter to slide on. 

The switch-rails are connected together by from 3 to 5 transverse wrought-iron 
lamp-rods, R R R\ Fig 4, full ins diam. These are fastened to the rails in 
rious ways; generally as shown in F’ig 6. The clamp-bars should be placed, if 
^ possible, at least as low as the tops 

- Gauge VT/C - ? Of the cross-ties, so as to avoid 

danger of their coming into con¬ 
tact with any portions of cars or 
engines that may become par¬ 
tially detached and drag on the 
Fig. G. track. 

One of the clamp-bars, R', is 
ar the toes, m and s , Fig 4. It projects beyond the track, and is jointed, as shown 
Figs 7 and 8, and connected with the lever, L, by which the switch is moved, 
le tie, T, Fig 4, to which the head-plates are fastened, is made longer than the 
hers, in order to give room for the switch-stand, M, Figs 4,7, and 8, which is bolted 
its upper surface. This tie should also be of larger cross-section than the others, 
rfectly sound, and well bedded; because upon it come severe strains due to the 
tssage of cars and engines across the space between the rail-ends, m and e, s and «, 
Fig 4. See second paragraph of Art 10, p 773. 


Fig. 5. 































































772 


TURNOUTS. 


Art. 5. The switch.lovers, and the switch.stands to which they are 

attached, are made in a great variety of forms. See Figs 7, 8,9, 14, 15, and 17. That 
shown in Figs 7 and 8 is the *• Tumbling-lever stand ” or Ground-lever 
stand,” and, in its numerous modifications, is very largely used. It is so arranged, 
that, whichever way the switch is set, the crank, C, is on the dead center, so that 
the lateral strains of passing cars or engines can exert no tendency to turn it. Tile 
“Greenwood” stand, made by the Pennsylvania Steel Co, Steeltou, Pa, has 



can be used for either a two-throw or a three-throw switch. i 

Tumbling switches are convenient because they occupy but little space. By means 
of a target or lantern, connected with the switch, they may be made to indicate to 
the engine driver the position of the switch. 

When the switch is set either way, the lever is padlocked to a staple driven into 
the tie and passing up through the slot in the handle of the lever. The lever is fre-« 
quently made with a weight of say 20 lbs on its free end, to aid in bringing it down 
to its proper position. Price of a tumbling-lever stand, 1886, $3.50 to $5.00, with¬ 
out target. 

Art. 6. Fig 9 represents a common form of the upright lever and stand. 
The switch-rod, R' Figs 4, 7 and 8, is generally attached at the lower end. A. of the lever, i 
The cast-iron frame, F, is fastened to the long tie, T, Fig 4, by large screws or 
spikes, which pass through its broad feet or flanges, B B. The top of the frame is I 
provided with two notches, N N, and staples, to which the lever is secured by a 
padlock. When this stand is to l>e used for a t/tree-throw switch, the frame has 
three notches and three staples. The upright stand may be used wherever it will 
not be in the way of passing trains. The target, T, at the top of the lever, by 
showing the position of the latter, indicates to the driver of an approaching engine 
which way the switch is set. 





MONKEY SWITCH 

Fig. 10. 


Art. 7. In the “ Monkey-switch,” Fig 10, the crank, oi, is moved hori¬ 
zontally through an arc of a circle by means of the lever, h h, about 3 ft long, which 
fits upon the square head,.s, of the vertical spindle or pin, s o. The switch-rod, R' Figs 
4, 7 and 8, is attached to the pin, iv. 

Many modifications of the monkey-switch are in use. The spindle, s o, is fre¬ 
quently made long enough to bring the lever to about the level of the hand; and 
the lever is permanently attached to the stand, and hinged near the spindle so at 
to hang down, out of the way, when not in use. To the top of the spindle is fre¬ 
quently attached a vertical rod of any desired length, and carrying at its top a targel 
which turns as the spindle does, and thus indicates the position of the switch. 
























TURNOUTS. 


773 


Art. H. All parts of the switch-stand, and the tie upon which it rests, should 
,| > perfectly rigid, because it is very important that they should hold the ends of 
j ie switch-rails exactly in line with those of the main line and turnout. They 
, lerefore, in view of the great strains to which they are subjected, must be strongly 
i, instructed, and frequently looked after. See Automatic switch-stand, Art 12, 
j id De Tout’s switch-stand, Art 14. 



Fig. 11. 


Art. 9. In Figs 1 and 4, d qrn is called the switch-angle. The dist, dm, 
igs 1, 4, and 11, required for the motion of the toes, is called the throw of the 
vitch. It must be equal at least to the width, dw. Fig 11, of the top of the rail, 
i ii addition to a width, win, sufficient to allow the flanges of the wheels to pass 
ong readily between b and e, Fig 1, and between r and w. The tops of the rails 
s re generally between 2 and 2)^ ins wide; and about 1% to 2^ ins suffice for the 
i inges. The throw, d in, however, is commonly about 5 ins. 

The gauge, Fig 6, p 771, of a railroad track, is the distance between the inner 
l des G G' of the heads of its two rails. Hence these inner sides are called the gauge 
ides of the rails. 

Art. 10. The stub-switch is cheaper in first cost than the improved safety 
i vitches, Arts 13,18, etc, but is less economical in the long run. 

As it is very essential that the toes of the switch-rails should never come into 
intact with the adjoining rail-ends, a space of about an inch must be allowed 
, the toes for expansion, and for “creeping” (see p 764). This renders the blows 
‘ passing trains very severe, and injurious to rolling stock, and to the rail-ends, 
lie latter are worn away rapidly and must be frequently renewed. From the same 
mse the tie under the head-plate is apt to become loose in its bed. 

In ordering fixtures for stub-switches, the exact section of rail, 
id gauge of track, should lie given. The cost of a stub-switch with switch- 
and, is, 1886, from $18 to $30, according to size, finish, character of stand, Ac, Ac. 











POINT SWITCH 


TURNOUTS 




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TURNOUTS. 


775 




Art. 12. The 44 Automatic” switch-staml, made by the Penna Steel 
: o, is shown in hor sectiou by Fig 14, and in vert section by Fig 15; and is designed 

*• — ! __ 


Fig. 14. 


Fig. 15. 

especially for split-switches like 
those shown in Figs 12 and 13. It 
is operated by a tumbling-lever, L, 
Fig 14. To the hor axis. A, of the 
lever, is fastened the beveled pin¬ 
ion, P. This engages in the teeth 
of the quadrant, Q, and moves it 
horizontally through a quarter of 
a circle, when the lever L is thrown from its position L to that 
of I/, shown by the dotted lines. The rod, K/, from the switch, 
is attached at X,or at X', to this quadrant; and the movement 
of the latter thus sets the switch, and at the same time turns 
he target, T, Fig 15, or lantern, fixed to the vert spindle, S. thus indicating the 
losition of the switch. The gearing is enclosed in a cast-iron case, as shown in Fig 
5. If a train on the turnout, moving in the direction of the arrow. Fig l (or “ trail- 
ng”), approaches a split-switch provided with the automatic stand, and set (through 
versiglit or otherwise) for the main track, as in Fig 12, the flange of the first wheel, 
ressing between rails X r and S' V', will push the switch-rails into the proper posi- 
ion, Fig 13, for the turnout; at the same time necessarily throwing the weighted 
:i ever over into the reverse position and turning the target so as to indicate that the 
witch is set for the turnout. A similar movement of the switch, in the opposite 
lirection, takes place if a trailing train on the main track approaches the switch 
vlien set for the turnout, as in Fig 13. Hence the term “automatic,” as applied to 
his stand. The switch is thus made a safety switch. 

Tlie cost of the automatic switch-stand (1886) is about $15. 


I Art. 13. The Lorenz safety-switch, designed by the late Wm. 
t' xtrenz, Esq, Ch Eng of the Phila & Reading R R, is a split-switch, in which the 
connecting-bar, R', nearest to the toes, is provided with a spring 1 . S, Fig 16, placed 
0 Sometimes between the rails, as there shown; sometimes outside of the track. This 
j: spring permits the moving of the switch-rails by the wheels of a trailing train, as 
« loes the automatic switch-stand. Figs 14 and 15; hut, after the passage of each wheel, 
jj the spring returns the switch-rails to their original position, ihe blow of the 
\ 






































































































































776 


TURNOUTS. 


switch-rails against the stock-rails, thus occasioned, is injurious to both, and liable 
to break the former. It is also urged as an objection to these switches that the 
spring may permit the switchman to force the switch-lever to its place, even 
although some obstacle, as a small stone, lodged between the switch-rail and stock- 
rail, prevents the two from coming into contact, so that both points remain an inch 
or so away from the stock-rails and in danger of being struck by the wheel-flangos 
of approaching trains “facing” the switch. See Art. 14. 

Art. 14. I»e Vout’s safety switch-stand, Fig 17, made by Penna 
Steel Co, is designed to remedy this. In this stand, the spring is placed in, and se¬ 
cured to, a semi-cylindrical iron spring-case or box, B; to the opposite sides of which 
are fixed two hor axles. One of these is shown at A. This axle passes through the 





switch-lever, L, near its fulcrum, F. It also passes through the inverted T-shaped 
slot, H, in the rigid bar, S, which, together with the bar, W, attached to the spring- 
case, is jointed, at J, to the switch-rod. It'. When the switch is properly set, either 
for the main line or for the turnout, the axle, A, is in the hor part of the slot, II, 
and immediately under the vert part, so that there is no obstrnction to the move¬ 
ment of the switch, and a trailing train will open a misplaced switch as explained 
in Arts 12 and 13. But when the lever is raised, for the purpose of setting the switch 
in the other position, the axle. A, rises into the vert part of the slot, as in the fig, 
lifting the spring-case with it. If now any obstrnction prevents the switch-rail 
from being pressed home, the rigid bar, S, by means of the axle, A, prevents the 
lever, L, from moving farther. Cost of I>e Vout’s stand, 1886, is about $12, 
without target. 

Art. 15. Theory would require that tin* lengths of the switch-rails, 

in split-switches, should vary with the radius of the turnout curve, and formerly 
they were so made. Where this radius is such that a No 10 frog (see Art 26) is re¬ 
quired, the switch-rails should, theoretically, be 28 ft long. But in practice a uni¬ 
form length of 15 ft (just half the usual length of the steel rail from which the 
switch-rails are cut) for all turnouts, gives the best results, combining ecouomy of 
manufacture with greater strength, and greater ease of handling, than are possible 
with ninch longer rails. 

Art. 16. It will be noticed that in point-switches (as also in the Wharton switch, 
Arts 18, Ac) there can be no such jar as that occasioned in the stub-switch by the 
long space between the toes of the switch-rails and the ends of the adjoining rails. 

Art. 17. It is important that the thin portions of each switch-rail should be 
carefully shaped so as to receive throughout a firm lateral support from the stock- 
rail when in contact with it. Otherwise the switch-rails are in danger of bending 
under the lateral pressure of passing trains. This might throw the point out from 
the stock-rail, endangering the train. 

The price of a 15-ft Ix>ren* switch, including the two point-rails with 
connecting-bars, spring, spring-fixtures, switch-rod, slide-plates,and rail-braces,but 
exclusive of stock-rails, lever, and stand, is, 1886, about $30 to $10, according to weight 
of rail, gauge, &c. 

l.oreii* switches abont 734ft long* are made for yard nse. Price, 
including the same items as in the full-sized switches, 1886, about $25 to $33. 

































































778 


TURNOUTS. 



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780 


TURNOUTS. 


Art. 22. Frogs. The frog is a contrivance for allowing the flange of the 
wheel on the rail ex, Fig 1, to cross the rail rz; and that of the wheel on r z, to cross 
ex. Tlie first contrivance for this purpose was a bar, approxi¬ 
mately of the shape of the rail, pivoted at the point where the center lines of the 
rails ex and r z cross each other, and free to move horizontally about this pivot, so 
that it could form a portion of ex when the train was passing to or from the turnout, 
or a portion of r z when the train was using the main track. Sometimes the pivot 



through one end of the bar, as in Fig 2b, and sometimes through its center, 
as in Fig 21. Such bare were generally moved by a rod 
(attached at n)and lever,similar to those used for switches; 
and they then, of course, required an attendant; but many 
attempts have been made to use such frogs by connecting 
them with the switch by means of rods, &c, so that the bar 
should move automatically when the switch was turned. 
Owing to the considerable distance (80 ft, more or less) be¬ 
tween the frog and switch, it has been found difficult to 
secure simultaneous movements of the switch and frog, and the contrivances referred 
to have not come into extensive use. Such bars, while they avoid the jar produced 
by wheels passing across the throat of the frog (Art 35), labor under the same dis¬ 
advantage as the stub-switch. Art 10, in requiring a liberal allowance of space be¬ 
tween their ends and those of the adjoining rails, to avoid any possibility of their 
coming into contact. 




Art. 22 . These bars were soon superseded by rigid cast-iron frogs. Fig* 
22 and 23. These were hardened by chilling, so as better to resist the action of 



passing wheels; but even with this precaution they wore out so much more rapidly 
than the rails, that the wings,wm and i c, and the tong ue, P, were capped 
with steel from inch to 1 inch thick, bolted or riveted to their upper surfaces. 


































TURNOUTS. 


781 


>8 

ft 




h* triangle, P, called the tongue of the frog, is the meeting-point of the two rails 
“ f* and/x, Fig 1; while the wings, w in and i c,are continuations of the rails e and r. 

The wings give support to thetreads of tlie wheels in passingover the spaces between 
'lie c “ ( e Point and w and t, which spaces are left for the passage of the flanges. 

The channel is called the mouth of the frog at a, Figs 22 and 23; and its 
tliroat at the narrowest part, w i. That part of the tongue back of u, Fig 23 or 
between u and g, is called its heel. 

The channel is made about 2 ms deep to prevent the flanges from touching its bottom. 
The projections, 1t, Fig 22, are for bolting the frog to the wooden cross ties. 
Although one side of the frog forms a part of the turnout curve, its shortness war¬ 
rants us in making both sides,# o, s t, Fig 23, straight. 

Art. 24. Guide-rails, or guard-rails, g g', Fig 1. Suppose wheels to be 
rolling from A toward B, Fig 1, on the main track; the switch-rails being in the 
dotted positions. On arriving opposite the frog, some irregularity of motion might 
cause the flanges of the wheels running along the rail, n, to press laterally against 
said rail. Consequently, after passing the throat, w i, Fig 22, they would press against 
the wing, tc; and passing between c and P, they would leave'the track; or strike 
the sharp end of P, breaking it, and endangering the train. To prevent this, the 
guard-rail, g , Fig 1, is placed so near the rail, b li (say \% to 2 ins from it), that the 
flanges at 6 h, while passing between it and g, prevent those at the opposite rail 
from pressing against the wing, tc, Fig 22, and from striking the point; and guide 
them safely along their proper channel, im. Similarly, if wheels be rolling from 
A toward D, Fig 1 (the switch-rails being in the positions, qm,ps), the centrifugal 
force due to the curve would cause the flanges to press against the rail, ex, and 
against the wing, wm, Fig 22, thus rendering the train liable to the same kind of 
accident as in the preceding case. This is prevented, in the same manner as before, 
by the guard-rail,#', Fig 1, which keeps the flanges in their proper channel, tec, 
Fig 22. 

The narrow flange-way between the guard-rail,#, Fig 1, and the rail, bh, 
should extend at least a foot each way from a point directly opposite the 
point,/, Fig 23, of the frog. In a distance of at least about 2 ft more at each of its 
ends the guard-rail should flare out to about 3 ins from the rail, b h, so as to guide 
the flanges into the narrow channel. The same with #'. 

Guard-rails have to resist a strong side pressure, and should 
be very firmly secured to the wooden cross-ties. This is usually done by bolting 
against them two or more stout blocks of wood, or of cast-iron, which, in turn, are 
bolted to the ties. 

Art. 25. The cast-iron frog, as first made, had no provision for 
fastening? it to the rails; but was simply bolted to the cross-ties. It was 
afterwards provided with a recess at each end, of the exact shape and size of the end 
of the rail. The rail ends were inserted into these recesses, and the frog was thus 
kept in line with the rail. In frogs made of rails, the same purpose is served by fish- 
or angle-plates, p 764, by which the ends of the frog are secured to those of the rails. 

Art. 26. The length, a g, Fig 23, of a cast-iron frog, usually varies from 
4 to 8 ft; and depends upon the angle, oft , at which the rails, e x and r z, Figs 1 and 
23, cross each other. This is called the frog-angle. This angle may be expressed 
either in degs and mins, or in the number of times the width of the tongue on any 
line, as o t. Fig 23, is contained in the distance, gf, from the point,/, to the center, 
g, of that line. This number is called the frog number. Thus, if the angle, 
oft, Fig 23, is such that the length, gf, is 3, 4, or 10, &c, times the width, ot, the 
frog is called a No 3, 4, or 10, &c, frog. Fig 23 is a No 3; Fig 22, No 5. Frogs are 
usually made of Nos 4 to 12; sometimes with half numbers, as 7%, 8%, &c. 

Art. 27. Draw two parallel lines, b b\ dd', for the top of rail, ex, Fig 23, and 
h h', k k', for that of rail, r z; crossing each other at the required angle. Then the 
intersection,/, of lines dd ' and hh' is the theoretical point of tlte frog. As 
this point would be too narrow and weak for service, it is in practice rounded off 
where the tongue is about 34 i*>eh wide,as shown. If the frog is to be simply abutted 
to the rail-ends, zx, Fig 23, as in some cast-iron frogs, the length,/#, need be only 
great enough to give a width, to, sufficient to accommodate the rail-ends, z and x, 
and the heads of the two spikes at v which confine them to the ties. If desired, a 
portion of the flange of each rail-end may be cut away 60 that the rail-heads come 
together; thus diminishing the width necessary for to, and, of course, the distance, 
/#. In the case of cast-iron frogs provided with recesses for holding the rail-ends, 
as in Art 25, the width, ot, and length,/#, must of course be greater. 

In frogs made of rails, the length must be such that the rail-ends, 
z and x. Fig 23, are far enough apart to give room for fitting to their inner sides the 
splice-plates by which they are connected with the frog. Where angle- plates are 
used, this distance must be greater than in the case of/s/t-plates. 





782 


TURNOUTS 






















































TURNOUTS. 


783 


nto recesses let into the sides of the throat-pieces and blocks. Tlie clamps 
ire prevented from slid in g by clips riveted on the flanges of the rails, 
treat care is taken to have all the adjoining’ surfaces in full contact 
fith eaeli other, throughout, so as to diminish the liability to wear. 

Art. S3. The Co also make stiff frogs in which the parts are held together by 
>olts passing through the rails and the throat-pieces; and others in which there 
re no throat-pieces, and in which the four rails forming the trog are riveted to a 
'rought-iron plate. (A spring-rail frog, of the plate pattern, is shown in Fig 27, 
.784.) All of these frogs can be made of any desired length and to any desired 
ngle. Tlie standard length of stiff frogs from be to dz, for Nos 4 to 8, 
tclusive, is 8 ft; Nos 9 and 10, 9 ft; Nos 1L and 12, 10 ft; No 15, 12 ft. Prices, 
886 , S 1 8 to $25, according to weight of rails and the length of frog. 

Art. 34. Figs 26 A and 26 B show the Mansfield patent elastic frog, 
1 lade by the Mansfield Elastic Frog Co, New Haven, Conn. These frogs have been 


rr xl 




Ol 



-I—■— u 

—" ’ Fig. 26 A^-* 



Fi g.26B 


argely used, with good results. Fig 26 B is a vert longitudinal section through the 
enter of the frog. Elasticity is given by two creosoted white-oak planks, 
) 0. Formerly india-rubber was used in place of these. The lower plank is 1]A 
ns thick ; the upper one about 2 ins, depending upon the ht of the rail used. Ill 
ire three rolled-iron plates, from % to y 2 inch thick. The point P, tongue T, and 
vings W W, are of hammered cast-steel; and their tops are ins above the sur- 
ace of the upper iron plate, to allow the wheel flanges to clear the latter. These 
various parts are fastened together by countersunk bolts of Ulster iron, as shown, 
it will be noticed that the point is secured laterally by being let into the upper iron 
date and upper oak plank, so as to rest on the second iron plate. The ends of the 
rog are notched, as shown at N N, Fig 26 A, to receive the webs and flanges of the 
rack-rail. Price, 1886, $40 for medium size. The Mansfield Co make also an 
ilastic steel-rail frog,spring-rail frogs, crossing frogs, safety switches, etc. 

Art. 35. The object in reducing the w idth of the channels 
,, r some distance each way from the point, f, Fig 23, so as barely to admit the 
flanges freely, is to allow the treads of the wheels to have as much bearing as pos¬ 
sible upon the wings and tongue while moving over the broadest part of the chan¬ 
nel near f. In frogs shorter than about No 4, it is difficult to secure sufficient 
bearim’- for the treads, even with the utmost allowable contraction of the channel, 
when the width of the tires is, as usual, about 6 ins. In the earliest frogs this diffi¬ 
culty was partially overcome by gradually raising the bottom of the 
Channel between the point and wings, so that the wheels, in traversing that part, 
ran upon their flanges instead of upon their treads. The jar occasioned by the 
tread in striking against wing and point, was thus avoided. This arrangement is 



depths 
























































XTV 


784 


TURNOUTS, 
















































TURNOUTS. 


785 


Art. 37. Ill Wood's self-acting' frog, vhich was largely and success¬ 
fully used for many years on the Camden & Amboy R R, under heavy trains moving 
at high speeds, and which, like those just described, was made entirely of rails; the 
point,/, Fig 28, was firmly fixed to the cross-ties;and both ot the wing-rails, t and s, 
while rigidly fixed at a proper distance from each other, were free to move, as a 
whole, about their heels, b and c, so that either t or s could be brought into contact 
with the point,/". It thus provided an unbroken bearing to trains on the turnout, 
as well as to those on the maiu track. There was no spriug; and the frog remained 
as it was set by the first pair of wheels of a train, until another train, using the 
other track, set it over the other way. 

If this frog is set as in Fig 28, and an engine moves from e toward x (so that the 
wings are moved by means of the guard-rail, g', and the wheel at the opposite track, 
ex), the motion of the wings has to take place while the weight of the engine itself 
is resting partly on the wing-rail, et\ and, consequently, their sliding transversely 
of the track is attended with great friction. The same effect is produced if t is in 
contact with f. and an engine moves from A to B. This, however, did not, in prac¬ 
tice, seem to interfere with the efficiency of the frog. 

Art. 38. Ill ordering frogs, give the frog angle or number, and the exact 
cross-section of the rail used on the road. 

For spring-rail frogs*, specify also whether the turnout is to the 
i right or left hand. In the case of stiff frogs, this is not necessary. 


Art. 39. Tlie laying-out of Turnouts. 

The words heel and toe are used in this article with reference, to the common or stub 
switch, Fig 4, in which the heels are at q and p, and the toes at m and s. In the 
Wharton, Lorenz, and some other switches, the positions of heel and toe will be seen 
to be the reverse of this. 

The formulas given in our early editions for finding frog dist, rad of turnout, Ac, 
were based upon the old practice of regarding the straight switch-rails, qm,p s, Fig 
4 ,as forming a tangent to the turnout curve, which last was considered as begiunmg 
at the toes, m and s, of the switch-rails. The modem practice is to curve 
the switch-rails so as to form a part of the turnout curve; the latter being supposed 
to begin at the heels, q and p, of the switch. This view of the case admits of simpler 

formulas. . , _ 

In each of the Figs 29. 30, and 31, wp z represents the main track. Ilie trog 
distance, pf is a straight line drawn from the theoretical point offrog, f, to the 
heel, p, of that switch-rail which, when opened, forms the inner rail of the turnout. 
Formerly, when the turnout curve was taken as starting at the toe. of the switch, the 
frog dist was a straight line from the theoretical point of frog to the toe, m, l'ig 4, 
of the outer switch-rail, q m, when opened. 

As already remarked, frogs are usually made of Nos 4 to 12; sometimes with halt 
numbers; and the turnout radii, &c, are made to conform to them. 

Scrupulous accuracy is not necessary in these masters. Thus, a deviation, either 
way of say 3 per cent in the length of the turnout radius from that given by the 
table or the formulas, will be almost inappreciable. So too, if a frog number should 
be used, intermediate of those in the first column of the table, the other dimensions 
may be found, approximately enough, by using quantities similarly intermediate. 
A rail almost always has to be cut in two in order to fill up the frog dist; and the 
exact length of the piece can be found by actual measurement at the time ot cutting it. 

Rem. When the turnout leaves a straight track, as in Fig 29, the frog angle 
is equal to the central angle./c o. When the main track is curved, and the turnout 
curves in the opposite direction (Fig30), it is equal to the sum (v/w) ot the 
central angles, fco,fnn\ and when the two curve in flic same direction (lug 
31), it is equal to the d iff ( nfc) of the central angles, fc o,fn o. 

51 




786 


TURNOUTS. 


Art. 40. To lay out a turnout, p x, Fig-29, from a straight 
track, p g. From' the column of radii in the table below, select one, co, suit¬ 
able for the turnout; together with the correspond¬ 
ing frog number, frog dist ,p / and switch length. 
Place the frog so that the main-line side of its 
tongue shall be at / z, precisely in line with the 
inner edge of the rail, wz, and its theoretical 
point, /, at the tabular frog dist, pf, from the 
starting-point, p. Stretch a string from q (oppo¬ 
site p) to/; and from it lay olf the three ordinates 
from the table; thus finding three points (in addi¬ 
tion to q and /) in the outer curve. Do not, how¬ 
ever, drive stakes at these points; but as each of 
them is found, measure off from it, inward, half 
the gauge of the track; and there drive stakes. 
Do the same front q and/. The five stakes will all 
then be in the dotted center line of the turnout, Fig 29; and will serve as guides to 
the work, without being liable to be displaced. The dimensions in the table below 
are found by tlie following formulas, the inaiu track being straight: 



Fi K . 20 


C' 


Tangent of 

half frog angle 

Frog No. 

Or, Frog No. 

Radius c o. 

Or, Radius c o. 

Or, Radius c o. 

Frog dist p f. 

Or, Frog dist p f.. 

Or, Frog dist pf.. 

Middle ord. 

Eacli side ord... 


= Gauge -t- Frog dist. 

= y Radius co -=r Twice the gauge. 

= Half the cotangent of half the frog angle. 

= Tw ice the gauge X Square of frog number. 

= (Frog dist pf-i- Sine of frog angle) — half the gauge. 
= (Gauge -r- Versed sine of frog angle) — half the gauge. 
= Frog number X Twice the gauge. 

: Gauge p q —■ Tangent of half the frog angle. 

= i Rad co + half the gauge) X Sine of frog angle. 

= x /i g al, ge. approx enough. 

= % mid ord = (or .188 of the) gauge, approx enough. 


Switch Length = 

approx enough 



Throw in ft X 10000 

Tangential dist for chords of 100 ft, for rad co of 
turnout curve. See table, p 72G to 728. 


TABLE OF TURNOUTS FROM A STRAIGHT TRACK. Fig 29. 

Gauge 4 ft 8 Yz ins. Throw of switch 5 ins. 

For any oilier gauge, the frog angle for any given frog number remains 
the same as in the table. The other items may be taken, approx enough, to vary di¬ 
rectly as the gauge. 


Frog 

umber 

Frog 

Angle 

Turnout 
Kudius 
e o 

Poll Aug of 
Turnout 
Curve 

Frog 

hist 

Pf 

Middle 

Ordinate 

Side 

Ords 

Stub 

Switch 

Length 


o r 

Feet. 

o / 

Feet. 

Feet. 

Feet. 

Feet. 

12 

4 46 

1356 

4 14 

113.0 

1.177 

.883 

34 

U 'A 

4 58 

1245 

4 36 

108.3 

1.177 

.883 

32 

11 

5 12 

1139 

5 2 

103.6 

1.177 

.883 

31 


5 28 

1038 

5 31 

98.9 

1.177 

.883 

29 

10 

5 44 

912 

6 5 

94.2 

1.177 

.8S3 

28 

9'A 

6 2 

830 

6 45 

89.5 

1.177 

.883 

27 

9 

6 22 

763 

7 31 

84.7 

1.177 

.883 

25 

z'A 

6 44 

6 S0 

8 26 

80.0 

1.177 

.883 

24 

8 

7 10 

603 

9 31 

75.3 

1.177 

.883 

22 

7^ 

7 88 

530 

10 50 

70.6 

1.177 

.883 

21 

7 

8 10 

461 

12 27 

65.9 

1.177 

.883 

20 

G 'A 

8 48 

398 

14 26 

61.2 

1.177 

.883 

18 

6 

9 32 

339 

16 58 

56.5 

1.177 

.883 

17 

5 14 

10 24 

285 

20 13 

51.8 

1.177 

.883 

15 

5 

11 26 

235 

24 32 

47.1 

1.177 

.883 

14 

4^ 

12 40 

191 

30 24 

42.4 

1.177 

.883 

13 

4 

14 14 

151 

38 46 

37.7 

1.177 

.883 

11 


Remark. The switch lengths in the Table merely denote the 
shortest length of fttnb switch that will at the same time form part of the turn¬ 
out curve, and give 5 ins throw. Pointed, or split-rail switches, like the 
Lorenz, &c, require only half this throw. In practice all kinds are frequently made 
much shorter than the table requires, thereby sharpening the beginning of the curve. 













































TURNOUTS, 


787 


Having de 
table, p 726, 


Art. 41. To lay out a turnout from a curved main track. 

There are two Cases: 

Case 1, Fig 30; when the two curves deflect in opposite directions. 

Case 2. Fig 31; when the two curves deflect in the same direction. 

ermined approx up6u a radius for the turnout curve, take from the 
ts corresponding deflection angle, and that for the main curve. In 
Case 1, find the sum of these two angles. In Case 2, 
find their difference. In the table, p 786, find the 
deflection angle (not the frog angle) nearest to the 
sum or diff just found. The frog number, 
switch length and frog distance pf in 
the table, opposite the deflection angle thus se¬ 
lected, are the proper ones for the turnout. 
Theoretically we should, in Case 1, add to the tab- 



f Fig-30 
Case I 



ni'' 


Fig. 31 
Case 2 


ular frog distance pf about half an inch per 100 feet for each degree of deflec- 
! tion angle of the easier of the two curves; and, in Case 2, deduct it; but this 
refinement is unnecessary. 

Deflection angle of turnout curve 
(in Case 1) = Tabular deflection angle (p786)— deflection angle of main curve, 
(in Case 2) = Tabular deflection angle tp 786) + deflection angle of main curve. 
For radius co of turnout curve, having its deflection angle, see 
table, p 726. 

Ex. Rad of main curve, 2865 ft. Rad selected approx for turnout, 716.8 ft. 
Here the defl angles (table, p 726) are, respectively, 2° and 8°. 

In Case 1; 8 0 + 2° = 10°. Nearest defl angles in table, p 786, 10° 50' and 9° 31'. 
If we select 10° 50', we have Frog No, 7}^; Switch L’gth, 21 ft; Frog Hist = 70.6 
ft + twice % in = say 70.7 ft; Defl Angle for turnout = 10° 50' — 2° = 8° 50'; Rad 
co (table, p 726), 649 ft. If we select 9° 31', we have Frog No, 8; Switch L’gth, 
22 ft: Frog Dist = 75.3 ft -f twice in = say 75.4 It; Defl Angle = 9° 31' — 2° = 
7° 31'; Rad c o, say 763 ft. 

In Case 2; 8° — 2° = 6°. Tabular defl angles, p 786, 6° 5' and 5° 31'. 6° 5' 
I gives Frog No, 10; Switch L’gth, 28 ft; Frog Dist = 94.2 ft — twice % in = say 94.1 
ft; Defl Angle = 6° 5' -f 2° = 8° 5'; Rad, say 709 ft. 5° 31' gives Frog No, 10^; 

I Switch L’gth, 29 ft; Frog Dist = 98.9 ft — twice % in = say 98.8 ft; Defl Angle = 
5° 31' + 2° = 7° 31'; Rad, say 763 ft. 

The frog; dist p f may also be found thus: 


Tangent of half fno — 


Gauge X Frog No 
Radius n o. 


Frog Dist P / = np X twice the sine of half fno. 

Place the frog with the main-line side of its tongue at fz, in line with the inner 
edge of tiie frog-rail, p z, of the main line, and with its theoretical point,/, at the 
dist. pf (found as above), from the heel, p, of the inner switch-rail. Stretch a string 
from/ to the heel, q , of the outer switch-rail. Measure the dist, qf, divide it into 
four equal parts, and lay off three ordinates, found thus: 

Middle ord = (Square of half qf) -s- twice the rad of turnout curve. Each side 
ord = three-fourths of middle ord. 

These three ords. and the points q and /, give us 5 points of the outer rail of the 
turnout curve; and from these we measure, inward, half the gauge,and drive 5 cen¬ 
ter guide-stakes, as in Art 40. 










788 


TURNOUTS. 


Art. 42. To find frog diets. Ac, by means of a drawing- to 
scale. The frog dint can generally be found near enough for practice, irom a ( 
drawing on a scale of about or y# inch to a foot. Aixl so in the many cases where p 
turnouts cross tracks in various directions, in and about stations, depots, Ac. 

Tigs 32 and 33 are intended merely to furnish a few general hints in regard to f 


A 


such drawings. For instance, the curves of a main track, as well as those of a turn¬ 
out, generally have radii too large to admit of being drawn on a scale of ^ it'd 1 to 
a foot, by a pair of dividers or compasses, hut they may be managed thus: Draw 
any straight line, ab , Fig 32, to represent by scale a 100-ft chord of the curve, divide 
it into twenty 5-ft parts, a 1, 1 2, 2 3, Ac, ami lay off by scale the 19 corresponding 
ordinates, 1 1, 2 2,3 3, Ac, taken from the table on page 730. By joining the ends 
of these, we obtain the reqd curve, uc b, of the main track ; and of course cau draw 


the inner line, y t. distant from it by scale the width of track, say 4 ft 8^ ins. Now 
let acb and y t , Fig 33, be a curved main track so drawn ; and let any point m be 
taken as the starting-point of the turnout, m v, Ac. On each side of m measure off 
any two equidistant points, n and n, in the same curve; and through m draw sg, 
parallel town. Then is ra^atang to the curve, y m f, at m. Having determined 
on the rad of the turnout curve, m v e, draw that curve by the same process as before; 
first laying off the angle, g m i, equal to the tangential angle of the curve, taken 
from the table, p 726. Then, beginning at m, lay off 5-feet dists along m t; and from 
them, as in Fig 32, draw the ords corresponding to the turnout curve. Through the 
ends of these ords draw the curve, mve. itself. Then the frog dist will be the 
straight dist from c to v, and can tie measured by the scale, within a few inches; or 
near enough for practice. The middle ord of the arc, m v, cannot be found correctly 
by so small a scale as ^ inch to a foot, but should be calculated thus: From the 
square of the rad take the square of half the chord, in v. Take the sq rt of the rein. 
Subtract this sq rt from the rad. If two other ords should be desired, half way be¬ 
tween m and v and the center one, they may each be taken as % of the center one. 
Make the switch-rail long enough to leave 2% ins at its toe between in n and m w. 

The frog angle at v will be equal to the angle, rvd, formed between the tang, vr, 
to the curve, acb; and the tang, v d, to the curve, mve. These tangs are found in 
the same way as mg; namely,for the tang, v r, lay off from v two equidistant points, 
h and h, on the curve, acb; and through v draw v r parallel to h h. Also, for v d , 
lay off from v any equidistant points, u and u, on the curve, mve, and through v 
draw vd parallel to them. This angle may be measured by a protractor. Or, if on 
the two tangs we make v 4 and v 4 equal to each other, and draw the dotted line 4 4; 
and from its center at 6 draw 6 v ; then 6 v divided by 4 4 will give the No of the frog. 
With care, and a little ingenuity, the young student will be able, by similar proc¬ 
esses, to solve graphically any turnout case that may present itself. The method 
by a drawing has great advantages over the tedious and complicated calculations 
which otherwise become necessary in cases where curved and straight tracks inter¬ 
sect each other in various directions. The drawing serves as a check against serious 
errors, which would lie detected at once by eye. None of the graphical measure¬ 
ments will be strictly accurate; but with care, none of the errors need be of prac¬ 
tical importance. The ordinates for bending rails so as to suit turnout curves can 
be found from the table, p 761. All of Art 42 may be done on the ground. 





















TURNOUTS. 


789 


Art. 43. An experienced track-layer, with a good eye. can place his own guide- 
itakes by trial on the ground; and by them lay his turnouts with an accuracy as 
practically useful as the most scrupulous calculations of the engineer can secure. 

The following example, Fig 34, of a turnout from a straight track, Y Z, exhibits a 
iommon case, in which all the w'ork may be performed on the ground, without pre¬ 



vious calculation. Let ivo be the tongue of a frog, with which the assistant has 
jeen directed to make a turnout from Y Z; and that he has received no instructions 
more than that the turnout must start at d, and terminate in a track, W, to be laid 
parallel to Y Z, and distant from it r x or r x, equal to 6 ft. 

Place the tongue of the frog by guess near where it must come, having its edge, 
b t, precisely in line with the inner or flange edge of the rail, b r. Then stretch 
a piece of twine along the edge, ov, of the frog, and extending to dg. Try 
by measure whether ve is then equal to ed; and if it is not, move the frog along 
the line, br, until those two dists become equal. Then is v the proper place for the 
point of the frog; b v is the frog dist; one-half of c e is the length of the middle ord 
of the turnout curve, dv, and if two intermediate ords are needed at s and s, each 
Of them will be % of said middle one. 

The frog being now placed, proceed thus : Place two stakes and tacks, x and x, at 
the reqd inter-track dist, rx and rx, of 6 ft from the rails, br. Then range by 
pieces of twine xx and v f, to find the point, n, of intersection. Then measure nv, 
ind make n in equal to it. Then is m the end of the reverse curve, v m, of the turn¬ 
out. The ords of this curve may be found as before; one-half of nk being the 
middle one, Ac. 

Rem. 11 may frequently be of use to remember that in any arc, as v m, of a circle; 
v n and m n being tangs from the ends of the arc; one-half of the dist, k n, is the 
middle ord, kz, of the curve; near enough for most practical purposes, whenever the 
length of the chord , v m, of the arc is not greater than one-half the rad of the circle 
of which the arc is a part. Or, w ithin the same limit, vice versa, if we make k n equal 
to twice kz , then will n be very approximately the point at which two tangs from 
the ends of the arc will meet. Also, the middle ord of the half arc, vz or zm, may 
be taken as ^ of the middle ord, k z, of the whole arc. 










Fig. 


790 


TURNTABLES, 



BY WILLIAM SELLERS & CO., PUILADA 

















































































TURNTABLES. 


791 


TURNTABLES. 


Art. 1. A turntable is a platform, usually from 40 to 60 ft long and 
about 6 to 10 ft wide (see Fig 1,) upon which a locomotive and its tender may 
be run, and then be turned around hor through any portion of a circle; and thus be 
transferred from one track to another forming any angle with it. The table is sup¬ 
ported by a pivot under its center; and by wheels or rollers under its two ends. 
Frequently other rollers are added between the center and ends. Beneath the plat¬ 
form is excavated a circular pit about 4 or 5 ft deep, having its circumf lined with 
a wall of masoury or brick about 2 ft thick, capped with either cut stone or wood. 
The diam of the pit in clear of this lining is about 2 ins greater than the length of 
the turntable. The lining is generally built with a step, as seen in Fig 1, tor sup¬ 
porting the circular rail on which the end rollers travel; or, instead of tins step, a 
detached support may be used for this circular rail, as at ti, Fig 11. At the center 
of the pit is a solid well-founded mass of masonry or timber, for the pivot to rest 
on, as seen in Fig 1. This, as well as the step for the end rollers, should be very 
firm, and perfectly level; otherwise the platform will be hard to work. The plat¬ 
form is frequently floored across for a width of 6 to 10 ft to furnish a pathway 
across the pit, without stepping down into it; especially when under cover of a 
building. At first they were floored over so as to cover the entire circular pit; but 
this increased not only their cost but their wt, so as to make them difficult to turn; 
besides causing much expense for repairs; with greater trouble in making them. 
It is therefore rarely done at present, except where want of space sometimes ren¬ 
ders it necessary in indoor turntables. 

For the minimum length of a tnrntable, add from 1% to 2 ft to 

the total wheel base (p 805) of the longest locomotive and tender for which it is to 
be used; but a turntable should be several feet longer than is necessary for merely 
allowing the engine and tender to stand on it; for the increased length enables the 
engine-men to move them a little backward or forward, so as to balance them chiefly 
upon the central support; and thus relieve the end rollers. By this means the fric¬ 
tion while turning is confined as much as possible to the center of motion; and is there- 































































































































































































































































































































































































































































792 


TURNTABLES. 


fore more readily overcome than if it were allowed to act at tlie circumf. The 
engine-men soon learn, by feeling, the proper spot for stopping the engine so as thus 

to balance the platform. . ^ . . , . w 

Art. 2. Figs 1 to 6 represent the Seller* cast-iron turntable of W m 
Sellers & Co., Pliila. It consists of two cast-iron girders of about ins average 
thickness, perforated by circular openings to save metal. One ot these girders is 
shown in Fig 1; and parts of one in Figs 3 and 4. Each girder is in two separate 
pieces, which are fastened, as shown in Figs 1, 3, and 4, to a hollow cast-iron cen¬ 
ter-box,” A B, Figs 2, 3, 4, and 6, by means of 2% inch screw-bolts, at/, Fig 3; and 
by hor bars, o o, of rolled iron about 3% ins square, fitting into sunk recesses on 
top of the boxing, and tightened in place by wedges, i i, screw-bolted beneath. 

Art. 11. The sides of the center-box are about ins thick. It is sus¬ 
pended from the steel Clip. C, by 8 screw-bolts 2 ins diam. On its lower side 
this cap has a semi-cylindrical groove extending across it, transversely of the 
track, as shown in Fig 6. This groove fits over a corresponding semi-cylindrical 
ridge on the top of the cast-iron socket,” s (so called), on which the cap thus 
rests. The socket, in turn, rests upon the upper one, u, of two annular steel plates, 
u and v, which form a circular box containing 15 steel conical anti¬ 
friction rollers, d d. Figs 2, 5, and 6. These are about 3 ins in length, and 
in greatest diam. They have no axles, but merely lie loosely in the lower part, v, 
of the circular box; filling its circumf with the exception of about % inch left for 
play. In the direction of their axes they have inch play. The lid, u, of the cir¬ 
cular box, rests upon the tops of the rollers, which separate it from t) by about ^ 
inch, v rests upon the top of the hollow cast-iron post, P, which, by means 
of its flaDges, is bolted to the cap-stone, M, of the foundation pier. 

In order to insure a perfect bearing of the revolving surfaces upon each other, 
and thus diminish the liability to abrasion, the rollers, d d, and the insides of the 
box in contact with them, are accurately finished, as are also the top and bottom of 
the roller-box, and the surfaces of the socket, s, and post, P. in contact with them. 
The rollers are oiled by means of the spaces shown by the arrows in Fig 2. 

Art. 4. Adjustment of tlie height of tlie table. By turning the 
nuts, N N, of the 8 screw-bolts which support the table, the latter may be raised or 
lowered 1 or 2 ins; the cap, C, socket, s, and roller-box, u v,” remaining stationary 
on top of the post, P. All turntables should have tlie means of making such 
adjustment. Before the nuts, N N, are finally tightened up, the blocks, w w, 
of bard wood, cut to the proper thickness for the desired lit of the table, are 
inserted between the cap, C, and the top of the center-box, A B, as shown. 

The ht of I be tabic should be such that each of the wheels at its outer 
ends shall be l /± inch, in the clear, from the circular rail on which the}' travel. 

Art. 5. At each of the outer ends of the table, the two girders are connected 
transversely by heavy cast-iron beams, called “ cross-girts.” These project 
beyond the girders, and carry the cast-iron end-wlieels, 20 ins diam, 2 at each 
end of the platform. The treads of these wheels are but about 3 or 4 ins below the 
bottoms of the girders, and the wheels therefore do not require any considerable 
depth of pit for their accommodation. In order that they may roll freely, their 
treads are coned, and their axles are made radial to the circular turntable pit. In¬ 
termediate transverse connection between the main girders, is secured by the 
woollen cross-ties notched upon them to support the rails, and frequently 10 
or 12 it long, for giving a wide footway across the pit. A lever, 8 or 10 ft long, 
fitting into a staple, is used for turning the table, not on account of friction, but as 
a handle for the workmen. 

Art. <». The semi-cylindrical shape of the joint between the cap, C, and the 
socket, s, permits the slight longitudinal rocking motion of the turntable which 
takes place when a locomotive comes upon it or leaves it; but prevents it from tip¬ 
ping sideways, as it was apt to do, when, as formerly, the cap rested directly upon 
the lid u of the roller-box, and the top of the post P was hemispherical, forming a 
ball-and-socket joint with a casting upon which the roller-box rested. 

Art, 7. Ten sizes of these turntables are furnished for locomotive use. 
The diameters are as follows: 70 feet, 65 feet, 60 feet, 56 feet, 54 feet,50 feet (two 
patterns), 45 feet, 40 feet, 30 feet and 12 feet. For prices, weights, dimensions of pit 
and foundations, timber required, etc; address William Sellers & Co, 1600 Hamilton 
St., Philadelphia. Machinery for turning, being considered unnecessary, is not at¬ 
tached, unless specially ordered. Its cost is extra. 

The entire cost of excavating and lining the pit; foundation for pivot; circular 
rail for end rollers, &c, complete for a 56-ft turntable will vary from $1200 to $2500 
in addition,'depending on the class of materials and workmanship; and whether the 
bottom of the pit is paved or not. 

Art, 8 . Tlie girders of turntables are now very generally made of 




TURNTABLES. 


793 


bt 

in 

!» 

J 

IS 

■to 

n- 

id 


dled-iron phttcs, each girder being in one piece; and the two main girders 
s then connected with each other, near their centers, by two cross-jsirders 
I or channel beams, or of plate-iron, one on each side of the central pivot. These 
>ss-girders, and the sides of the main girders between them, form a sort ofrectan- 
lar box, corresponding to the cast-iron center-box of the Sellers table, Arts 2 and 
The arrangement of the center bearing apparatus, and the manner in which it 
connected with the cross-girders, differ in the tables of the several makers. 

Art. 9. In the plate-iron turntables made by the Edge Moor Iron Co, Wil- 
tigton, Del, the cross-girders, G G, Fig 8, are of plate-iron ; and, at their ends, have 
nges. F F, of angle-iron, by which they are riveted to the main-girders, E. In the 
-ft tables the cross-gii'ders are 26 ins apart in the clear. 



r Enlarged transverse section of roller- 
box &c, on center line of Fig 8. 



f Art. 10. The bolts, II, by which the table is suspended from the cap, C, are 
in number, and are arranged in two rows of three bolts each, one row being on 
r ;h side of the post, P, and between it and one end of the turntable. To each cross- 
der is riveted a short hor bar of angle-iron, J J. Through the hor flange of each 
1 these two bars pass the 3 bolts, II, of one row. The hor flange, bearing all the 
1 of its girder and load, rests upon the heads of these 3 bolts. To prevent the 
1 n«-e from yielding under its load, vert struts of angle-iron are riveted to the web 
1 tl7e cross-girder between the bolts. Two of these struts are shown at K K, Fig 7. 

1 ieir ends abut against the upper flange of the cross-girder, and against J J. 

The 6' bolts, II H. pass up through the flanges of the cap, C (three bolts through 
;-h flange), and their nuts rest upon its top. 

Art. 11. The cap, C, is held in place on the socket, s, by means of 
nges which extend down from it on both sides, as shown in Fig 7. 

The rollers, dd, and the roller-box, uv , are those made by Wm. Sellers & Co, 
lila. Art 3. 

The ht of the table may be adjusted, within a range of 2 or 3 ins, by 
leans of the nuts, N N, as in the Sellers table, Art 4. 

The lower part, v, of the roller-box, instead of resting directly upon the post, P, 
in the Sellers table. Art 3. rests upon an iron casting, L, which, in turn, rests upon 
e post, P, and is held in place upon it by a lug which projects down into it. 

The post is built up of plate- and angle-irons riveted together, and may be a hol- 
w truncated square pyramid, as shown, or of other shapes. Those presenting the 
ape of a cross in hor section have the advantage of being accessible for painting 

id inspection. _ , , , 

Art. 12. The main girders are braced by hor diag rods, whose ends 
e shown at Q, Fig 8. These rods are provided with sleeve-swivel turn-buckles, by 

hich they may be tightened or loosened. 

The remarks in Art 5 on the end wheels of the Sellers table apply also to 
ose of the Edge Moor, except that in the latter the wheels are 25 ins in diarn, and 
o cross-girts which carry them are of angle-irou. The wheels are fastened to their 
les which turn in brass bearings enclosed in cast-iron journal-boxes. Each box 
provided with means for raising or lowering it; and the brasses are rounded on 
p so ^ to insure a uniform bearing, no matter what inclination may be given to 
le’axle by the vert adjustment of the journal-boxes. The brasses may be readily 

















































































794 


TURNTABLES. 



replaced when worn out. The lower part of the journal-box is filled with oil .an 
waste for lubrication. The several parts are made large, in order to lessen the fri' 
tion per square iuch, and to withstand the shocks to which they are subjected. 

The Edge Moor tables are made 50, 54, 50, 
and fiO ft long, and are designed to turn the 
heaviest “ Consolidation ” locomotive and ten¬ 
der without subjecting any part of the iron 
work of the table to a greater tensile or com¬ 
pressive strain than 10000 lbs per square inch. 

Art. 13. The approximate weights 
»n<l prices of Edge floor turn¬ 
tables are given on the accompanying table. 


Length. 

Finished wt. 

Price, 1886, 
on cars at 
Edge Moor. 

50 ft 

54 ft 

56 ft 

60 ft 

1S900 lbs 
22600 lbs 
24300 lbs 
26400 lbs 

$1100 

$1450 

$1500 

$1600 


Average cost, 1886, say 6 cts per lb. 

Art. 14. Fig 9 shows the center bearing arrangement of the turntable paten 
and furnished by Mr. C. O. If. Fritzsche, 71 Broadway, New York. The 1 
side of the fig is in section; the right side is in elevation. 


Here each cross-girder, H, is a heavy rolled-iron I beam, to each end of 
which are riveted flanges, F, of angle-iron. These flanges are bolted to the webs of 
the main girders, E, in order that iron “ adjusting-plates ” of any requim 
thickness may be inserted between either angle-flange and the web of the main 
girder which it supports. This permits a transverse adjustment of the table, so that 
the center of gravity of its transverse cross-section may be brought immediately 
over the vert axis of the post, P. 

Art. 15. Four rolled-iron plates, of which two (ora pair) are shown at Z Z, 
extend from one cross-girder to the other, and are fastened to the webs of the latter 
by angle-pieces, Y Y. The lower edges of these plates rest upon rolled-iron washers, 
X. Each washer supports two of the four plates, Z Z. and rests, in turn, upon one 
of the nuts, NN, of the single inverted U-bolt. II. The upper part of this U-bolt 
rests in a cast-iron saddle, S; the neck, V, of which enters the top of the cast-iron 
post, P, and rests upon the upper one of two steel discs, d, which, finally, rest on the 
bottom of the cylindrical cavity in the top of the post. The object in the 
use of the single IT-bolt is to reduce, as far as possible, the number of 
points of support, and thus to reduce, also, the uncertainty as to the proportion of 
the total load sustained by each. Its shape, and the manner in which it rests in its 
saddle, S, render the table a rigid mass revolving on the center post. 














































































































































TURNTABLES. 


795 


J Art. 16. Oil is applied through openings in the side of the saddle-casting, 
5. These openings communicate with a vert hole through its center, and thus with 
i similar hole through each disc. The two faces of the discs in contact with each 
" »ther, form a segment of a sphere ; and each face has three radial gutters, extending 
rom its cen to its circumf. Channels are cut in the sides, and around the lower 
“ dge, of the cylindrical cavity in the top of the post. Thus all the parts which 
evolve upon one another can be kept bathed in oil. 

Art. 17. Each leg of the U-bolt passes through, and is held in place by. a cast- 
ron box, B. These boxes are held in position by the edges of the four inner 
mgles (corresponding to Y Y, but not shown in the fig) between the plates Z Z. 

Sometimes the two washers, X, four plates, Z Z, and the two boxes, B. are omitted, 
md two cast-iron boxes of another shape are substituted in their places, one 
or each leg of the U-bolt. These boxes are not fastened to the cross-girders, but the 
ipper flanges of the latter rest upon the upper corners of the boxes, which are fitted 
It o them. The cross-girders, in such cases, are prevented from spreading apart, by 
el tolts passing through both of them, close to the cast-iron boxes. 

Art. 18. The following gives the weights of iron in several of these turn- 
ables now in use; with the wt of locomotive and tender which they 
urn ; the max load on the end carriage; and that on the pivot pier. This last, 
)f course, includes the wt of track and cross-ties in addition to the other wts 
tamed. 


Diam of 
turntable, 
ft. 

Wt of iron in 
turntable, 
lbs. 

Wt of loco 
and tender, 
lbs. 

Max load on 
end wheels, 
lbs. 

Max load on 
centre pivot, 
fbs. 

50 

17500 

124000 

57000 

149000 

50 

20500 

150000 

64000 

177000 

55 

22500 

124000 

61000 

154000 

65 

25500 

161000 

68000 

194000 

60 

27500 

160000 

74000 

206000 


Prices, accompanied by strain sheets based upon the wheel loads of loco and ten« 
ier, furnished upon application. 

Art. 19. Tlie Oreenleaf turntable, Clements A.Greeuleaf, M E, Indi- 
tuapolis, Ind, patentee and manufacturer; is made in a variety of forms. Its dis- 
tinguishing feature consists in a series of 27 cylindrical steel roll- 
jers, R R, Fig 10, 2/ g ins diam, with their axes vert. These rollers are arranged 
in a circle around the post, P, P, near its base. They have no a^les, but are held in 
a circular cast-steel box attached to the cross-girder, as in the fig, or in a circular 
groove in the center-box when this last is of cast-iron. The post is made cylindrical 
externally, near its foot, as shown, in order to give a full bearing to the vert sides of 
the rollers. Their object is to prevent the tipping of the turntable, while 
turning, even although the locomotive is not exact!y balanced on the table (Art. 1) 
ami thus to prevent the end wheels from bearing on their circular rail. 

Art. 20. Conical rollers, d d, in a roller-box, u v, are used, as in 
the Sellers (Art 3) and other tables. The cap, C, rests and fits upon a cast- 
iron hemisphere, S, a cylindrical dowel on the bottom of which enters and 
fits the space in the center of the roller-box. Oil is supplied to the roller-box 
/through the vert passage, Q, and its branches. 

Art. 21. The post here shown is made of eight 60-lb steel 
T rails, P P P, &c, with their flanges outward. To these flanges is riveted an 
iron-plate, 0 0, % inch thick, forming a conical shell or covering for the post. 
,The steel casting, L (at the top of which is the path for the vert rollers, R), is held 
to the wrought-iron base-plate, K, by long rivets which pass between the feet of the 
rails, P, of the post. A wrought-iron “ tire,” T, is shrunk around the ends of the rails. 
When this form of post is used, the arrangement for hanging- tlie table 
to the cap is similar to that in the Edge Moor table, Art 10. Hollow cast- 
iron posts, however, are more largely used, and are recommended as being bet¬ 
ter and cheaper. 

With cast-iron posts a cast-iron center-box is used, closely surrounding the post, 
and furnished with flanges, by which it is riveted to plate-iron cross-girders; but 
the arrangement of the cap, C, hemisphere, S, and rollers, is the same as in Fig 10. 

Art. 22. In all cases the table is suspended from the cap by 8 

bolts, arranged in a circle. The holes in the cap are made a little larger 

in diam than the bolts, to allow the slight hor movement of the center-box, caused 
by the tipping of the table when an engine comes upon it, or leaves it, because the 


















796 


TURNTABLES. 


l 



vert rollers, R R, prevent this motion from taking place near the foot of the post, 
as it does in other tables. 

A rt. 215. These tables are made both of wrought- and of cast-iron, and are largely 
used (especially on Western roads) with very satisfactory results. 

A Greenleaf wrought-iron turntable, 60 ft long, weighs 28000 tbs, and 
costs, 1886, SI680, say 6 cts per ft); cast-iron, same length, 32000. lbs, $1800, say 
4 cts per lb. A center, complete, with cast-iron center-post and center-box, costs 
about $ 600 . 

Art. 24. The wrought-iron turntable made by the Union Bridge Co 
(late Kellogg & Maurice), of Athens. Pa, also has vert rollers 

travelling around the post near its base; but the rollers are only two in number * 
They are held in place by vert axles, and are fixed one on each side of the post, or so 
that a hor line, joining their centers, is at right angles to the longitudinal axis of 
the turntable. They thus aid in preventing sideways tipping, but not in balancing 
the locomotive longitudinally. They therefore bring no great strains upon the post 
or its foundation. 

Art. 25. Mr. M. T. Stock, Chicago, Ill, furnishes a turntable in which 
the main girders are heavy rolled I-beams, trussed on their upper sides with a cen¬ 
tral vert strut and inclined tension-rods like Fig 52, p 514, inverted. Price of a 40-ft 
table, 1886, $800; 60 ft, $1500. 



























































































TURNTABLES. 


797 


Wooden turntables, with none but two common wheel rollers at each 
;nd of the platform, are sometimes resorted to from motives of original cost. 
They are, however, much harder to turn, generally requiring two men, aided 
)y wheel work ; and are more liable to get out of order; and more expensive to 
•epair. They are made of a great variety of patterns, both as regards the 
'irders, and the central pivots, end rollers, &c. Frequently an addition is made 
I )f 8 to 12 small rollers travelling on a circular rail of 6 to 12 feet diameter, 
(round the pivot as a center. These are intended to sustain the whole weight; 
he end rollers being so adjusted as to touch their rail only when the platform 
•ocks or tilts as the engine enters or leaves it. Therefore, there is less resistance 
rom friction than when, as in Figs 11, there are only the end rollers r. In this 
ast case, the engine and tender cannot be balanced so precisely upon the 
slender central pivot, as to prevent a great part of the weight from being 
;hrown upon the end rollers; thus materially increasing the frictional re¬ 
sistance. 


In plan, these wooden platforms are sometimes in shape of a cross; that is, in 
[addition to the main platform for the engine, there is another transverse or at 
right angles to it, also extending across the pit; and having end rollers travel¬ 
ling on the circular rail. Thus, in Figs 11, (which show one of the many modes 
[of framing a table which has only a central pivot/, and end rollers r,) the main 
platform rests on the girders c, which are strengthened below by braces a; while 
the transverse one rests on the timbers o, o, which must be imagined to extend 
across the pit. One-half, or one arm, of this transverse platform is intended to 
carry the wheel work R,xx, for turning the platform; and the other arm 
serves merely as a balance to it: therefore, neither of them requires to be 
very strong. It is important to connect the four ends of the two platforms by 
ifour beams, as the whole structure is thereby materially stiffened. In the 
figures the wheel work fixx is for convenience improperly shown as if it stood 
upon the main platform. 



The figures need but little explanation. They represent an actual 45-foot 
platform, which has been in use for some years. The convex foot / of the 
central pivot, about 6 inches diameter, should be faced with steel: and should 
rest on a steel step ss. This should be kept well oiled; and protected from 
dust by a leather collar around p, and resting on gg. Its upper part, about 4 
inches diameter, is cut into a screw with square threads about inch thick, 
for a distance of about 15 inches. It works in a female screw in the strong 
cast-iron nutyy; and serves for raising the whole platform when necessary. 
When not in use for this purpose, it is keyed tight to the platform, (by a key at 



















































































































798 


TURNTABLES. 


its head n,) so as to revolve with it. Strong screw-bolts ii connect the severa 
timbers at the center of the platform. 


It 

eel 


R is a light cast-iron stand supporting two bevel wheels about 1 foot diameter 
which give motion by means of an axle d, inches diameter to two simila: 
ones below, shown more plainly at W and Y. These last give motion by the 
axle x to the pinion e , (6 inches diameter, and 2^4 inches lace,) which turns tin 
platform by working into a circular rack t, (teeth horizontal, 1 inch pitch; 3}/. 
inches face,) which surrounds the entire pit. This rack is spiked to the undei 
side of a continuous wooden curb H, which is upheld by pieces F, a few fee 
apart, which are let into the wall J J, which lines the pit. The short bean 
M N, (about 6 feet,) which carries the lower wheelwork, is suspended stronglj 
from the beams of the transverse platform above it. Instead of the two lowei 
bevel wheels VV Y, and the horizontal axle x, a more simple arrangement is t< 
place the pinion e at the lower end of the vertical axle d ; and let it work int< 
a rack with vertical teeth at u , on the inner face of the stone foundation of tin 
circular rail. For this purpose the stand R should be directly over u. Then 
are two cast-iron rollers r, 2 feet diameter, 3 inch face, under each end of th< 
main platform ; and one under each end of the secondary one. 


ift 

lii 

'ra 


l lit 


Although this kind of platform necessarily has much friction, yet one mat f 
can generally turn a 45-foot one by means of the wheelwork, when loaded ™ 
with a heavy engine and tender. Indeed, he may do it with some difficulty s 
by hand only, while all is new and in perfect order; but when old, and tht to 
circular railway uneven and dirty, it requires two men at the winches to do it 
with entire ease. ] 


12 



As before remarked, the resistance to turning is diminished by employing s 
set of from 8 to 12 rollers or wheels r, 

Figs 12, about a foot to 15 inches in diam¬ 
eter, so arranged as to form a circle 8 to 12 
feet diameter around the pivot. When this 
is done, the main girders of the platform 
are placed 8 to 12 feet apart; and long 
cross-ties are used for supporting the 
railway track. Also, the main girders 
are sometimes trussed by iron rods, as 
in the swing bridge on page593; but in¬ 
stead of one post ac , it is best to have 
two, 6 or 8 feet apart at foot, and meeting 
at top. The width of platform must then be sufficient to allow the engine to 
pass the posts on either side of it. Ten feet will suffice. 


3 

Ji 


tii 

at 

ii 

it 

ai 




Fig 12 shows the arrangement of these rollers r, which revolve upon a cir¬ 
cular track s: while the platform rests on their tops by the track u. The 
rollers r are held between two wrought-iron rings o. o, about 3 inches deep, 
14 inch thick, which also are carried by the rollers. From each roller a radia r 
tie-rod t, 1 inch diameter, extends to a ringnn, which surrounds the pivot n , 
closely, but not tightly, so as to revolve independently of it. These tie-rods 5 
keep the rings oo at their proper distance from the pivot, so that the rollers 
cannot leave the rails a and w. Between each two rollers, the rings oo should 
be strengthened by some arrangement like a, to prevent change of shape. The i ( 
pivot p may be as in Figs 11. There must, of course, be the usual two rollers 
under each end of the platform, for sustaining the engine as it goes on or off; j 
but during the act of turning the platform, the whole weight should rest on 
the central rollers. Such a platform of 50 feet length can, if carefully made, 
be turned, together with an engine and tender, by one man, by means of a 
wooden lever 12 to 15 feet long,inserted in a staple for that, purpose; and there¬ 
fore may dispense with the transverse platform for sustaining wheelwork. 


Such rollers as have just been described, in connection with friction rollers, 
Fig 5, form perhaps the best arrangement for a large turning 
bridge. At. least one end of a platform must be provided with a catch of 
stop for arresting its motion at the moment it has reached the proper spot. 













ENGINE-HOUSES, ETC. 


799 






a 




# , f v r 1,1 V1 ,u JO vv-VA HI j ILIC UdI lfl 

rtpa out of its recess by the ring F, until it passes the casting; when it is again 
id upon the coping cc, and moves with the platform; or, if required, the 
inge at m allows it to be turned entirely over on its back. When there is a 
•ansverse platform, the proper place for the stop is at that end which carries 
le turning gear; as it is there handy to the men who do the turning. If there 
1 only a main platform, the stop may be placed midway of the rails. Some- 
mes a spring* catch is placed at each end of the platform; and each catch 
i loosened from its hold at the same instant by a long double-acting lever. All 
he details of a platform admit of much variety. 


common mode is shown at Fig 13. It consists of a wrought-iron bar mn, 4 
et long, 3 inches wide, and % thick; hinged at its end m, which is confined to 

the top of the platform. Its outer end n is 
formed into a ring V for lifting it. A strong 
casting ee (or in longitudinal section at tt,) 
about 15 inches long, 3 inches wide, and 1 
inch thick, is also firmly bolted to the top of 
the platform ; and the stop-bar mn rests in its 
recess r, while the platform is being turned. 
A similar casting a a is well bolted to the 
wooden or stone coping cc, which surrounds 
the top of the lining wall of the pit. When 
the stop-bar reaches this last casting, as the 
platform revolves, it rises up one of its little 
inclined planes tl, and falls into the recess of 
a a, bringing the platform to a stand. When 
the platform is to be started again, the bar is 


13. 

o 


o \ 


m 


m jv 





Tl ( ) 

£ 

£ 

o 


o 



XV 



Instead of the friction rollers , Fig 5, friction 
balls 5 or 6 inches diameter, of polished steel, are 
sometimes used. Tiie pivots also are made in 
many shapes. 


Fig. 14. 


Platforms like on, Fig 14, revolv¬ 
ing around one end o as a center of mo¬ 
tion, are sometimes useful. The shaded space is 
ie pit. If an engine approaching along the track W, is intended to pass on to 
ny one of the tracks 1, 2, 3, 4, the platform is first put into the required posi- 
ion, and the engine passes at once without detention. If the platform is long, 
t will be necessary to have roller-wheels not only under the moving end a, but 
t one or two other points, as indicated by the roller rails c c. . 


Engine houses, of brick, cost from $1000 to $1200 per engine stall, exclu- 
ive of the foundations. 


The eost of a complete set of shops of brick, for the thorough re¬ 
pair of about 20 locomotives, and of the corresponding number of passenger 
md other cars; together with suitable smith shop, foundry, car shop, boiler 
■hop, copper and brass shop, paint shop, store rooms, lumber shed, offices, in¬ 
completely furnished with steam power, lathes, planing machines, scales, and 
dl other necessary tools and appliances, will he about from $75000 to $100000 ex¬ 
clusive of ground. A large yard, of at least an acre, should adjoin the buildings. 
\ moderate establishment, for the repairs of a few engines only, may be built 
ind furnished for $25000. 






























800 


WATER STATIONS. 


WATER STATIONS. 

Water stations are points along a railroad, at which the engines sjfop t 
talce in water. Their distance apart varies (like that of the fuel sta 
tions, which accompany them,) from about 6 miles, on roads doing a very larg 
business: to 15 or 20 miles on those which run but few trains. Much depends 
however, upon where water can be had. It has at times to be conducted it 
pipes for 2 or 3 miles or more. The object in having them uear together is t 
prevent delay from many engines being obliged to use the same station. T< 
prevent interruption to travel, they are frequently placed upon a side track 
A supply of water is kept on hand at the station, usually in large wooden tub 
or tanks, enclosed in frame tank-houses. The tank-house stands near the track 
leaving only about 2 to 4 feet clearance for the cars. It is two stories high; th 
tank being in the upper one ; and having its bottom about 10 or 12 teet above t h 
rails. In the lower story is usually the pump for pumping up the water into th 
tank; and a stove for preventing the water from freezing in winter.* 

The tanks are usually circular; and a few inches greater in diameter at th< 
bottom than at the top, so that the iron hoops may drive tight. Tlieii 
capacity generally varies from 6000 to 40000 gallons, (rarely 80000 or more, 
depending on the number of engines to be supplied. A tender-tank holds 
from 1000 to 3000 gallons; and an engine evaporates from 20 to 150 ga 
Ions per mile, depending on the class of engine; weight of train; steepness o 
grade, Ac. Perhaps 40 gallons will be a tolerably full average for passenger, anc 
80 for freight engines. The following are tile contents of tanks 
of different inner diameters, and depths of water. U. S. gallons of 231 cubi< 
incites; or 7.4805 gallons to a cubic foot. 


ft 

r 

It 

ft 

in 

CJ 

ta 

II 

Ir 

tv 

in 

ti 

01 

ft 

\V 

u 

V 

In 

In 

tl 

ei 

k 


Diam. 

Depth. 

Contents. 

Diam. 

Depth. 

Contents. 

Ft. 

Ft. 

Gallons. 

Cub. Ft. 

Ft. 

Ft. 

Gallons. 

Cub. Ft. 

12 

8 

6767 

995 

24 

12 

40607 

5429 

14 

9 

10363 

1385 

26 

13 

51628 

6902 

16 

9 

13535 

1810 

28 

14 

64481 

8621 

18 

10 

19034 

2545 

30 

15 

79310 

10603 

20 

10 

23499 

3142 

32 

16 

96253 

12868 

22 

11 

31277 

4181 

34 

17 

115451 

15435 


Cypress or any of the pines answer very 
may be about 2^ inches thick for the smaller 
largest. The bottoms may be the same. The 

chineiy to suit the curve precisely. Nothing is then needed between tbestav 
to produce tightness. A single wooden dowel is inserted between each two near 
the top, merely to hold them in place while being put together. The bottom is 
dowelled together; and simply inserted into a groove very accurately cut, about 
an inch deep, around the inner circumference of the tub,’at a few inches above 
the bottoms of the staves. 

One of 20 feet diameter, and 12 feet deep, may have 9 hoops of good iron ; placed 
several inches nearer together at the bottom of the tank than at the top. Their 
width 3 inches; the thickness of the lower two,^iinch ; thence gradually dimin¬ 
ishing until the top one is but half as thick. The lower two are driven close 
together. These dimensions will allow for the rivet-holes for riveting together 
the overlapping ends; and for a moderate strain in driving the hoops firmly 
into place.f Three rivets of % inch diameter, and 3 inches apart, in line, are 
sufficient for a joint of a lower hoop. One of 34 feet diameter, 17 deep, may 
have 12 hoops; the lower ones 4 inches by with three %-inch rivets to a 
lower hoop-joint. 

The bottom planks of the tank must bear firmly upon their supporting joists, 
or bearers. 

A tank must have an Inlet-pipe by which the water may enter it; a waste- 
pipe for preventing overflow; and a discharge or feed-pipe 7 or 8 
inches diameter, in or near the bottom; through which the water flows out to 
the tender. The inner end of the discharge-pipe is covered by a valve, to be 
opened at will by the engine man, by means of an outside cord and lever. To 


I 


*A frame tank-house, 18 feet square, with stone foundations for both house 
anil tank, will by itself eost $400 to $600. A brick or stone one somewhat more. 

tSuch a tank, put tip in its place, will cost, about $400. Geo. J. Burkhardt’s 
Sons, North Broad St below Cambria, Philadelphia, make tanks their specialty ; 
and are provided with machinery which secures perfect accuracy of joint in 
every part. Their work is sought from great distances. 


































I WATER STATIONS. 801 

s outer end is generally attached a flexible canvas and gum-elastic hose about 
or b inches diameter, and 8 or 10 feet long, through which the water enters the 
mder-tank. Or, instead of a hose, the feed-pipe may he prolonged by a metal- 
e pipe, or nozzle, sufficiently long to reach the tender; and so jointed as, when 
ot in use, to swing to one side, or to be raised to a vertical position, (in the last 
ise it is called a drop,) so as not to be in the way of passing t rains. 

Ihe same tank may supply two engines on different tracks, at once. The 
inks are very durable. 

The patent (rosi-proof tank of John Burnham, Batavia, 
linens, is simply an ordinary tank, in which the water is prevented from 
eezing by means, 1st, of a circular roof which protects a ceiling of joists, be- 
veen which is a layer of mortar; 2d, by an air-space obtained by a similar ceil- 
ig beneath the timbers on which the tank rests. Although the sides are en- 
rely unprotected, no house is necessary ; but merely strong posts and beams 
l a stone foundation, for the support of the tank.* 'The supply pipes are in 
ixes made of boards and tar-paper. 

Tanks are frequently made rectangular, with vertical sides of 
ists lined with plank, and braced across in both directions by iron rods. They 
•e more apt to leak than circular ones. They have been made of iron: but 
ood seems to be preferred. 

The water for supplying the tanks, may be pumped by hand, steam, 
>rse, wind, hydraulic ram, or otherwise, from a running stream; from a pond 
ade by damming the stream if very small or irregular; from a cistern below 
le tank; or from a common well. Many roads doing a business of 10 or 12 
igines daily in each direction,depend entirely upon wells; and pump by hand ; 
merally two men to a pump. Those doing a very large business, when the 
ipplv cannot be obtained by gravity, mostly use steam. The windmill is 
e mosCeconomical power; and when well made, is very little liable to get out 
order. Of course it will not work during a calm ; but this objection may be 
tviated in most cases by having the tanks large enough to hold a supplj' for 
veral davs.f Steam, however, is most reliable. 

The following table will give some idea of the power required in 
steam engine for the pumping. In ordering an engine, specify not 
> number of horse-powers, but the number of gallons it must raise in a given 
imher of hours, to a given height; with a given steam pressure, (say about 60 
80 lbs per square inch.) The pump should be sufficiently powerful not to have 
work at night; and should be capable of performing at least 25 per cent, more 
an its required duty. 

A fair average horse should pump in 8 hours the quantities 
ntained in the first 3 columns; to the height in the 4th column ; or sufficient 
supply the number of locomotives in the 5th column, with about 2000 gallons 
3h. Two men should do about one-third as much.j; 


A Cub. Ft. 

Lbs. 

Gals. 

Ht. Ft. 

No. of 
Locos. 

Cub. Ft. 

Lbs. 

Gals. 

Ht. Ft. 

No. of 
Locos. 

I 1600 

100000 

11968 

100 

6 

4571 

285714 

34194 

35 

17 

2000 

125000 

14960 

80 

7 l A 

5333 

333333 

39893 

30 

20 

1 2667 

166666 

19946 

60 

10 

6400 

400000 

47872 

25 

24 

I 3200 

200000 

23936 

50 

12 

8000 

500000 

59840 

20 

30 

[ 3555 

222222 

26596 

45 

i M 

10667 

666667 

79787 

15 

40 

f 4000 

250000 

29920 

40 

15 

16000 

1000000 

119680 

10 

60 


A reservoir, with a stand-pipe, or water column, is preferable 
!to the ordinary tank, when the locality admits of it ; being less liable than the 
jump to get out of order; and being cheaper in the end. The reservoir is sup¬ 
posed to be filled by water flowing into it by gravity; and to have its bottom at 

*The U.S. Wind-Engine and Pump Co, of Batavia, Ill, make a specialty of the 
construction and erection of these tanks, complete in every detail, ready for use. 
They also make windmills. 

f Andrew J. Corcoran, No. 76 John St, New York, furnishes excellent machines. 
He also, when desired, provides pumps, Ac, complete. The cost of windmill 
alone, for railway stations, varies from about $450 for 18 feet diameter; to $1500 
for 36 feet diameter, at the factory. 

JThe cost of a direct acting steam pump, with its boilers, Ac, 
fixed in place, readv for work, and capable of the duty of the above table, may 
be roughly set down at about $450; twice the duty, $600; 4 times, $750; 6 times, 
$900 ; 10 t imes, $1300; 20 times. $2000. Add cost of engine-house. Made by Henry 
R. Worthington, 86 Liberty St, New York ; Geo. F. Blake Manufacturing Co,44 
Washington St, Boston ; and by many establishments in most of our large cities. 

52 


























802 


WATER STATIONS. 


least about 8 feet above the rails; or at any greater height whatever that the 
ground and the height of the water may require. It may be excavated in the 
ground; lined with brick or masonry in cement; with a bottom of concrete; 
or it may be built above ground, according to the locality. It may be roofed 
and covered in, or not; and it may be near the tracks, or at a considerable dis¬ 
tance from them, according to circumstances.* From its bottom, an iron pipe 
from 8 to 12 inches diameter, is carried (generally underground,) to within a few 
feet of the track. At that point it turns vertically upward to about 8 or 10 feet 
above the track, forming a stand-pipe, or water-column: from the 
upper end ot which the water flows (through either a hose or a jointed nozzle,) 
as in the case of a tank. Several such pipes, or one larger one, may be laid, for 
the supply of two or more engines at once, through as many stand-pipes. Where 
the pipe makes its bend, and becomes vertical, is a valve for opening and closing 
it; and which may be worked by a hand-wheel placed at such a height as to be 
easily reached by the engine man.f A valve on the principle of those for street 
pipes, page 301, is best. 

On some of the more important lines, the tenders of fast trains scoop 
up water, while running', from a long trough, or “track tank’ * 1 

laid between the rails. The tanks are about % mile long. They must of course be 
level, and they therefore require a level track. 

As originally introduced in England, by Ramsbottom, the trough was of cast 
iron, in lengths of about 6 ft. These were bolted together by means of flanges at 
their ends. The ends were not in contact with each other, but were separated bj 
a strip of vulcanized rubber. 



Our figure is a vertical cross-section of the standard track tank of the Pennsyl 
vania Railroad, 1884. It is of inch rolled plate-iron, the sheets of which are 6 
ins long. The lengths overlap each other 2 ins; leaving 5 ft as their showing lengtl 
The sheets are cut slightly tapering, so that at one end of each length the trough i 
^5 in.deeper than at the other,and the tops are thus kept flush with each other through 
out. The joints are double riveted with % inch rivets, about 1 ins from center t 
center, and staggered. At each end of the trough, the bottom slopes upward, and 
in a length of 6 ft, comes to the level of the tops of the sides. The cross-ties an 
notched, as shown, to receive the trough, which is loosely held to them by tw< 
spikes, S ami S. in each tie. The heads of the spikes fit over the horizontal flange: 
ot the 1X 1}^ inch angle bars, A and A. M and M are mouldings of Xy% X \ 
inch bar-iron. The angles and the mouldings are in lengths of 15 ft, and are rivete < 
to the sides of the trough continuously throughout its length. 

The scoop on the tender is lowered into the trough, and raised from it 
by means of a lever on the fireman’s platform, and is not permitted to touch tli 
bottom of the trough. 

The trougli is supplied with water by means of pipes leading from an ad 
jacent tank. The supply is regulated by a man in charge. 

To prevent the water from freezing' in winter, steam is led to the trougl 
from the boiler of the pumping engine, through iron pipes laid under ground along 
side of the track. These pipes are provided with branches which introduce tli 
steam to the trough at every 40 ft of its length. The steam-pipes are protected b; 
wooden boxes, and are furnished with valves for regulating the supply of steam. 


* An uncovered rrsbrvotr 50 ft. diarn by 12 ft deep, lined with brick or masonry, will usually cos 
from $2500 to $:i500, according to circumstances. 

1 Thk prick of a cast-iron water-column, of 6-inch bore; with bed-plate; holding-down bolt! 
and washers; connecting pipes; swing-joint with copper arm 9 ft long: valve; haud-wheel; &e 
complete, ready to set up, (by Isaac S. Cassiu & Co, Philada, in 1886,) is $475 at the shop less 20 pe 
cent discouut. 


I 

i 

I 

1 

I 

1 

1 

5 




I 

Of 

os 

ab 


l 

at 

of 

in 


St 









































FENCES, ETC. 


803 


onJF' ap0ra * ion from Locomotives. In addition to what is said on page 
bOO, in the passage preceding the table, we may state that the evaporation is 
usually troiu 6 to 7 lbs of water to 1 ft> of fair coal. Hence if we take the average 

qaa*» ® s ’ or sa ^ a g allon water to 1 ®> of coal, and assume, as on page 
800, that a passenger engine evaporates an average of 40 gallons per mile, and 
a freight engine 80 gallons, we shall have very nearly 2% tons of coal consumed 
per 100 miles by the former; and 4^ tons by the latter. The evaporation from 
a heavily tasked powerful engine may amount to 150 gallons or more per mile; 
but such is an exceptional case. 





Theoretical thickness near bottom of sheet-iron water 
tanks, single riveted; safety 4; ultimate strength of the iron 40000 lbs per 
square inch, but reduced say one-half by punching the rivet holes. Although 
safe against the pressure of the water, some are plainly far too thin for handling. 


Depth in 
Feet. 


INNER DIAMETER IN FEET. 


5 


10 


15 


20 


25 


30 


35 


40 




THICKNESS IN INCHES. 


1 

.0026 

.0052 

.0078 

.0104 

5 

.0130 

.0260 

.0391 

.0520 

10 

.0260 

.0521 

.0781 

.1042 

15 

.0391 

.0781 

.1172 

.1562 

20 

.0521 

.1042 

.1562 

.2084 

25 

.0651 

.1302 

.1953 

.2604 

30 

.0781 

.1562 

.2344 

.3124 


.0130 

.0051 

.1302 

.1953 

.2604 

.3255 

.3906 


.0156 

.0781 

.1562 

.2344 

.3125 

.3906 

.4687 


.0182 

.0911 

.1823 

.2734 

.3645 

.4557 

.5470 


.0208 

.1042 

.2083 

.3125 

.4166 

.5208 

.6250 


Railroa<l track scales are made by Rielile Bros, office, 4th and Chestnut 
8ts, Phila, and by Fairbanks & Co, St Johusbury, Yt. Price-list of Rielile track 
Scales. Discount, 1886, about 45 per cent. The capacities are in tons of 2000 
llis or 2240 lbs, as may be desired. 


Capac- 

Length. 

Price. 

Capac- 

Length. 

Price. 

ity. 

ft. 

$ 

ity. 

ft. 

$ 

10 

12 

350 

65 

40 to 65 

1850 

15 

12 to 15 

400 

75 

40 to 85 

2200 

20 

12 to 16 

600 

100 

50 to 112 

2800 

30 

20 to 32 

850 

150 

60 to 123 

3200 

40 

30 to 40 

975 

150 

100 to 150 

3700 

50 

40 to 50 

1100 





Post-an<l-rail fences, in panels 8% ft long; 5 rails; usually cost between 
0 to 100 cents per panel, including the putting up ; or from $512 to $1280 per mile 
if road fenced on both sides, with 1280 panels. 

/ Fence-posts are usually of chestnut, cedar, or white oak. They last about 10 years 
in an average. The usual size is 2 to 3 ins thick X 6 to 7 ins wide, 8 ft long, 5 ft 
ibove ground. Their cost varies greatly; say from 5% to 25 cts each; average, 10 
to 15. 

{ Worm fences seven rails high, with two rails on end at each angle, cost about 
£th less. Labor $1.75 per day. The scarcity or abundance of timber chiefly in- 
luences the price ; as is also the case with ties. 

Tlie Glidden Parked Steel Wire fence, made by I. L. Ellwood & Co., 
’it De Kalb, Illinois, has (1882) come into extensive use; many thousands of miles 
)f it having already been put up. Its cost per mile of single row of fence, put up, 
ncluding the wooden posts and all labor, will usually rauge from $150 to $250, de¬ 
fending on the height of fence, the varying market price of wire, labor, &c. 

A way-station house, 30 X 60 feet, surrounded by a platform 12 feet 
wide, protected by projecting roof; for passengers and freight; will cost from 
$6000 to $10000, according to finish and comple.eness, at eastern city prices. 









































804 


COST 0«F RAILROADS. 


Approximate average estimate for a mile of single-track 

railway. Labor $1.75 per day. 

Grubbing and clearing , (average of entire road,) 3 acres at $50.$ 150 

Grading; 20000 cab yds of earth excavation, at 35 cts .. 7000 

“ 2000 cub yds of rock excavation. at $1.00 . 2000 

Masonry of culverts, drains, abutments of small bridges, retaining-walls, <£c; 

400 cub yds, at $8, average . . 3200 

Ballast; 3000 cub yds broken stone, at $1.00. 300C 

Cross-ties; 2640, at 60 cts, delivered . 1584 

Rails ; (60 lbs to a yard ;) 96 tons, at $30, delivered .. 2880 

Spikes ... 15< 

Rail-joints . 301 

Sub-delivery of materials along the line . 30( 

Laying truck. . 60( 

Fencing : (average of entire road.) supposing only % of its length to be fenced.. 45( 

Small wooden bridges, trestles, sidings, road-crossings, cattle guards, die, d-c . 100( 

Land damages . 100( 

Engineering, superintendence, officers of Co, stationery, instruments, rents, 
printing, law expenses, and other incidentals . 238< 

Total .|2600( 

A<l<l for depots, shops. Engine-houses, Passenger and Freight Stations, Platforms 
Wood Sheds, Water Stations with their tanks and pumps, Telegraph, Engines, Cars, Weigh Scales 
Tools, he, Ac. Also for large bridges, tunnels, Turnouts, Ac. 






























LOCOMOTIVES 


805 



ROLLING STOCK 






50 

00 

00 

JO 

«l 

.'SI 

181 

« 

’<10 

1 


LOCOMOTIVES. 

Dimensions, Weights, Ac, of Locomotives. 

The following lists of the dimensions, weights, Ac, of some of the principal sixes 
if locomotives made by the Baldwin Locomotive Works, Pliila; Burnham, Parry, 
Williams & Co, proprietors; will give an idea of the present usual proportions 
#f locomotives ami tenders as marie in the United States. 

In the designation of the class, the first number (K, 10, Ac) is the total number 
if wheels of the locomotive. The second (20, 26, Ac) is an arbitrary number indi¬ 
cating the diameter of the cylinders. The letter (C, D, or E) indicates the number 
4, 6, or 8, respectively) of dmunp'-wheels. 

The wlieel-base is the distance from center to center of the front and back 
vheels. For minimum length of turntable, add to 2 ft to the total 
vheel-lmse of locomotive and tender, which is = wheel-base of locomotive + wheel- 
lase of tender-f distance between centers of front tender wheel and hind engine wheel. 

Under 44 Service,” I* means passenger; E, freight; M, mixed; S, switching. 


:# 

■9 


For gauge of 4 ft 8 1-2 ins. 
Dimensions. 


Class. 

Service. 

Type. 

Cylin¬ 

ders. 

Driving 

Wheels. 

Wheel-base. 

Extreme Igth 
loco and 
tender. 

Extreme 

width. 

Height 

of top of 
chimney 
above 
top of 
rails. 

a 

a 

S 

Stroke. 

© 

55 

a 

cfl 

5 

Locomotive. 

Ten¬ 

der. 

Loco 

and 

tender. 

Driv’rs 

Total. 




Ins. 

Ins. 


Ins. 

Ft. 

In. 

Ft. 

In. 

Ft. 

In. 

Ft. 

In. 

Ft.In. 

Ft.In. 

Ft. In. 

i— 14 —C 

p ) 


( 10 

20 

4 

45 to 51 

5 

6 

16 

4 

5 

1 

27 

6 

41 3 

7 9 

12 

4 

i—20—C 

P F >■ 

3 3 

1 13 

22 

4 

49 to 57 

7 

0 

20 

6 

14 

2 

42 

0 

47 8 

8 4 

13 

0 

30—C 

“ ) 

< t 

( 18 

22 

4 

61 to 66 

8 

6 

22 

5 

14 

0 

44 

4 

54 6 

8 6 

13 

6 

)— 26—D 

FM) 

A, Zs 

(16 

24 

6 

51 to 56 

12 

6 

22 

8 

1.3 

5 

42 

2 

54 9 

9 4 

13 

0 

)—28—D 

“ £ 


< 17 

24 

6 

51 to 56 

12 

10 

23 

0 

13 

1 

42 

11 

55 1 

8 11 

13 

0 

)—32—D 

“ ) 

- XJ 
£ 

( 19 

24 

6 

54 to 60 

13 

6 

23 

8 

13 

11 

44 

10 

55 9 

9 0 

13 

6 

i—18—D 

F ) 

• _ 

( 12 

18 

6 

37 to 41 

10 

0 

16 

0 

10 

11 

33 

0 

41 8 

8 4 

12 

6 

1—26—D 


Sr 

1 lfi 

24 

6 

45 to 51 

14 

2 

21 

6 

13 

5 

42 

2 

54 6 

9 0 

13 

0 

i—32—D 

“ 5 

- to 

(19 

24 

6 

54 to 60 

15 

2 

22 

6 

13 

n 

43 

10 

56 2 

9 2 

13 

8 

)—34—E 

“ ] 

a 

C 20 

24 

8 

48 to 50 

14 

9 

22 

10 

14 

4 

46 

2 

58 6 

9 6 

14 

0 

)—36—E 

“ 5 

O? 0 

i 21 

24 

8 

48 to 50 

14 

9 

22 

10 

14 

4 

46 

2 

57 3 

9 8 

14 

0 

t—26—C 

S 

• 2*" 

16 

24 

4 

48 to 54 

7 

6 

7 

6 

13 

5 

34 

6 

47 2 

9 0 

13 

0 

>—28— D 

s 


17 

24 

6 

45 to 49 

11 

0 

11 

0 

13 

1 

35 

5 

48 4 

9 0 

13 

4 


Weights, Ac. 


Class. 


8.14.C 
8.20.C 
8.30.C 

10.26.D 

10.28.D 

10.32.D 

8.18.D 
8.26. D 
8.32. D 

10.34.E 
10.36.E 

4.26.0 

6.28.D 


v 

cn 


“ 1 
s 
s 


a> 

CM 

tr> 


v . 

3 3 

< o 


— 

(C 


^ 3 

: so 


a cs -. 
03 B 

or o 
: 2 - 


Weights in working order. 


Locomotive. 


Greatest on 
1 pair of 
drivers. 


Ibs 

(10500 
1 17500 
( 27000 12.0 


a 

o 

*-> 

4.6 

7.8 


(18000 
{ 19000 
(22000 

(11000 

1 21000 
( 25000 


8.0 

8.5 

9.8 

4.9 
9.4 

11.2 


( 23000 10.3 
$ 23000 10.3 

2800ojl2 5 
24000110.7 


On all 
drivers. 


Ibs 

21000 

35000 

54000 

58000 

63000 

72000 

33000 

63000 

80000 

88000 

96000 

56000 

69000 


c 

c 

-*-» 

9.4 

15.6 

24.1 

25.9 

28.1 

32.1 

14.7 

28.1 

35.7 


Total. 


lbs 

36000 

56000 

82000 


0 

o 

♦3 

16.1 

25.0 

36.6 


78000 34.8 
84000; 3 7.5 
96000 42.9 

40000 17.9 
75000 33.5 
94000 i 42.0 


39.3 104000 46.4 
42.9 112000 50.0 

56000 25.0 
69000 30.8 


25.0 

30.8 


Tender 

loaded. 


fits 

26000 

34200 

55500 


3 

O 

-«-> 

11 6 
15.3 
24.8 


47800 21.3 
51600 23.0 

59300126.5 

216001 9.6 
4780021.3 

59300.26.5 

59300 26.5 
63400 28.3 


47800 21.3 103800 
51600 23.0 120600 


Total 
loco and 
tender. 


Ibs 

62000 

90200 

137500 

125800 

135600 

155300 

61600 

122800 

153300 

163300 

175400 


0 

2 

27.7 

40.3 

61.4 

56.2 

60.5 

69.3 

27.5 

54.8 

68.4 

72.9 
78.3 

(6.3 

53.8 


Capacity of tender. 



O 

Water. 

O © 

O 

O 

* 



Ibs cords 

Gals of 
231 cuin 

lbs 

8000 

1.25 

1000 

8300 

9000 

1.60 

1400 

11700 

12300 

2.90 

2400 

19900 

11800 

1.70 

2000 

16700 

12000 

1.80 

2200 

18400 

12500 

2.00 

2600 

21700 

8500 

1.50 

1200 

10000 

11800 

1.70 

2000 

16700 

12500 

2.00 

2600 

21700 

12500 

2.00 

2600 

21700 

13000 

2.20 

2800 

23300 

11800 

1.70 

2000 

16700 

12000 

1.80 

2200 

18400 



















































































806 


LOCOMOTIVES 




For gauge of 3 ft. 
Dimensions. 


Class. 

Service. 

Cylin¬ 

ders. 

Driving 

Wheels. 

I 

Dr 


Wheel-base. 



)S§ 

a> S 

s * . 
t « £ 

s-« 

H 

Extreme 

width. 

Height 

of top 61 
chimucy 
above to[ 
of rails. 

Diam. 

Stroke. I 

o 

£ 

a 

n 

3 

^ocon 

vers 

aotive. 

Total. 

Tender. 

Loco 

aud 

tender. 



Ins. 

Ins. 


i,,- 

Ft. 

Ins. 

Ft. Ins. 

Ft. 

Ins. 

Ft. 

Ins. 

Ft.In. 

Ft.Ins. 

Ft. Ius. 

8—18— C 

P 

12 

16 

4 

43 

7 

2 

18 3 

11 

4 

35 

8 

44 7 

7 4 

12 0 

8—22 —C 

it 

14 

18 

4 

45 

8 

2 

20 1 

13 

2 

40 

5 

48 6 

7 10 

12 4 

8 —20—D 

F 

13 

18 

6 

37 

12 

0 

17 10 

11 

4 

35 

3 

47 1 

7 4 

12 2 

8 —22—1) 

it 

H 

18 

6 

39 

12 

0 

18 4 

13 

2 

38 

8 

45 5 

7 8 

13 0 

10—24—E 

it 

15 

18 

8 

37 

11 

4 

17 10 

13 

5 

39 

11 

49 5 

8 1 

12 5 

10—26—E 

II 

16 

•20 

8 

37 

11 

9 

18 1 

13 

0 

40 

9 

50 0 

8 4 

13 7 

4 1# f! 

g 

12 

li> 

4 

37 

6 

o 

6 0 



6 

o 

37 0 

7 10 

13 0 

6 —22—D 

II 

14 

18 

6 

37 

9 

. 6 

9 6 


9 

6 

43 6 

7 4 

a i 


Weights, Ac. 


Class. 

Service. 

8—18—C 

P 

8 —22—C 

it 

8 —20— D 

F 

8 —22—D 

it 

10—24—E 

it 

10—26—E 

<t 

4—18—C 

S 

6 —22—D 

it 


Weights in working order. 


Locomotive. 




Greatest 
on 1 pair 
of drivers. 

Ou 

drive 

all 

1 ' 8 . 

Total. 

Ten 

load 

der 

ed. 

loco 

tend 

ind 

er. 








w 


cn 

Sis 

a 

3 

lbs 

a 

3 

lbs 

a 

o 

-*-> 

lbs 

a 

2 

fibs 

a 

o 

■** 

13000 

5.8 

24000 

10.7 

37000 

16.5 

31000 

13.8 

68000 

30.4 

16000 

7.1 

33000 

14.7 

48000 

21.3 

35000 

15.6 

83000 

37.0 

13000 

5.8 

39000 

17.4 

46000 

20.5 

31000 

13.8 

77000 

34.4 

14000 

6.2 

42130 18.8 

50330 

22.5 

35000 

15.6 

85330 

38.1 

12000 

5.4 

48000 

21.3 

56000 

25.0 

37000 

16.5 

93000 

41.5 

15000 

6.7 

58000 

25.9 

67000 

29.9 

38500 

17.2 

105500 

47.1 

15000 

6 7 

28000 

12.5 

28000 

12.5 



28000 

12.5 

15000 

6.7 

48000 

21.3 

48000 

21.3 



48000 

21.3 


Capacity of tender 

or tank. 



•6 

Water. 

£ 

O c 

o 

o 

fee 



lbs cords 

Gals of 
231 cu in 

lbs 

9000 

1.3 

1200 

1000 

9775 

1.4 

1400 

1170 

9000 

1.3 

1200 

1000 

9775 

1.4 

1400 

1170 

10300 

1.5 

1500 

1245 

9775 

1.4 

1600 

1328 

1800 

.5 

400 

332 

•2000 

.5 

500 

415 


Standard passenger (Class N) and freight (Class I) locomotive! 
of Pennsylvania llailroad, 1885. Gauge, 4 ft ‘J ius. 


Dimensions. 


Class. 

Cylin¬ 

ders. 

Driving 

Wheels. 

Wheel-base. 

Extreme l’jfth 
loco and ten¬ 
der. 

Extreme 

width. 

Height 

of top of 
chimney 
above top 
of rails. 

Diam. 

Stroke. 

O 

| Diam. 

1 

Locomotive. 

Tender. 

Loco 

and 

tender. 

Drivers 

Total. 


Ins. 

Ins. 


Ins. 

Ft. Ins. 

Ft. Ins. 

Ft. Ins. 

Ft. Ins. 

Ft. Ins. 

Ft. Ins. 

Ft. Ins. 

N 

17 

24 

4 

62 

8 6 

23 2 

14 10 

45 2 

54 7 

9 0 

15 0 ' 

1 

•20 

24 

8 

50 

13 8 

21 6 

15 4 

47 7 

56 0 

9 3 

15 11 


Weights, Ac. 


Class. 

Weights in working order. 

Capacity of tender. 

Locomotive. 


Locomo¬ 
tive and 
tender. 


Water. 

Grea 
on 1 
of dri 

test 

mir 

vers. 

On 

drivt 

111 

irs. 

Toti 

il. 

Tend 

er. 

Co 

al. 

N 

I 

lbs 

30800 

22200 

CD 

a 

o 

18.8 

9.9 

lbs 

57700 

80500 

00 

a 

2 

25.8 

35.9 

lbs 

91300 

92700 

00 

a 

o 

40.8 

41.4 

lbs 

50500 

56650 

CO 

a 

o 

22.5 

25.3 

lbs 

141800 

149350 

CD 

a 

2 

63.3 

66.7 

lbs 

8000 

8000 

09 

a 

o 

3.6 

3.6 

Gals of 
231 cu ins. 

2400 

3000 

lbs 

20000 

25000 








































































































































































LOCOMOTIVES. 807 


The following are the principal dimensions, «fec, of five recent and ex- 
;eptionalIy larg’e types of locomotives: 


ht 

lop 

\ 

“ Decapod."* 
Largest loco 
built by the 
Baldwin Wks 
up to 1887. 

“ El Goberna- 
dor,’’ Centl 
Pacific It R. 
1881, cvls 21 
X 36. 

“ Class K.” 
Engine “ No. 
10. &c, Peuna 
R R 

“ Class P.” 
Penua R R. 

Phila & 
Reading R R. 


lelvice. 

Freight. 

Freight. 

Fast Pass'r. 

Passenger. 

Fast Pass'r. 

— 

»/nge. 

4 ft sy ins 

4 ft 8ins 

4 ft 9 ins 

4 ft 9 ins 

4 ft S]/ 2 ins 

11. 

driving wheels: 






# 

Number. 

10 

10 

4 

4 

4 

t 

Diam. 

45 ins 

57 ins 

78 ins 

68 ins 

68 ins 

i 

’ruck wheels: 






0 

Number. 

2 

4 

4 

4 

4 

J 

Vheel-base: 






9 

Locomotives: 






1 

Drivers. 

17 ft 0 ins 

19 ft 7 ins 

7 ft 9 ins 

7 ft 9 ins 

7 ft 0 ins 


Total. 

24 ft 4 ins 

28 ft 11 ins 

22 ft ry 2 ins 

22 ft V/ 2 ins 

21 ft 1 in 


Tender. 

14 ft 5 ins 

15 ft 2 ins 

15 ft 4 ins 

15 ft 4 ins 

16 ft 10 ins 


Loco and ten- 







der.. 

49 ft 2 ins 

52 ft 7 ins 

47 ft 8 ins 

47 ft 8 ins 

50 ft 8J4ins 

dr 

Sxtreme length: 







Loco and ten- 






■ 

der.. 

63 ft 3)/£ ins 

63 ft 8 ins 

58 ft 6 ins 

58 ft 7 ins 

58 ft 8%ins 


leight: 







Top of chimney 







above top of 







rail. 

14 ft 6 ins 

16 ft 2 ins 

15 ft 0 ins 

15 ft 0 ins 

14 ft 4 ins 


Veight in work- 







ing order, lbs 






wol 

Locomotive: 






!780] 

Greatest on 1 






m 

TOO 

pair of dri- 






.« 

vers. 

27000 

25800 

33600 

34700 

36000 

ISO 

On all dri- 






to 

vers. 

134000 

121600 

59000 

69450 

71950 

U50 

Total. 

148000 

152000 

92700 

100600 

103850 

— 

Tender. 

82000 

85650 

56300 

56300 

G5000 

es 

Loco and ten- 







der. 

230000 

237650 

149000 

156900 

168850 


Capacity of ten- 







der: 







Coal: lbs. 

16000 

10000 

12000 

12000 

12000 

it 

Water: 






of 

Gals of 231 






J 

cubic ins... 

3600 

3000 

2400 

2400 

3570 


»p-—- - --——- 

i. From the above lists it will be seen t hat the weight of road locomo¬ 
tives per foot of tlieir total wheel-base varies from .9 ton in light 
-jassenger engines for 3 ft gauge, to 2.5 tons in very heavy freight engines for 
< standard gauge. 

The cost of locomotives and tenders is about from 9 to 12 cts per 
xmnd of locomotive alone, varying with the details of the specifications, &c. 

Steel driving’ tires are usually of open hearth steel, and are from 2 y 2 
“,o 3^ inches thick; occasionally 4 inches. Flanges average 1 % to inches 
leep, inches thick. Prices, delivered in New York : rough, 5 cts per lb; 
' iored, 5% cts. Standard Steel Works, makers, Lewistown, Pa; office, 220 S 4th 
5 t, Phila. The inner diameter of the tire is usually made -j^ 1 ^ (or inch per 
• bot) less than diameter of the wheel center upon which it is*t<> go. Heating the 
ire increases its inner diameter so that it can be placed upon the wheel center. 
Its contraction, or tendency to contract, in cooling, binds it fast. 

The treads are re-turned in a lathe as often as from T 3 6 to x / x inch wears off from 
heir thickness. (=% to x / 2 inch in diameter) and are abandoned when worn down 
,:o about inches thick. On passenger engines, they run about 60000 miles 
oetween turnings; on freight engines, 35000. 

j * Built 1886 for Northern Pacific R R, Nos 500,501; Extreme width 10 ft. 

> The original “Decapod,” built by same works in 1S84 for Dorn Pedro IIR R, Brazil 
'5 feet 3 inches gauge), and described in our last edition, was a little lighter. 












































808 


LOCOMOTIVES. 


Driving axles are of iron, or of a softer steel than the tires. They are nsiL 
ally from 5 to 734 ins diam. 

Passenger engines usually carry fuel and water sufficient for 40 o 
50 miles; some, 50 to GO. Freight trains, enough for 20 to 25 miles. Hoads, i 
divisions, with steep grades require the fuel and water stations to be nearer togetlu 
than where the grades are easy. 

Performance of Locomotives. 

The following gives the loads (exclusive of locomotive and tender) which tb 
above described Baldwin engines will haul, at their usual speed,on astraigl ni 
track and on different grades varying from a level to 3 ft per 100 ft, or 158. , 
ft per mile. The loads are based upon the assumption that the so-called “adllt ! 
Sion” of the locomotive is one-fourth of the weight on all the drivers, and tin'», 
the condition of road and cars is such that the frictional resistance of tl r 
cars does not exceed 7 lbs per ton of 2240 lbs of their weight. These are ordinaril rl 
favorable conditions. The adhesion is seldom less than one-fifth, or more than 01 m jj 
third, of the weight on the drivers. 

The resistance of cars to motion, on a level track, and with cars an| 
track in fair order, is usually taken at about from 6 to 8 lbs per ton ol 2240 lb 
With everything in perfect order, it may fall as low as 5, or even 4 lbs per ton. 0 - 
the other hand, if the wheels are not truly round, and it the journals are not we f 
lubricated, it may greatly exceed 10 or 12 lbs. See p 374 e. 

Gauge 4 ft 8 1-2 ins. J 

Loads in tons of 2240 lbs (exclusive of locomotive and tender). 


Class. 

* 

Service. * 

Type. 

On a grade of 

0 per 
center 
0 ft per 
mile. 

% per 
cent = 
26.4 ft 
per 
mile. 

1 per 
cent — 
52.8 ft 
per 
mile. 

per 
cent = 
79.2 ft 
per 
mile. 

2 per 
cent = 
105.6 ft 
per 
mile. 

2 ^ per 
cent = 
132 ft 
per 
mile. 

3 per 
cent — 
158.4 ft 
per 
mile. 

8—14—C 

P ) 


( 500 

220 

130 

85 

60 

45 

35 

8—20—C 

P P l 

“Arne- 

■{ 1000 

435 

255 

170 

125 

95 

75 

8—30—C 

“ ) 


( 1550 

675 

395 

265 

200 

150 

115 

10—26—1) 

F M ) 


1685 

740 

435 

305 

220 

175 

135 

10—28— D 

“ y 

“ 10- 

■{ 1830 

7*0 

460 

320 

235 

185 

150 

10—32 -D 

“) 


( 2090 

915 

540 

375 

270 

215 

175 

8—18—D 

F 1 


1 930 

405 

240 

160 

120 

95 

75 

8—26—D 

“ [ 

4 ‘ MO- 

friil ** 

-{ 1830 

800 

475 

330 

245 

195 

155 

8—32—1) 

“ i 


(2330 

1026 

610 

412 

315 

245 

195 

10—34— K 

“ i 

‘•Con- 

/ 2560 

1130 

670 

465 

350 

275 

220 

10—36—E 

“ ; 

solidu- 

X 2740 

1205 

720 

495 

370 

290 

235 

4—26—C 

S 

Lion 

1635 

720 

430 

300 

225 

175 

140 

6—28 — 1) 

tb 


1865 

820 

490 

340 

255 

200 

100 


ei 

:i 

hi 

I 

1 

1 ? 

f 

r 

ri 

8 

I 

a 

t 


Gauge 3 ft. 

Loads in tons of 2240 lbs (exclusive of locomotive and tender). 


Class. 

* 

Service. » 

Type. 

J 

Oil a grade of i 

0 per 
cent — 
0 ft 
per 
mile. 

H per 
cent = 
26.4 ft 
per 
mile. 

1 per 

cent — 
52.8 ft 
per 
mile. 

114 per 
cent — 
79.2 ft 
per 
mile. 

2 per 
cent — 
105.6 ft 
per 
mile. 

2^ per 
cent— 
132 ft 
per 
• mile. 

3 per 
cent = 

158.4 ft 
per 
mile. 

8—18—C 

P 

“ Ame- 

050 

252 

148 

103 

73 

55 

43 

8—22—C 

(( 

rican * 

790 

308 

182 

123 

90 

68 

53 

8—20— D 

F 

“ 10- 

900 

353 

215 

145 

112 

83 

66 

8—22—1) 

(t 


990 

392 

235 

162 

120 

92 

74 

10-24—E 

U 

44 Mo¬ 
rrill " 

noo 

460 

275 

191 

142 

116 

89 

10—26—E 

(4 


1450 

580 

350 

243 

181 

142 

114 

4—18—C 

S 

‘ L/on- 

825 

335 

205 

145 

110 

90 

75 

6—22—D 

bt 

tion ” 

1320 

540 

330 

235 

180 

145 

120 


* See foot note, p. 809. 


















































on 


he tractive 
©we** of a loco- 
otive, in lbs 


LOCOMOTIVES. &()9 

Square of diam of .. Single length of . . ^ verago steam pres- 
oue piston in ins. * stroke in ins. ^ sure in the cylinders 
__ in lbs per sq inch 

Diameter of driving-wheel in inches 


It it does, the tractive power 


3$,, 


rovided the result does not exceed the adhesion, 
equal to the adhesion. 

The initial steam pressure in the cylinders is always less than the 
liler pressure; and the disproportion increases with the speed. Thus, at 8 or 10 
iles an hour, the boiler pressure may be about 110 lbs per square inch; and the 
Under pressure from 90 to 100 lbs, while at a speed of 30 or 40 miles, the propor- 
) e on may be as 110 to 00 or 70 lbs. The average cylinder pressure is ascertained by 
eansof an indicator applied to the cylinder; and its proportion to the initial 
•essure depeuds upon how early in the stroke the supply of steam from boiler to 
“blinder is cut off; or, in other words, upon the extent to which the steam is used 
®: pansively . 

The power and speed of locomotives, and their consump¬ 
tion ot fuel and water, vary greatly with circumstances, such as grades and 
irvature; condition of track and rolling stock; number of cars in train: diam- 
ers, number and distance apart, of car wheels: manner of coupling the cars; skill 
locomotive runner and fireman, &c, &c. The following records of actual perform- 
lce will serve as indications: # 

Raid win engines. Anthracite passenger engine, class 8—28—C, 
American ” type, hauls 9 loaded passenger cars, 216 tons besides weight of engine 
id tender, 52 tons, up a grade 1771 ft long, averaging 107 ft per mile, at 10 miles 
ir hour. At the foot of the grade is a curve of 225 ft radius, on which the grade 
100 ft per mile. A similar engine hauls 4 passenger cars, 1 parlor car and 1 bag- 
ige car, 145 tons, engine and tender, 59 tons, 59 miles over a nearly level road, with 
w and easy curves, in 1J4 hours. Boiler pressure about 125 lbs per square inch, 
jur such runs consumed 10000 lbs of coal, llitiimiiious passenger en- 
ine, class 8—30—C, “American” type, hauls 4 passenger cars, 1 sleeper, and 3 
iggage and mail cars, 150 tons, engine and tender, 61.4 tons, 11.55 miles in 28 min¬ 
es, up continuous grades, mostly of about 70 It per mile, and over nearly continu- 
ts reversed curves of from 1° to 7°. Freight eng'ines, class 10—30—D, “10 
heel ” type, haul the following trains: with bituminous coal, 18 cars, 337 tons, 
igine and tender, 65 tons, up a grade of 79.2 ft per mile, with a curve of 819 ft 
dius, 885 ft long; 48 cars, 386 tons, engine and tender, 65 tons, up a grade of 62 ft 
;r mile, with 4° curves ; 40 cars, 785 tons, engine and tender, 65 tons, up a grade 
21 ft per mile, on 5° curves. Freight engines, class 8—26—D, “Mogul” 
pe, haul 45 cars, 300 tons, engine and tender, 55 tons, up grades of 83 ft, per mile, 
ith a 2° curve; starting on the grade; boiler pressure about 130‘lbs; also, 37 cars, 
15 tons, engine and tender, 55 tons, up a grade of 85 ft per mile, with curves of 9° 
id 10°. Freight engines, class 10—34—E, “ Consolidation ” type, haul 90 
,rs, about 2000 tons, engine and tender, 68 tons, 45.5 miles in 4 hours 21 minutes, 
T er a nearly level road with easy curves, consuming 1.8 to 2.7 lbs of bituminous 
»al per loaded car per mile; also,with anthracite coal,33 cars, 264 tons,engine and 
nder, 68 tons, up a grade of 96 ft per mile, 12 miles long, with many curves of 573 
radius, at from 10 to 20 miles per hour; also, with anthracite, 25 loaded 4-wheel 
ml cars, 235 tons, engine and tender, 68 tons, up a grade ot 126 ft per mile. 
Central Pacific It R, 1883. Engines with 4 drivers, 56 ins diam, cylinders 
X 24 ins. Weight on drivers, 47550 lbs = 21.25 tons; weight of engine and ten- 
>r, in working order, 133000 lbs = 59.4 tons. Average train, 51 freight cars, 8f0 tons, 
axiroum grades, 52 ft per mile. Sharpest curve, 5°. Runs of 84 and 145 miles. 
)tal distance run, 3000 miles. Average speed about 10 miles per hour. Maximum 
>iler pressure 125 lbs per square inch. Bituminous coal consumed 73 lbs per train 
ile; or 30.8 train miles per ton of coal; or 11 ton miles per lb of coal. Water 
aporated 41 gallons per mile; 4.71 lbs per lb of coal. 

On the Boston Albany R R, passenger engines with 4 drivers 
% ins diam, cylinders 18 X 22 ins, with 6 passenger cars, run between Boston and 
>rtngfield, 97 miles, in about 4% hours, consuming 8000 lbs coal, 400 lbs wood for 
;hting, and evaporating 47800 lbs of water; or say 6 lbs per lb of coal. Boiler 
essure 130 to 150 lbs. 


* In the Baldwin Works classification (“8—30—C” etc) the first number (8, 10 etc) 

the total number of wheels of the loco. The second indicates arbitrarily the diam 
the cvls; thus. 12 means 9 ins; 14, 10; 16, 11; 18, 12; 20, 13; 22, 14; 24, 15; 26, 
;; 28, 17; 30, 18; 32, 19; 34, 20; and 36, 21 ins diam. The letter (C, D, or E) indi¬ 
ces the number (4, 6, or 8, respectively) of driving wheels. In the “Service” col- 
mn of opposite table, P means passenger; F, freight; M, mixed; and S, switching. 











810 


LOCOMOTIVES. 


Average of main line of Pliila A- Reading R R, 1884. 


Trains. 

Average 

No. 

of cars. 

Weight of train. 

Coal consumed. 

speed. 

Ca’-s. 

L. & T. 

Total. 

Kind. 

per 

train-mile 

Per 

ton-mil 




miles 
per hour. 


tons. 

tons. 

tons. 


lbs. 

lbs. 

Passenger. 

30 

8 

157 

66 

223 

A nth 

68.4 

.31 

Freight. 

Coal : 

30 

40 

835 

72 

907 

Waste 

103.8 

.11 

Up (empty). 

15 

165 

578 

72 

660 

“i 

138 


Down (full). 

12 

145 

1320 

72 

1392 

“ 5 



il 


e«l 

m 


« 0 ( 

»tf 




0 


w 


“ Coal consumed ” includes amount used in firing up. “ Ton ” = 2240 lbs. T1 
“ waste” is finely broken anthracite, or “culm,” the refuse of collieries, A - 
burned on the special grates designed for it by Mr. John E. Wootten. 

On the Pliila & Reading R R, in 1883, the cost of fuel per freiglit traiifri 
mile was, for anthracite, 12.4 cts; for “ culm,” 2.3 cts. Tne fast passenger engim 
ran 55 miles in 76 minutes, with 15 full passenger cars, consuming 53 lbs culm pt 
minute = 73 lbs per mile, and evaporating 55 gallons of water per minute = 76 ga i 
Ions per mile. 

The average consumption of bituminous coal by passenge 

engines, on 9 roads, in 1882, was 52.44 lbs per mile run ; greatest, 67 ; least, 34. 

Wood fuel. A ton (2240 lbs) of good anthracite or bituminous coal is aboi 
equal to 1^4 cords of good dry, hard, mixed woods (chiefly white oak); or to 2 con 
of such soft ones as hemlock, white, and common yellow pine. Much of the inferii 
bituminous coal of Illinois is hardly equal (per ton) to a cord of average wood. 

A cord is 4 X -A X 8 ft, or 128 cul) ft. A cord of good dry, white oak, (next to liicl 
ory, the best wood for fuel,) weighs 3500 lbs or 1.563 tons. Dry hemlock, white, < ' 
common yellow pine, (all of them inferior for fuel,) about .9 ton. Perfectly gree 
woods generally weigh about £ to x /± more than when partially dried for locomotivlr 
use; in other words, a cord of wood, in its partial drying, loses from x /± to ton o « 
water, and still contains a large quantity of it. Since this water causes a grer »t, 
waste of heat, green wood should never be used as fuel. The values of woods as fu< 
are in nearly the same proportion as their weights per cord when perfectly dry. 

The run of freight and passenger trains throughout the 1 
States is 20 to 40 miles per cord, and 30 to 60 miles per ton ; or say 40 to 80 pounc “ 
of coal per mile; but very heavy freight trains will burn from 100 to 200 poum ,| 
or more of coal per mile ; = 22.4 to 11.2 miles per ton. Much depends upo 
the adaptation of the engine to the kind of fuel used. A good coal-burner may be ha 
for wood, and vice versa; so that trials with the same engine may give very erroneon 
results as to the comparative merits of the two kinds of fuel. When wood is used,aboi 
.2 cord; or when coal, about x /z cord °f wood, must be used for kindling, and gettin 
up steam ready for running; and this item is the same for a long run as for a shoi 
one; so that long roads have in this respect an advantage over short ones, in ecoi 
omy of fuel. Wood has the disadvantage of emitting more sparks ; and is, moreove 
nearly twice as heavy as coal, for the performance of equal duty; and is, therefor* 
more expensive to handle. It also occupies 4 or 5 times as much space as coal. 

Up grades greatly increase the consumption of fuel. Thus, on a road 95 mile 
long, with grades mostly of less than 6 ft per mile, and with very few exceeding 1 
ft per mile, with coal trains of 734 tons descending, and 291 tons (empty)ascendin ,i 
at about 10 miles per hour each way, the coal consumption per 100 miles for ear 
ton of total train (including engine and tender) was 14.5 lbs descending, and 36.6 11 
ascending. 

On first-class roads a passenger engine will average about 3500< 
miles per year, or say 100 miles per day; a freight engine 25000 miles per yeai Pi 
or say 70 miles per day. 

Locomotive expenses per 100 miles run, will average about as follows 


I) 


i; 


Passenger. 

Fuel. 3.00 

Water. 1.00 

Oil, waste, &c .70 

Repairs . 4.00 

Engineer and fireman. 5.00 

Putting away, cleaning, and getting out 1 50 
Locomotive superintendence.30 


Freight. 
6.00 
2.00 
.90 
8.00 
6 00 
2.00 
.50 


*1 


$15.50 


$25.40 






































CARS. 


811 


This is all that is usually stated in annual reports of expenditures; but inasmuch 
an engine in active service, even under a judicious system of repairs, generally 
comes worthless, (except as old iron,) in say 16 years on an average,an additional 

lOWAUCfi of* illwillt K liPP ft /mi f lui tirw-f oncf ulwinf t-.. <2!UAA r... . - <1 Ky 


ow 

les 


(CAvt-pu its uiu iron,j m say id years on an average, an auuitional 
ance of about 6 per ct on the first cost, or about $500 to $800 = say $2 per 100 

i, should be made annually for depreciation of eacll engine. 

CAltS. 

sual dimensions, weights, anti capacities. Approx prices 
in 1886. For 4 ft 8)^ in gauge. 


III 

4c 


Length 

of bony. 

ft. 

Width. 

ft. 

Height 

above rail. 

ft. 

W'eight, 

empty. 

lbs. 

Nominal ca¬ 
pacity , in 

passengers, or 
lbs. 

Approx cost, 
1886. 

$ 

ssenger. 

44 to 52* 

9^ to 10 

14 

40000 to 50000 

50 to 60 

4000 to 6000 

in 

rlor. 

50 to till* 

9 *4 to 11 

« i 

5C000 to 60000 

30 io 40 

8000 to 12000 

Dei 

eper. 

*• * 

it 

44 

60000 to 72000 

25 to 50 

10000 to 15000 


vgage, mail, and 








xpress. 

45 in 55* 

UK 

if 

40000 to 50000 



8*1 

s and cattle. 

28 to 36>£ 8 to” 9% 

11 to 13t 

20000 to 28000 

40000 to 50000 

400 to 550 


ndola. 

• 4 

8 to 9)4 

6 to 7^t 

16000 to 22000 

44 

300 to 450 


.ttoim. 


%% to 9 

3^ to 4 y 2 i 

14000 to 19000 

30000 to 40000 

4 4 

4, 

il, 8 wheels. 

22 to 35 

8 to 9 

7 to 

18000 to 22000 

35000 to 50000 

400 to 450 


‘ 4 wheels. 

12 to 16 

6 to 8 

to 7 t 

5600 to 9600 

11000 to 20000 

200 to 250 

>rd 

tup. 

i 

14 

8 

6M 

9000 

18000 

100 


Add 6 ft in all for the two platforms. These are usually from 3 ft 6 ins to 4 ft above the rails 
Add from 1 to 2 ft for projecting brake rod and handle. 

Add about 1 ft for projecting brake rod and handle. 

[>n narrow gauge (3 ft and 3 % ft) roads, there is but little uniformity in 
• building. The dimensions and capacities of the cars are now not much less than 
io >se of corresponding cars for standard (4 ft 8)/£ ins) gauge ; while the weights and 
Sts are usually from 15 to 20 per cent less. The following are about the averages : 
fae 3senger, 45 ft long, 8)^ ft wide, $3000. Parlor, 45 ft long, 8)^ ft wide, $9000, 
ggage, &c, 35 ft long, 8)4 ft wide, $1800. Freight and coal, 24 to 30 ft long, 7 to 
K ft wide, $200 to $400. 

adi Dimensions, Arc, of iron-frame cars, built by the United States Tube 
ml King Stock Co; office, No 3 Broad St, New York. 


i 







Length. 

Width. 

Depth 

of body. 

W'eight, 

empty. 

Nominal 

capacity. 


ft. 

ft. 

ft. 

lbs. 

lbs. 


For 4 ft 8)4 ins gauge. 


and cattle. 

jdola. 

tform. 

1 , 8 wheels. 


c and cattle 

idola. 

tform.. 

1 , 8 wheels. 


33 H 

8 hi 

6)4 

21500 

60000 

34 to 34Mto S% 


16000 to 19000 

50000 to 60000 

34 

8 


14000 

60000 

34 Vi 

8 


18900 

60000 


For 3 ft gauge. 


28 

7 

6 

18000 

40000 

li 

44 

2 

10500 

44 

ii 

14 


9500 

44 

41 

44 

2 

10400 

44 


Cost. 


•‘So 

Q o 

§|* 

■5 2 s 

t. o § 

, 

c8 -4J GO 
0> Cft u 
*- t- e« 
CC o 


ur or data respecting cars, &c, we are indebted to Jackson & Sharp Co and Dure & Co, Wilmington, 
Allison Mfg Co, makers of cars and dealers in wheels and axles, Phila; Harrisburg (Pa) Car 
Co ; Erie (Pa) Car Works, him ; Youngstown (O) Car Mfg Co ; Michigan Car Co, Detroit; Mid¬ 
town (Pa) Car Works ; J G Brill & Co, Phila.; Gilbert Car Mfg Co, Troy, N Y. 

L’he average life of a passenger car is about 16 years. Average annual 
'pairs, including painting, $300 to $700; for mail and express cars, $150 to $300; 
iglit cars, $75 to $150; 4-wheel coal cars, $20 to $30. On the Phila & Reading, in 
13. repairs were as follows: passenger cars,$0,156 per 100 passenger-miles; freight 
s, $0,133 per 100 ton-miles; coal cars, $0,078 per 100 ton-miles. 

\llowing 125 lbs per passenger, a full car-load of passengers (50 to 60 in number) 
uld weigh but from 6250 to 7500 lbs, or say 3 tons; while the cars themselves 
igh say 20 tons, or nearly 7 tons of dead load to 1 paying ton of 
tsseiigers. But, as a general rule, passenger trains are not more than halt¬ 
ed; thus making the proportion about 13 to 1. In England, the cars are heavier 
































































812 


CARS. 


in proportion to the number of passengers carried; and the dead load on some lin< _ 
is from 20 to 30 times the paying load. The above table shows that when frcig'li j 
Cars are loaded to their nominal capacity, there is but about 1-13 ton of <lca J 
load per ton of paying load ; or, with cars half loaded, 1 to 1. 

From the table, p 814, it will be seen that the average cost, in the United State 
of moving a passenger one mile is 2j^ times that of moving a ton of freight orT 
mile, while the receipts per passenger-mile are not quite double those per freigh L 
ton-mile. 

The resistance of cars to motion, on a level track, and with cars anj 

tradk in fair order, is usually taken at about from 6 to 8 lbs per ton of 2240 lb < 
With everything in perfect order, it may fall as low as 5, or even 4, lbs pertoi^ 
On the other hand, if the wheels are not truly round, and if the journals are nt 
well lubricated, it may greatly exceed 10 or 12 lbs. See p 374e. 

The wheels for passenger and freight cars are usually about 3 ’ 
to 33 ins diam ; and those of coal cars 26 to 28. The cast-iron wheels made by Messi 
A. Whitney & Sons, of Philada, weigh as follows, per single wheel. Cone, 1 in 32: 


Diameter 

or 

Wheels. 

Usual Gauge, 

Narrow Gauge. 

Single Plate. 

Double Plate. 

Single Plate. 

Double Plate. 

4 in 
Tread. 

4% in 
Tread. 

4 in 
Tread. 

4 y x in 
Tread. 

3 ]4 in 

Tread. 

3$£ in 
Tread. 

3f£ in 
Tread. 

3^ in 
Tread. 

20 in. 

fits 

260 

fits 

270 

fits 

lbs 

fils 

/ 200 
(250 
245 
(265 

< 320 
(330 
( 285 
■{ 335 
(.350 
( 320 

< 365 
(400 
(405 
( 445 

tbs 

205 

255 

255' 

275) 

330 

340 

300 

350 

360 

335' 

380 

415 

425) 

465 


Bis 

Bis 

22 in. 

24 in. 

26 in. 

28 in. 

30 in. 

33 in. 





(345 

1355 

/ 360 
(375 

/ 405 
( 440 

(445 

(485 

(510 

(545 

585 

685 

360) 

370/ 

380 ( 
395/ 

425) 
460 f 

470 ( 
510/ 

540 1 
575 / 

640 

365 

380 

440 

480 

(525 
■{ 545 

I 560 

380 

400 

460 

505 

555) 

575 y 

590 j 

340 

345 

380 

460 

350 

360 

395 

470 

36 in. 





38 in. 
















The weights by other makers do not differ from these materially. 

The diam of car or engine wheels does not include the flanges; but is th. 
least diam from tread to tread. 

I'rice of cast chilled wheels, in Phila, in 1886, about 2 cts per lb. 

(■ood chilled cast wheels. :t:t ins diam, will ran about 50,00< 
miles; usually about 40.000; rarely 60,000 or much higher. 42-inch wheels (rareU 
used) average about one-third higher. On first-class roads about 1 to Vyi per cen 
of all the car-wheels are cracked or broken annually. 

In the steel-tired “paper car-wheels” of the Allen Paper Car-Wheel Co 
office 240 Broadway, New York, the hub is of cast-iron or steel'; the “center,” oi 
main body of the wheel, of compressed straw-board confined between two circulai 
plates of rolled iron; and the tire is or rolled steel. The bolts, which confine the 
iron plates to the paper center, pass also through flanges on the outside of the hut 
and the inside of the tire. To secure elasticity, the bolt-holes in the flange of tin 
tire are slightly elongated, and the circular iron pla tes are made of a little less diair 
than the inside of the tire, so that the plates and tire do not come into contact, bui 
the weight of car and load is transferred from the hubs to the tires through tht 
paper centers only. These wheels are now largely used on passenger, parlor, anc 
sleeping cars, and on the trucks of locomotives. The principal sizes for 4 ft 8% in 


- TV S 



































































CARS. 


813 


I,: paper w 

veighs 1085 


- oxl" 
,'it r 

■s m v 
"IbiJ 


-l «r 

n 32; 


,j„S?auge are 33 inch and 42 inch; treads, for either diam, 3% and 4^ ins. A 42-inch 
heel costs, 1886, from $85 to $95, according to width of tread; and 
1085 tbs; of which 165 Tbs are paper; 550 Tbs tire; 160 Tbs side-plates; 180 
bs hub; and 30 lbs bolts. The steel tires on 42-incli wheels run about 100,000 miles 
etween turuings, and about 400,000 miles before having to be abandoned. A new 
ire is then placed upon the old center, at a cost of about $55. Allowance must be 
lade for the fact that these wheels are generally under sleeping or parlor cars or 
r8t-class passenger cars, on through trains which make few stops; and that they 
re therefore subjected to less of the destructive action of the brakes than are coiti¬ 
on wheels. Besides, the great majority of the latter are used under freight cars, 
here they have rough usage, due to the inferior character of the springs on such 
ars, &c. The average cost of turning steel-tired wheels is about $1 to $1.25 per 
[, nnum per pair. Axles are said to run several times longer with paper wheels than 
0 ;ith cast-iron ones. 

Wheels of cast- and of wrouglit-iron with steel-tires are being largely used ex- 
erimentally under high-class cars. They run longer than chilled cast-iron wheels, 
ut are more costly. 

Axles. Standard dimensions adopted by the Master Car Builders’ and 
laster Mechanics’ Associations in 1879: Length, total, 6 ft 11*4 ins; between 
mbs, 4 ft 0% in ; each wheel-seat, 7 ins ; each journal, 7 ins. I>iam, at middle, 
% ins; at'hubs, 4% ins; at journals, 3% ins. Weight, finished, 347 lbs per 
ixle. The diam at middle was increased to 4V£ ins by the Master Car Builders’ 
Association in 1884. This change of course increased the weight slightly. 
j s Cost of wrought-iron axles, hammered or rolled. 1886, 2% to 3 cts per lb. 

••4 For Standard Railway Time, see p 396. 






814 RAILROAD STATISTICS. 


BAILROAD STATISTICS. 


Art. 1. In the following, most of the figures for 1880 are based upon the U. S 
Census for that year; those for 1884, upon Poor’s Manual. 

IX THE UNITED STATES. 


1880. 

1884. 



7174 

3977 

87S01 

125152 

5191 

. 

71403 


12335 


12282 


46S00 


4112 

. 

17412 

24587 

12330 

17993 

4475 

5911 

375312 

798399 

80138 

4761 


418 


51561 

55330 

4530 

6925 

65392 

70143 

368514 

357367 

1641 

1653 

4740 

4018 

230 

429 

6611 

6100 

.0251 

.0236 

.0129 

.0112 

.2483 

.2709 ' 

.7169 

.6588 

.0348 

.0703 

.1136 

.1102 

4019 

3970 

.0076 


.0171 

.. 

.6078 

.6508 

.0504 

.0385 


C — 


Plant. 

Miles built in one year.. 

(In 1881, 9789; in 1882, 11596.) 

Miles in operation. 

Gauge, Percentage of all, 1880. 

3 ft. 5. 

4 ft % l /2 ins.66.S % 

4 ft 9 ins.11. ' 

5 ft (Southern gauge).11. a 

Cost of road, exclusive of rolling stock, « 

per mile, in dollars. 

total, in millions of dollars. 

Rolling stock in operation. 

Number of locomotives. 

“ passenger cars. 

“ baggage, mail, and express cars. 

“ freight cars. . 

“ other “ . 

Cost of rolling stock, 

per mile of road, in dollars. 

total, in millions of dollars. 

Cost of road and equipment. 

per mile, in dollars. 

total, in millions of dollars. 


it 

t« 


Operation. 

For one year. 

Passengers carried one mile, per mile of road.. 

Tons of freight carried one mile, per mile of road. 

Gross earnings, 

per mile of road, from passengers, dollars. 

“ “ freight. “ .. 

“ “ mails, &c. “ . 

“ total. “ . 

perpassenger-mile,from passengers, “ . 

“ ton-mile, from freight. “ . 

passenger earnings -s- total earnings.. 

freight “ -f- “ . 

mail, Ac, “ “ . 

gross earnings -f- total investment.. 

Expenses. (For details, see Art 3.) 

per mile of road...dollars. 

cost of moving freight, per ton-mile. “ 

(Penna It R, 1883, $.0056.) 
cost of moving passengers, per passenger-mile.. “ 
(Penna R R, 1883, $.0163.) 

expenses -s- gross earnings. 

Xet earnings. 

Net earnings -r- total investment. 





































































RAILROAD STATISTICS. 


Art. 2. TNITED STATES BY DIVISIONS, 1SS4. 


815 


St 

-52 



Eastern 

States. 

Middle 

States. 

Southern 

States. 

Western 

States. 

Pacific 

States. 

To tal, 
U. S. 

Plant. 

iles in operation. 

6405 

18256 

19826 

72704 

7961 

125152 

ost of road and equipment 
per mile, dollars. 

52166 

92306 

42338 

48418 

68549 

55330 

Operation. 

For one year. 

ross earnings per mile, 

9142 

12177 

3524 

5199 

4348 

6100 

xpenses per mile, $. 

6564 

7951 

2322 

3339 

2016 

3970 

xpenses-^Gross earnings. 

.718 

.653 

.659 

.642 

.601 

.651 


Art. 3. Items of total annual expenses for maintenance and opera- 
>n of all the railroads of the United States in 1880. 


■pairs of road-bed and track. 

■liewals of rails (total $17243950). 
‘ “ ties (total $10741577)... 

■pairs of bridges. 

“ buildings. 

“ fences, crossings, &c. 

legraph expenses. 

.. 


30 liutenance of road and real estate. 


25 


pairs, &c, of locomotives. 

“ “ passenger, baggage, and mail cars. 

“ “ freight cars. 


pairs, &c, of rolling stock. 

Including renewals and additions.) 


US 


ssenger train expenses. 

eight “ “ . 

el for locomotives. 

iter supply, oil, and waste. 

iges of locomotive runners and firemen. 

ents and station service and supplies. 

® laries of officers and clerks.-. 

vertising, insurance, legal expenses, stationery, and 

printing. 

mages to persons and property. 

ndries. 


« 


„„ inning and general expenses, 
.gregate annual expenses.. 


$ per 
mile 
of road. 

per cent 
of total. 

per cent 
of earn¬ 
ings. 

451 

11.23 

6.82 

197 

4.89 

2.97 

122 

3.04 

1.85 

102 

• 2.55 

1.55 

87 

2.17 

1.32 

17 

.42 

.25 

41 

1.01 

.62 

152 

3.77 

2.29 

1169 

29.08 

17.67 

249 

6.19 

3.76 

120 

2.99 

1.82 

257 

6.40 

3.89 

.626 

15.58 

9.47 




137* 

3.41 

2.07 

330 

8.21 

4.99 

374 

9.31 

5.66 

70 

1.74 

1.06 

310 

7.72 

4.69 

451 

11.23 

6.82 

139 

3.46 

2.10 

123 

3.06 

1.87 

40 

.98 

.60 

250 

6.22 

3.78 

2224 

55.34 

33.64 

4019 

100.00 

60.78 





Each of these items is. however, subject to great variation, not. only on diff roads, 
t on the same road, from year to year. A road witn many bridges, deep cuts, high 























































































RAILROAD STATISTICS. 


816 


ambkts, &c, to keep in repair, will have heavier maintenance of way than one wlii 
has but few; and this item may be but small one year, and twice as great the ne; 
Fuel may be cheap on one road, and dear on another; thus materially affecting t 
item of motive power. And so with the other items. Sometimes maintenance 
way exceeds motive power and cars together; at others, conducting transportati 
is fully half the total expense. 

The total annual expenses on railroads in the United State- 

usually range between (55 and 130 cents per train mile; that is, per mile actual li 
run by trains. Also, between 1 and 2 cents per ton of freight, and per passengi 
•tarried one mile. When a road does a very large business, and of such a charact 
that the trains may be heavy, and the cars full, (as in coal-carrying roads,) the ♦ 
pense per train mile becomes large; but that per ton or passenger small; and vi It 
versa, although on coal roads half the train miles are with empty cars. 

Art. 4. Ciross annual earnings per mile, per passengi 
mile, and per ton mile, of some of the principal U S rai 
roads in 1SSO. 

s 


Length 

miles. 

From 
passrs per 
mile of 
road. 

From 
passrs per 
passr 
mile. 

From fi t 
per mile 
of road. 

n 

From t i) 
per to „ 
mile. 

Pennsylvania R R. 

New York Central & Hudson River... 

Baltimore & Ohio. 

Central Pacific. 

Chicago, Burlington, & Quincy. 

Philadelphia & Reading. 

Union Pacific.. 

Wabash, St Louis, <te Pacific. 

Atchison, Topeka, & Santa Fe. 

Average of United States. 

1806 

994 

1487 

2447 

1805 

780 

1215 

1730 

1398 

87801 

$4700 

6651 

1812 

2237 

1532 

3429 

2624 

1220 

1144 

1641 

$.0242 

.0200 

.0206 

.0303 

.0240 

.0201 

.0320 

.0271 

.0606 

.0251 

$15615 

21794 

10310 

4577 

7202 

17200 

7154 

4382 

3974 

4740 

$.008 
.008 
.008 
.024 ' 
.011 i 
.016 ‘ 
.019 ' 
.008 1 
.020 ' 
.012 r 

1 

Art. 5. Annual earnings and expenses of some of the prii 1 
cipal railroads of the United States in 1SS0. » 


Length 

miles. 

Gross 
earnings 
per mile 
of road. 

Expenses 
per mile 
of road. 

Ex pern 
V gros 
earnini 

Pennsylvania R R. 

New York Central & Hudson River. 

Baltimore & Ohio. 

Central Pacific. 

Chicago, Burlington, & Quincy. 

Philadelphia & Reading. 

Union Pacific.?.. 

Wabash, St Louis, & Pacific.. 

Atchison. Topeka, & Santa Fe. 

Total, United States. 

1806 

994 

1487 

2447 

1805 

780 

1215 

1730 

1398 

87801 

$20,315 

28,445 

12,122 

6,814 

8.734 

20.629 

9,778 

5,602 

5.118 

6,611 

$12,267 

17,969 

7,035 

3,340 

4,454 

11,754 

4,507 

3,942 

2.408 

4,019 

.585 
.609 
.571 
.470 
.497 
.568 
.42f 
.67 i’ 
.458 
.608 


The following table of expenses in past years will serve for comparison with t 
above. 



















































t nex 
tl 


i! :6( 


RAILROAD STATISTICS. 


817 


Table of Annual Expenses of some U S Railroads.* 


Names of Companies 


« 

a 

<4 


: '08'* 
H8 


:# 

,'i24 


nil' 

iie 

m 


a 

« 

« 

(t 

« 


i»! 


“ 1867 

ena & Chicago, 1859 



i»iigh Valley, I860., 

1862 

1868 and 1869....about. 

1872 

imore & Ohio, main stem, 1859 
“ “ “ “ I860 

“ “ “ “ 1865. 

“ “ “ “ 1866., 

“ “ “ “ 1872. 

t Tennessee & Georgia, 1872. 
nphis & Charleston, 1860. 

f ial rgia Central, 1872. .... 

I' 1 na Central, main line from Phila to Pittsburg, o58 miles, 

1859, exclusive of State tonnage tax. 

1860, 

1861, 

1868, tonnage tax repealed 

1869, “ ■ •« •“ . J ...about... 

1872, 

la & Reading, 1859. 

“ 1860 

“ 1868, 365 miles of main road and branches... 

“ 1869 

“ 1872 

th Pennsylvania, 1860, 54 miles long. 

« 1862. 

“ 1867. 

" 1868. 

“ 1872. 

necticut; average of all the railroads, 1861. 
jisachusetts; “ “ “ “ 1861. 


averages of 19 years previous 


“ 1860 

la, Wilmington & Baltimore, main stem, 1859 
“ “ “ “ 1860. 

“ “ “ “ 1861. 

“ “ “ “ 1867.. 


v York; all the It R in the State, average,! 1859 
‘ “ “ “ “ “ 1861 


v Jersey R K and Transportation, 1861 
lisville & Nashville, 1861. 
la & West Chester, 1861, 27 miles 
“ 1862 


la, Germantown <fc Norristown, 1861, 20 miles... 
“ “ 1862.. 


1867. 

York & Erie, 1861 

“ “ 1867, with its branches, 784 miles in all. 

York Central, 1861. .. 

“ *• 1867, with its branches, 696 miles in all.... 

lish R R, averages for 1856-7-8 .. 
h “ 

(« « « « « u 


Annual reports often omit the lengths of the roads and branches; and as these frequently vary 
|t year to year, it is possible that the table may contain some errors in the first, column. 

528 miles in operation. Total exps equalled 1.56 cts per passenger or ton carried 1 mile. 

| ht of cars, equal to 1.1!) tons jier passenger; and to 1.71 tons per ton of freight. 

53 


Dead 



































































































818 


RAILROAD STATISTICS 


Art. 6 . Statistics of several U. S. narrow-gange railroads for 188 
from Poor’s Manual. 


-_- 

Gauge, Feet. 

Length, Miles. 

Rolling Stock. 

Cost of road and 

equipment per mile. 

Gross Annual Earn¬ 

ings per mile of 
road. 

Annual Expenses 

per mile of road. 

Locomotives. 

Passr Cars. 

Mail, &c, Cars. 

Freight Cars. 

Bridgton & Saco River, Maine...'. 

2 

16 

2 

2 

1 

16 

$12167 

$1112 

$ 834 

Profile & Franconia Notch, N H. 

3 

14 

3 

7 

.... 

6 

15430 

1346 

640 


3 

6 





13645 

2868 

264 2 

Bradford, Bordell & Kinzua, Pa. 

3 

3!) 

5 

5 

2 

69 

14922 

1793 

1717 

Denver <fc Rio Grande. Col. 

3 

1685 

239 

115 

71 

5676 

35000 

3519 

2573 



The weights of the steel rails used on narrow gauge roads vary from 30 to 40 
per yard. 

Art. 7. Miles of railroad in the world at the close of 1883. Amer 

(U S, 125152), 143335. Europe, 114313. Total, 279856. 

In Great Britain, in 1883, there were 16000 miles of railroad. Gross ea 
ings for half year, $10130 per mile. Expenses for half year, §5364 per mile. 1 
penses -*• Gross earnings = .53. 


































GLOSSARY OF TERMS, 


819 


or 1| I 


to 401 
Ameri 


■-eirj, 

ile. ' 


GLOSSARY OF TERMS. 


s bacus; the flat square member on top of a column. 

« bsciss or abscissa; any portion of the axis of a curve, from the vertex to any point from which 
,j. ie leaves the axis at right augles, and extends to meet the curve itself; said line beiug called an 
t; nate. An absciss and ordinate together are called co-ordinates, 
ipclivity ; an upward slope, or ascent of ground, Ac. 
lit; a horizontal passage into a mine, Ac. 

dze; a well-known curved cutting instrument, for dressing or chipping horizontal surfaces, 
<i\'.ternating motion; up and down, or backward and forward, instead of revolving, Ac. 
ngle-bead, or plaster bead: a bead nailed to projecting augles in rooms, to protect the plaster on 
r edges from injury. 

' ngle-block; a triangular block against which the ends of the braces and counters abut in a Howe 
Jige. 

* ngular velocity. See p 365. 

•* meal; to toughen some of the metals, glass, Ac, by first heating them, and then causing them to 
very slowly. This process however lessens the tensile strength. 

titiclinal axis; in geology ; a line from which the strata of rocks slope away downward in.oppo- 
directions, like the slates on the roof of a house; the ridge of the roof representing the axis. 
yex; a point in either chord of a truss, where two web members meet. 

yron; a covering of timber, stone, or metal, to protect a surface against the action of water flow- 
over it. Has many other meanings. 

'bor. See Journal. 

'chitrave; that part of an entablature which is next above the columns. Applies also when there 
uo columns. Also, the mouldings arouud the sides and tops of doors and windows, attached to 
er the inner or outer face of the wall. 

rris; a sharp edge formed by any two surfaces which meet at an angle. The edges of a brick are 
ses. 

shier; a facing of cut stone, applied to a backing of rubble or rough masonry, or brickwork, 
strayal; a small moulding, about semi-circular or semi-elliptic, and either plain or ornamented by 
dug. 

t is ; an imaginary line passing through a body, which may be supposed to revolve around it; as 
|diam of a sphere. Any piece that passes through and supports a body which revolves ; iu which 
is it is called an axle, or shaft. 
tie-box. See Journal-box. 

xletree; an axle which remains fixed while the wheel revolves around It, as in wagons, fee. 
zimuth. The azimuth of a body is that arc of the horizon that is included between the meridian 
le at the given place, and another great circle passing through the body. 

acki.ng; the rough masonry of a wall faced with finer work. Earth deposited behind a retaining- 
1 , &c. 

alance-beams; the long top beams of lock-gates, by which they are pushed open or shut. 
a Ik; a large beam of timber. 

allast; broken stone, sand or gravel, Ac, on which railroad cross-ties are laid. 
all-cock; a cistern valve at one end of a lever, at the other end of which is a floating ball. The 
rises and falls with the water in the cistern; and thus opens or shuts the valve. 
all-valve. See Valve. 

argeboards; boards nailed against the outer face of a wall, along the slopes of a gable end of a 
se, to hide the rafters, &c ; and to make a neat finish. 
ascule bridge ; a hinged lift-bridge furnished with a counterpoise. 

atter, (sometimes affectedly batir.) or talus ; the sloping backward of a face of masonry. 
ay; on bridges, Ac, sometimes a panel; sometimes a span. 
ead; an ornament either composed of a straight cylindrical rod ; or carved or cast iu that shape 
iny surface. 

earing ; the course by a compass. The span or length in the clear between the points of support 
i beam, Ac. The points of support themselves of a beam, shaft, axle, pivot, Ac. 
ed-mouldings; ornamental mouldings on the lower face of a projecting cornice, Ac. 
ed-plate; a large plate of iron laid as a foundation for something to rest on. 

' •eetle; a heavy wooden rammer, such as pavers use. 
t ell-crank. See Crank. 

tench-mark; a level mark cut at the foot of a tree for future reference, as being more permanent 
n ?i stftkft. 

term, or herme; a horizontal surface, as irfor a pathway, and forming a kind of step along the face 
sloping ground. In canals, the level top of the embankment opposite and corresponding to the 
rpath is called the berm. . 

Bessemer steel is formed by forcing air into a mass of melted cast iron ; by which means the excess 
carbon in the iron is separated from it, until only enough remains to constitute cast steel. The 
■bon is chemically united with the steel, but mechanically with the iron. 

Beton; concrete of hydraulic cement, with broken stone and bricks, gravel, Ac. 

Bevel; the slope formed by trimming away a sharp edge, as of a board, Ac. Edges of common 
iwing rulers and scales are usually bevelled. See 13, p 613. 

Bevel year; cog-wheels with teeth so formed that the wheels can work into each other at an angle. 
Bilge; the nearly flat part of the bottom of a ship on each side of the keel. Also, the swelled part 
a barrel, Ac. To bilge is to spring a leak in the bilge, or to be broken there. 

Bitts; the small boring points used with a brace. 

Blast-pipes; iu a locomotive; those through which the waste steam passes from the cylinder into 
- smoke-pipe, and thus creates an artificial draft in the chimney, or smoke-pipe. 

Boasting ; dressing stone with a broad chisel called a boaster, and mallet. The boaster gives a 
toother surface after the use of the point, or the narrow chisel called a tool. 

Body ■ the thickness of a lubricant or other liquid. Also, the measure of that thickness, expressed 
the number of seconds in which a given quantity of the oil, at a given temperature, flows through 
jiven aperture. 








■820 


GLOSSARY OF TERMS 


Bolster; a timber, or a thick iron plate, placed between the end of a bridge audits seaton 
abutment. 

Bond; the disposing of the blooks of stone or brickwork so as to form the whole into a firm st 
ture, by a judicious overlapping of each other, so as to break joint. Applies also to timber, Ac 
various ways. 

Bonnet; a cap over the end of a pipe, &c. A cast-iron plate bolted down as a covering over 
aperture. 

Bore; inner diameter of a hollow cyliuder. 

Borrow-pit; a pit dug iu order to obtain material for an embankment. 

Boss; au increase of the diameter at auy part of a shaft for any purpose. A projection in sh 
of a segment of a sphere, or somewhat so, whether for use or for ornament; often carved, or cast 

Box-drain; a square or rectangular drain of masonry or timber, under a railroad, Ac. 

Brace ; a kind of curved haudle used for boring holes with bitts. The head of the brace renin 
stationary, being pressed against by the body of the person usiug it, while the other part with 
bitt is turned round by his hand. Also, au incliued beam, bar, or strut, for sustaining compressi 

Bracket; a projecting piece of board. Ac, frequently triangular, the vertical leg attached to 
face of a wall, and the horizontal one supporting a shelf, Ac. Often made in ornamental sha; es 
supporting busts, clocks, Ac. Also, the supports for shafting ; as pendent, wall, and pedestal brack' 

Brake ; an arrangement for preventing or diminishing motion by means of friction. The frict 
is usually applied at the circumference of a revolving wheel, by means of levers. On railroads, 
car-brakes should be worked by steam, as those of Loughridge, Westinghouse, and Creamer. A 
such a haudle as that of a common pump. 

Brass is composed of copper and zinc. 

Brasses; fittings of brass iu many plummer-blocks, and in other positions, for diminishing 
friction of revolving journals which rest upou them. 

Braze; to unite pieces of iron, copper, or brass, by means of a hard solder, called spelter sold 
and composed, like brass, of copper and zinc, but in other proportions. 

Break joint ; to so overlap pieces that the joints shall not occur at the same place, and thus j 
duce a bad bond. 

Breast-summer; a beam of wood, irou, or stone, supporting a wall over a door or other openii 
a kind of lintel. 

Breast-wall; one built to prevent the falling of a vertical face cut into the natural soil; in ( 
tinction to a retaining-wall or revetment, which is built to sustain earth deposited behind it. 

- Breech; the hind part of a cannon. Ac. 

Bridge- or bridge-piece, or bridge bar; a narrow strip placed across an opening, for support; 
something without closing too much of the opening. 

Bronze is composed of copper and tin. 

Bulkhead; on ships, Ac, the timber partitions across them. Also, a long face of wharf para 
to the stream. 

Buoy ; a floating body, fastened by a chain or rope to some sunk body, as a guide for finding 
latter. Sometimes also used to indicate chaunels. shoals, rocks, Ac. 

Burnish ; to polish by rubbing; chiefly applies to metals. 

Bush; to line a circular hole by a ring of metal, to prevent the hole from wearing larger. A 
when a piece is cut out, and another piece neatly inserted into the cavity, the last piece is sometii 
said to be bushed in; sometimes it is called a plug. 

Butt-joint; one in which the ends of the two pieces abut together without overlapping, and 
joined by one or more separate pieces called covers or welts, which reach across the joint and 
fastened to both pieces. 

Buttress; a vertical projecting piece of brickwork or masonry, built in front of a wall 
strengthen it. 

Caisson; a large wooden box with sides that may be detached and floated away. 


:l 

1 


!l 

l: 

it 

i 

k 

(i 


Caliber; the inner diameter, or bore. 

Calipers ; compasses or dividers with curved legs, for measuring outside and inside diameters. 

Calk, or caulk; to fill seams or joints with something to prevent leaking. 

Calking iron, a tool Tor forciug calking into a joint. 

Camb, or cam, or t riper : a piece fixed upon a revolving shaft in such a manner as to produce i 
alternating or reciprocating motion in something iu contact with the cam. An eccentric. 

Camber ; a slight upward curve given to a beam or truss, to allow for settling. 

Camel; a kind of barges or hollow floating vessels, which, when filled with water, are fastenet 
the sides of a ship; and the water being then pumped out, they rise by their buoyancy ; and lift I 
shin so that she can float in shallower water. 

Cantilevers ; projecting pieces for supporting an upper balcony. Ac. 

Cants, rims, or shroudings ; the pieces forming the ends of the buckets of water-wheels, to prei 4 
the water from spilling endwise. 

Capstan; a long hollow rope-drum surrounding a strong vertical pivot, upon the head of whic' 
rests, and around which it turns. Its topis a thick projecting circular piece, having holes around 
outer edge or circumference, for the insertion of the ends of levers ; or capstan-bars. It. is a kind 
vertical windlass. 

Case-harden ; to convert the outer surface of wrought iron into steel, by heating it while in cont . 
w'th charcoal. 

Casemate; in fortification ; the small apartment in which a cannon stands. 

Castors : rollers usually combined with swivels; as those used under heavy furniture, Ac. 

Causeway ; a raised footway or roadway. 

Cavetto : a moulding consisting of a receding quadrant of a circle. 

Cementation ; the process of converting wrought iron into steel, by heating it in contact with ch i 
coal. This process produces blisters on the steel bars ; hence blister steel. These are removed, 1 1 
the steel compacted, by reheatiug it, aud then subjecting it to a tilt-hammer. It is then tilted st ' 
or shear steel. Or if the blister steel is broken up; remelted; and then run into ingots or blocks 
is called cast, or ingot steel; which is harder and closer-grained than tilted steel. It may be softet i 
aud thus become less brittle, by auuealiug. The iugots may be converted into bars by either roll ] 
or hammeriug, the same as shear aud blister. 

Center; the supports of au arch while being built. 

Center of gravity. See p. 347, Ac. 

Center of gyration. See Radius of Gyration p 440. 




GLOSSARY OF TERMS. 


821 


1.1 

"«4 


»ii 


St SI 


atom 

\ter of oscillation, or of vibration. See Rem 2, of Pendulums, p 365. 

111 liter of percussion, in a moving body, is that point which would strike an opposing body with 
^fer force than any other point would. If the opposing body is immovable, it will receive all the 
of a rigid moving body which strikes with its center of percussion. See Pendulum, page 365. 
i spool; a shallow well for receiving waste water, filth, &c. 

xnifer ; means much the same as bevel; but applies more especially when two edges are cut away 
to form either a chamfer-groove, (see 14, p 613, of Trusses.) or a projecting sharp edge. 
b eks; two fiat parallel pieces confining something between tnem. See w, at 15, of Pigs 'Zl'/i, of 
fses, p 583. 

rt H tiling, chill-hardening, or chill-casting; giving great hardness to the outside of cast-iron, by 
ng it into a mould made of iron instead of wood. The iron mould causes the outside or skin of 
listing to cool very rapidly; and this for some unknown reason increases its hardness. This pro 
s frequently confounded with case-hardening. 

8 ock ; any piece used for filling up a chance hole, or vacancy. 

* nek; the arrangement attached to the revolving shaft, arbor, or mandril of a lathe, for holding 
hing to be turned. 

■I urn-drill; a long iron bar. with a cutting end of steel; much used in quarrying, and worked by 
;c ig it nd letting it fall. When worked by blows of a hammer or sledge it is called a jumper. 
ua, or cyma: a moulding nearly in shape of an S. When the upper part is coucave, it is called 
a recta ; when convex, a cima reversa. See page 151. 
ack valve. See Valves. 

amp ; a piece fastened by tongue and groove, transversely along the end of others, to keep them 
warping. A kiud of open collar, which, being closed by a clamp-screw, holds tight what it sur¬ 
ds. See Cramp. 

up boards ; short thin boards, shingle-shaped, and used instead of shingles. 
aw , a split provided at the end of an iron bar, or of a hammer, Ac, to take hold of the heads of 
t«) or spikes for drawing them out; as in a common claw-hammer. 

eat ; a piece merely bolted to another to serve as a support for something else; as at 7, 8, 10, 

613, of Trusses. Often used on shipboard for fastening ropes to, as at 11. Also a piece of 
1 nailed across two or more other boards, for holding them together, as is often done in tempo- 
8 doors, Ac. 

evis. See Shackle. 
ck. See Ratchet. 

it p; a fastening like that on the tops of the Y’s of a spirit level; being a kind of half collar opening 
hiuge. 

itch ; applied to various arrangements at the ends of separate shafts, and which by clutching or 
ling into each other cause both shafts to revolve together. A kind of coupling. 
ck ; a kind of valve for the discharge of liquids, air, steam, Ac. 

'efficient; or a Constant of friction, safety, or strength, Ac, may usually be taken to be a num- 
which shows the proportion (or rather the ratio) which friction, safety, tensile strength, Ac, bear 
;ertaiu something else which is not generally expressed at the time, but is well understood. Thus, 

^ i we say that the coeflf of friction of one body upon another is yL, Ac. it is understood that the 
ion is in the proportion of y^th of the pressure which produces it. A coeflf of safety of 3, means 
the safety has a proportion or ratio of 3 to 1 to the theoretical breaking load. A eoefl of 500 lbs, 
20 tons, Ac, of tensile strength of anv material, denotes that said strength is in the proportion 
O lbs, or of 20 tons, Ac, to each square inch of transverse section. Ac. Same as Modulus. 
r^S\ffer dam; an enclosure built in the water, and then pumped dry, so as to permit masonry or 
work to be carried on inside of it. 
g; the tooth of a cog-wheel. 
liar; a flat ring surrounding anything closely. 

liar-beam; a horizontal timber stretching fiom one to another of two rafters which meet at top; 
ibove the main tie-beam. See 21. p 613. 

ncrete; artificial stone formed by mixing broken stone, gravel. Ac, with common lime. When 
anlic cement is used instead of lime, the mixture is called beton. The terms “ lime concrete •’ 
‘•cement concrete ” would be convenient. 

nnecting-rod; apiece which connects a crank with something which moves it, or to which it 
motion. 

ill nsole: a kind of ornamental bracket, somewhat in shape of an S ; much used in cornices, Ao, 
upporting ornamental mouldings above it. 

iping ; flat plates of stone, iron. Ac, placed on the tops of walls exposed to the weather. 

. rhel; a horizontal projecting pieoe which assists in supporting one resting upon it which projects 
farther. 

re; anything serving as a mould for anything else to be formed around. A term much used In 
dries. . 

nice; theoruameutal projection at the eaves of a building, or at the top of a pier, or of any other 
cture. 

tier-bolt, or key-bolt; a bolt which, instead of a screw and nut at one end, has a slot cut through 
.car that eud, for the insertion of a wedge-shaped key or cotter, for keeping it iu its place. Some- 
les the euds of these keys are split, so as to spread open after beiug iuserted, so as not to be jolted 
0 , of place. . 

1 ounlerfort; vertical projections of masonry or brickwork built at intervals along the back of a wall 
(jtreugtheu it; and generally of very little use. 

c!i ounter-shaft; a secondary shaft or axle which receives motion from the principal one. 
t ountersunk. See Ream. 

» ounter-weight; or counter-balance; any weight used to balance another. 

ouplings; a term of very general application to arrangements for connecting two shafts so that 
I*.! y shall revolve together. 

ji 'over; see “ butt-joint." .... , . , . „ . 

'over • in re-rolling iron and steel from piles of small pieces, a large bar or slab, called a cover, of 
U same width and length as the pile, is employed to form the bottom of the pile, and a similar slab 
flthe top. The covers serve to bold the pile together: and, after rolling, they form unbroken top 
3l bottom surfaces of the finished plate, bar, rail, 1 beam, Ac. 


lw* 





822 


GLOSSARY OF TERMS 


Crab ; a short shaft or axle, which serves as a rope-drum in raising weights ; and is revolved ei f 
»>' cog-wheels, a winch, or by levers or handspikes, inserted in holes around its circumference li f 
windlass, or capstan, of which it is a variety. It may be either vertical or horizontal. It is o | 
■et in a frame, to be carried from place to place. Also the whole machine is called a crab. 

Cradle; applied to various kinds of timber supports, which partly enclose the mass sustained. { 

Cramp; a short bar of metal, having its two ends bent downward at right angles for insertion 1 
two adjoining pieces of stone, wood, Ac, to hold them together. Much used at the ends of copiug-sto |l 
Also a similar bent piece, with a set-screw passing through oue of the bent ends, for holding th ;> 
tight between it and the other end. This last is also called a clamp. 

Crane; a hoisting machine consisting of a revolving vertical post or stalk; a projecting jib ; i| 
a stay for sustaining the outer end of the jib. The stay may be either a strut or a tie. There 
also cog-wheels, a rope drum or barrel, with a winch, ropes, pulleys. Ac. In a crane the post, I 
and stay do not chauge their relative positions, as they do in a derrick. 

Crank; a double bend at right angles, somewhat like a Z, at the end of a shaft or axle, and forn 
a kind of handle by which the axle may be made to revolve. Sometimes, as in common grindsto > 
this crank is formed of a separate piece removable at pleasure. That part of this piece which has 
square opeuing in it for fitting it to the square end of the axle, is called the crank-arm ; and theo 
part the crank-handle. A bell-crank consists of 4 bends at right angles at the center of an axle, f( 

lug in it a kind of U. A double crank consists of two bell cranks arranged thus, The bem | 

the U forms the crank-wrist. The term bell-crank is applied also to those used in fixing common dw 
ing house bells: and to larger ones on the same principle. A crank-pin is a pin projecting from r l 
volving wheel, disk, or other body, and serving as a crank-handle. A crank shaft is a shaft wl 
has a crank in it, or at its end. A cranked shaft has it in it only. A ship or other vessel is sai i 
be crank when its breadth is so small in proportion to its depth as to make it liable to upset easily 
when the same liability is caused by want of sufficient ballast. 

Crest; that top part of a dam over which the water pours. 

Cross-cut saw; a large horizontal saw worked by two men, one at each end. 

Cross-head; a piece attached across the eud (or near it) of another piece, and at right angles t 
so as to form a kind of T or cross. Often seen on piston rods, which they serve to keep in plac< 
resting on the slides, or guides. 

Crowbar; a bar of iron used as a lever for various purposes; often pointed at one end. 

Crown, or contrate wheel; a cog-wheel in which the teeth stand not upon its outer circumferenc 
usual, but upon the plane of its circle. 

Curb; a broad flat circular ring of wood, iron, or stone, placed under the bottoms of circular w; 
as iu a well, or shaft, to prevent unequal settlement; or built into the walls at intervals, for the g: 
purpose. Has many other meanings. 

Cut-off; an arrangement for cutting off the steam from a cylinder before the piston has made 
full stroke. Also a channel cut through a narrow neck of land, to straighten the course of a rive 

Cutwater, or starling; the projecting ends of a bridge pier, Ac, usually so shuped as to allow wa 
Ice, Ac, to strike them with but little injury. 

Damper; a door or valve to regulate the admission of air to a furnace, stove, Ac. 

Dead load; the cars, engine, Ac, in a train ; non-paying load. 

Dead-load; in a bridge, the weight of the bridge itself, with flooring, roof, Ac ; as distinguis 
from the live load of passing traius, vehicles, pedestrians, Ac. 

Dead points : those two points in the revolution of a crank, where the crank arm Is parallel w 
the rod which connects it with the moving power; and at which said rod exerts no tendency to t> 
the crank. 

Declination, of the sun, or of a star, is its angle north or south of the earth's equator at the ti 
of observation. 

Declivity; a downward slope or descent of ground. Ac. 

Dentils ; blocks constituting ornaments in a cornice ; placed at short intervals apart, they resen ! 
teeth. When, instead of mere blocks, they are handsomely carved in various shapes, they are ca i 
modillions. 

Derrick; a kind of crane, differing from common ones, chiefly in the fact that the rope or cl i 
which forms the stay may be let out or hauled iu at pleasure, thus raising or lowering the iuclina 
of a jib; thereby enabling the raised load to be placed vertically at the required spot. This cat ; 
be done with a crane, which, therefore, is not as well adapted for laying heavy masonry, especi 
at great heights. 

Diaphram; a thin plate or partition placed across a tube or other hollow body. 

Die; that part of a stamp that gives the impression. Dies are also two flat plates of hardened s 
on an edge of each of which is hollowed out a semicircular half of a short female screw. When t 
plates are put in contact they form a complete female screw, like that in a nut; and being stroi 
held together hy an iron boxing called the die-stocks, which have long handles tor revolving them.t 
constitute a mould or cutter for forming threads on n male screw. Also the main body of a pedes 

Dip; in geology, either the angle which the slope of a stratum forms with a horizontal ; or 
direction by compass, toward which it slopes. In surveying, the inclination at which an unbalai 
compass-needle rests on its pivot after being magnetized. 

Disk: a flat circular piece. 

Dock; nn artificial enclosure, either partial or total, in which ships and other vessels are pit 
for being loaded or unloaded, or repaired. The first is a wet dock ; the last a dry one. 

Dcg-iron; a short bar of iron, forming a kind of cramp, with its ends bent down at right am 
and pointed, so as to hold together two pieces into which they are driven. Often used for tempot 
purposes. It is also called a dog-iron when only oue end is bent down and pointed for driving, 








GLOSSARY OF TERMS 


823 


olfedi 

T 3 


, 

iep 

IBdfc 

[rim 

lick m| 

id Hi 
Hie,! 

lie 


'■ Iron 
iialtli 
ilii 

eul 


her end being formed into an eye or a handle by which the piece into which the other end is driven 
ay be hauled or towed away. 

Donkey-engine; a smalt steam engine attached to a large one, and fed from the same boiler. It is 
ted for pumping water into the boiler. 
sd.| Double crank. See Crank. 

Double keys. See K. of Trusses ; page 613. 

Dovetail; a joint like 20. page 613; it is a poor one for timber when there is much strain, 

iing then apt to draw out more or less. 

Dowel; a straight pin of wood or metal, inserted part way into each of two faces which it unites. 
Draft; the depth to which a floating vessel sinks in the water; in other words the water it draws. 
Draught; a drawing. A narrow level stripe which a stonecutter first cuts arouud the edges of a 
>ugh stone, to guide him in dressing off the face thus enclosed by the draught. 

Draw-plate; a plate of very hard steel, pierced with small circular holes of different diameters, 
trough which in succession rods of iron are drawn, and thus lengthened out into wire. Sometimes 
le holes are drilled through diamond or ruby, &c, instead of steel. 

Drift; a horizontal or inclined passage-way, or small tunnel, in mines, &e. To float a way with a 
irrent. Trees, Ac, carried along by freshets. 

Drip ; a small channel cut under the lower projecting edge of coping, &c, so that rain when it 
taches that point will drip or fall off, instead of finding its way horizontally beneath to the wall, 
hich it would make damp. 

Drop; short pieces of nearly complete cylinders, placed at small distances apart, in a row like 
ieth, as an ornament to cornices, Ac. 

Drum; a revolving cylinder around which ropes or belts either travel or are wound. When nar- 
ow and used with belts they are called pulleys. 

Dry-rot; decay in such portions of the timber of houses, bridges, Ac. as are exposed to dampness, 
specially in confined warm situations. The timber in cellars and basement stories is mere liable to 
'• than in other parts, owing to the greater dampness absorbed by the brickwork from the ground, 
lontact with lime or mortar hastens drj- rot. The ends of girders, joists, Ac, resting on damp walls, 
lay be partially protected by placing pieces of slate or sheet iron under them. The painting or tar¬ 
ing of unseasoned timber expedites internal dry rot. A thorough soaking of timber in a solution of 
8 grains of quicklime to 1 gallon of water is said to be a preventive of dry-rot; but the best process 
or that purpose is saturation with creosote or carbolic acid. 

Dyke; mounds of earth, Ac, built to prevent overflow from rivers or the sea. A kind of geological 
rregularity or disturbance, consisting of a stratum of rock injected as it were by volcanie action, be- 
ween or across strata of rocks of another kind. A levee. 

Eccentric; a circular plate or pulley, surrounded by a loose ring, and attached to a revolving 
haft, and moving around with it, but not having the same center; for producingan alternate motion. 
)ften used instead of a crank, as they do not weaken the axle by requiring it to be bent. There are 
□ any modifications. 

Escarpment; a nearly vertical natural face of rock or soil. 

Escutcheon; the little outside movable plate that protects the keyhole of a loek from dust. 

Eye; a circular hole in a flat bar, Ac, for receiving a pin, or for other purposes. 

Eye and strap; a hinge common for outside shutters, Ac, one part consisting of a* iron strap one 
| s iJ:nd of which is forged into a pin at right angles to it; and the other part, of a spike with an eye, 
hrough which the pin passes. When the eye is on the strap, and the pin on the spike, it is called a 
look and strap. Such hinges are sometimes called “ backflaps." 

Eye-bolt; a bolt which has an eye at one end. 

Face-wall; one built to sustain a face cut into natural earth, in distinction to a retaining-wall, 
vhich supports earth deposited behind it. 

Fall; the rope used with pulleys in hoisting. 

False-works; the scaffold, center, or other temporary supports for a structure while it is being 
iouilt. In very swift streams it is sometimes necessary to sink cribs filled with stone, as a base for 
false-works to foot upon. 

Fascine* ; bundles of twigs and small branches, for forming foundations soft ground. 

Fatigue; of materials ; the increase of weakness produced by frequent bending; or by sustaining 
heavy loads for a long time. 

Faucet; a short tube for emptying liquids from a cask, Ac; the flow is stopped by a spigot. Tha 
wider end of a common cast-iron water or gas pipe. 

Feather; a slightly projecting narrow rib lengthwise of a shaft, and which, catching into a corre- 
pouding groove in anything that surrounds and slides along the shaft, wilt hold it fast at any required 
j^part of the length of the feather. Has other applications. * 

•J Feather-edge; when one edge of a board, Ac, is thinner than the other. 

Felloe , or felly; the circular rim of a wheel, into which the outer ends of the spokes fit; and which 
• Is often surrounded by a tire. 

ij. Felt; a kind of coarse fabric or cloth made of fibres of hair, wool,coarse paper, Ae, by pressure, 
jr and not by weaving. 

j. Fender; a piece for protecting one thing from being broken or injured by blows from another; 
frequently vertical Umbers along the outer faces of wharves, to prevent injury front the rubbing or 
vessels. 

Fender-pile* ; piles driven to ward off accidental floating bodies. 

Ferrule; a broad metallic ring or thimble put around anything to keep it front splitting or breaking. 
A small sleeve. 

Fillet; a plain narrow flat moulding in a cornice, Ae. •See Platband. 

Fish ; to join two beams, Ac, by fastening other long pieces to their sides. 

Flag*; broad flat stones for paving. 

Flange ; a projecting ledge or rim. 

Flashing*; broad strips of sheet lead, eopper, tin, Ac, with one edge inserted into the joints of 
brickwork or masonry an inch or two above a roof, Ac; and projecting out several inches, so as to be 
flattened down close to the roof, to prevent rain from leaking through the joint between the roof and 
the brick chimney, Ac, which projects above it. 

Flasks; upper and lower; the two parts of the box which contains the mould into which melted 
iron is poured for castings. 

Flatting; causing painting to have a dead or dull, instead of a glossy finish, by using turpentine 
instead of oil in the last coat. 

Fliers ; a straight flight of steps in a stairway. 

Floodgate; a gate to let off excess of water in floods, or at other times. 




824 


GLOSSARY OF TERMS. 


Flume; a ditch, trough, or other channel of moderate si ze for conducting water. The ditches t 
culverts through which surplus water passes from an upper to a lower reach of a canal. 

flush; forming an even continuous line or surface. To clean out a liDe of pipes, sewers, gutter ®e 
4c, bv letting on a sudden rush of water. The splitting of the edges of stones under pressure. 

Fluies; various substances used to prerentthe instantaneous formation of rust when welding tw ji 
pieces of hot metal together. Such rust would cause a weak weld. Borax is used for wrought iron Mt 
a mixture of borax and sal ammoniac for steel; chloride of tiuc for xinc; sal ammoniac for coppe i 


See also p 615. 


! 


ll 


or brass; tallow or resin for lead. 

fly-wheel; a heavy revolving wheel for equalising the motion of machinery. 

Foaming; an undue amount of boiling, caused by grease or dirt in a boiler. 

Follower; any cog-wheel that is driven by another; that other is the leader. 

Forceps; auy tools for holding tilings, as by pincers, or pliers. 

Forebay, or penstock ; the reservoir from which the water passes immediately to a water-wheel. 

Forge; to work wrought iron into shape by first softening it by heat, and then hammering it iut ffc 
the required form. 

Forg e-hammer; a heavy hammer for forging large pieces; and worked by machinery. 

Foxtail ; a thin wedge inserted into a slit at the lower end of a pin, so that as the pin is drivei 
down, the wedge enters it and causes it to swell, and hold more firmly. 

Frame; to put together pieces of timber or metal so as to form a truss, door, tr other structure 
The thing so iramed. 

Friction-raUec*; hard cylinders placed under a body, that it may be moved more readily than to 
Sliding. 

friction-wheels ; wheels so plaeed that the journals of a shaft may rest upon their rims, and thu: 
he enabled to revolve with diminished friction. See page 374 a. 

Frieze; in architecture, the portion between the architrave and cornice. The term is often applies 
when there is no architrave. 

Fulcrum; the point about which a lever turns. 

Fur rings ; pieces placed upon others which are too low, merely to bring their apper ytrfar«3 up It 
a required level; as is often done with joists, when one or more are too low ; a kind of chuck. 

Fuze, or fuse; to melt. A slow match, which, by burning for some time before the fire reaches the 
powder, gives the men engaged in blasting time to get out of the way of Hying fragments of stone. 

Gasket; rope-yarn or hemp, used for stuffing at the joints of water-pipes, he. 

Gearing; :t train of cog-wheels. Now much supplanted by belts. 

Gib ; the piece of metal somewhat of this shape, l-I, often used in the same hole with a wedge 

shaped key for confining pieces together. In common use for fastening the strap to the stub-end of 
the connecting-rod of an engine. 

Gin ; a revolving vertical axis, usually furnished with a rope-drum, and haring one or more long 
arms or levers, by means or which it is worked by horses walking in a circle around it- Used for 
hoisting. Cotton-gin. a machine for separating cotton from its seeds. 

Girder; a Veaia larger than a common joist, and used for a similar purpose. 

Glacis; in fortiticatioD, an easy slope of earth. 

Gland. See Stuffing-box. Also, a kind of coupling for shafts. 

Glue; a cement for wood, prepared chiefly from the gelatine furnished by boiling the parings of 
hides. Good glue will bold two pieces of wood together with a force of from tub to 150 k,s per sq m. 

Govern or ; two balls so attached to an upright revolving axis as to fiy outward by their centrifugal 

force, and thus regulate a valve. 

Grapnel ; a kind of compound hook with several tarred points, for finding things in deep water. 

Grillage; a kind of network of timbers laid crossing each other at right angles; frequently placed 
on fbe bead* of piles, for supporting piers of bridges, and other masonry. See p 641. 

Grom; an arch formed by two segmental arches or vaults intersecting each other at right angles. 
Also, a kind of pier built from the shore outward, to intercept shingle or gravel. 

Groove; a small channel. A triangular one is called a 

chamfered groove. 

Grmend-eweU; waves which continue after a storm has ceased; or caused by atoms at a distance. 

Grout ; thin mortar, to he poured into the interstices between stones or bricks. 

Gudgeons; the metal journals of & horizontal shall, such as that of a water-wheel. For 
speeds 


i>iam, ins > _ I' Weight in firs on one gudgeon 
if of cast-iron J jq 

For wrought-inon, add one-twentieth. 

Gun-metal. or bronze; a compound of copper and tin, sometimes used for cannon. Also, a quality 
of cast iron fit for the same purjvxe- 

Gussets; plain triangular pieces of plate iron, riveted by their vertical and horizontal legs to the 
sides, tops, and bottoms of box-girders, tubular bridges, Ac, inside, for strengthening their angles. 

Gtsys ; ropes or chains used to prevent anything from swinging or moving about. 

Gyrate; to revolve around a central axis, or point. 

Hairing ; to notch together two timbers which cross each other, so deeply that the joint thickness 
■hall equal only that of one whole timher. 

Hammer dress ; to dress the face of a stone by slight Mows of a hammer with a cutting edge. The 
patent hammer for sneb purposes bss several sneb edges placed parallel to each other, each oi which 
may be removed and replaced at pleasure. 

Hand lever ; in an engine, a lever to be worked by hand instead of hy steam. 

Handspike ; a wooden lever for working a capstan or windlass ; or other purpose*. 

Hand wheel ; a wheel used instead of a spanner, wrench, winch, or lever of auy kind, for renewing 
Bots, or for raising weights, or for steering with a rudder, Ac- 

Hangers, or /xmdent brockets; fixtures projecting below a ceiling, to support the journals of long 
lines of shafting; and for other purpose. Should be “ self-adjusting." 

Hasp; a piece of metal with an opening for folding it OTer a staple. 

Hatchway ; a horizontal opening or doorway in a floor, or in the deck of a Teasel. 

Haunches ; the parts of an arch from the keystone to the skewback. 

Head-block ; a block on which a pillow-block rests. 

Header; a stone or brick laid lengthwise at right angles to the face of the masonry. 

Heading ; in tunnelling, a small driftway or passage excavated in advance of the main body of the 
tunnel, but forming part of it; for facilitating the work. 

Headway ; the clear height overhead. Progress. 

Heel-poet; that on which a lock gate tarns on its pivot. 

Helve ; the handle of an mxe. 








GLOSSARY OF TERMS 


825 


i) 


'»i 


it 


.1 

'll 

St 


11 Hinge; those commonly used on the doors of dwellings are called butts, or butt hinges. (Rye and 
trap, .) Rising hinges are such as cause the door to rise a little as it is opened, and thus cause. 
l9 jle door to shut itself. 

Hip roof, or hipped roof; one that slopes four ways; thus forming angles called hips. 

Hoarding; a temporary close fence of boards, placed around a work in progress, to exclude 
ragglers. 

Holding-plates, or anchors; strong broad plates of iron sunk into the ground, and generally sur- 
>uuded by masoury ; for resisting the puli of the cables of suspension bridges ; and for other simi- 
,r purposes. 

Hook and strap. See Eye aud strap. 

Horses ; the sloping timbers which carry the steps in a staircase. s 

Housings ; in rolling mills, &c, the vertical supports for the boxes in which the journals revolve. 
Hub, or nave ; the central part of a wheel-, through which the axletree pusses, and from which 
>K|ie spokes radiate. 

Impost ; the upper part of a pier from which an arch springs. 

Ingot ; a lump of cast metal, generally somewhat wedge shaped. A pig of cast iron is an ingot. 
Invert; an inverted arch frequently built under openings, in order to distribute the pressure more 
venly over the foundation. 

Jack; a raising instrument, consisting of an iron rack, in connection with a short stout timber : 
hich supports it, and worked by cog-wheels and a winch. A screw-jack is a large screw working 
l a strong frame, the base of which serves for it to stand on ; and which is caused to revolve aud 
ise, carrying the load on top of it, by turning a nut, or otherwise. 

Jack-rafters, or common rafters; small rafters laid on the purlins of a roof, for supporting the 
hingling laths, &c. 

Jag-spike ; a spike whose sides are jagged or notched, with the mistaken idea that its holding power 
t thereby much increased. If a spike or bolt is first put into its place loosely, and then has melted 
ittd run around it, the jaggiug does assist; but not when it is driven into wood. 

Jambs; the sides of an opening through a wall, &c; as door, window, and fireplace jambs. 
Jamb-linings ; the facing of w oodwork with which jambs are covered and hidden. 

Jaw; au opening, tiften V-shaped, the inner edges of which are for holding something in place. 
Jettie, or jetty ; a pier, mound, or mole projecting into the water; as a wharf-pier, &c. 

Jib; the upper projecting member or arm of a crane, supported by the stay. 

Jig-saw ; a very narrow thin saw worked vertically by machinery, and used for sawing curved 
iroaments in boards. 

Joggle; a joint like that at 3 or 4, &c. p 613, of Trusses, for receiving the pressure of a strut at 
igbt angles or nearly so. Also applied to squared blocks of stone sometimes inserted between 
ourses of masonry to prevent sliding, &c. 

Joist ; binding joists are girders for sustaining common joists. The common ones are then called 
ridging joists. Ceiling joists are small ones under roor trusses, or under girders, and for sustain- 
ng merely the plastered ceiling. 

Journal-box ; a fixture upon w hich a journal rests and revolves, instead of a plummer-block. 
Journals ; the cylindrical supporting ends of a horizontal revolving shaft. Their length is usually 
ibout 1 to \% times their diam. In lines of shafting 4 diams. To find the diam, see Gudgeon. 
Jumper; a drill used for boring boles in stone by aid of blows of a sledge-hammer. 

Kedqe; a small anchor. . . .. . .... 

Keepers ; the pieces of metal or wood which keep a sliding bolt in its place, and guide it in sliding. 
Kerf; the opening or narrow slit made in sawing. 

Key-bolt. See Cotter-bolt. 

Keystone; the center stone of an arch. 

Kibble; the bucket used for raising earth, stone, &c, from sharts or mines. ...... 

King-post, king-rod ; the ceuter post, vertical piece, or rod, in a truss; all those on each side of ft 
ire queen-posts, or queen-rods. Frequently called simply kings and queens. , . 

Knee ; a piece of metal or wood bent at an angle; to serve as a bracket, or as a means of uniting 

,wo surfaces which form with each other a similar angle. . _ _ 

Lagqing, or sheeting ; a covering of loose plank ; as that placed upon centers, and supporting the 
vrchstoues. Also, an outer wooden casing to locomotive boilers and others. 

Landing ; the resting-place at the end of a flight of stairs. 

place^on^piece'upori another, with the edge of one reaching beyond that or the other. 
Lao-welding • welding together pieces that have first been lapped ; in distinction to butt-welding. 
Lead (pronounced leed.) in steam-engines, a certain amount or opening; of’the: port-valve befoie 

, sasssssi or wheeled - 

thos C e ed frequently e pl 0 ace P d in front of the driving-wheels. 
S^’a^artKojeSing^er like a shelf; a rock so projecting. A narrow strip of board nailed 

wedge-shaped hole in a block 

Sw, used for unloading vessels out from the shore. 

lilich pin; a pin near the end of an axle, to hold the wheel on 

Y^mZn- *a dev ice forVegiTlat!ng*thi^movement* oM.he main or port valve in a steam-engine.^ 

Lintel ; a horizontal b ^ a ™ ^ Cr w 0 °r^or'masonry jt'is cal led a“breMtTuninJer. or bressummer. 
wide span, and supporting heavy b ° rk ° T . , concealed within the thicknws of the door, are 

JSakSSS S7t" .the fete of a door, rim locks. It must be remein- 

fre„u«nlt, M the top. of .oof. of depot., he, provided .1th he 

l !“L „; 'the ,h.po Of , rl,»„,b; ofton Z JeS .M for v.rl.u. porpo.o,, ,«ch .. for 

£S;'“«r*'i or fr . 

Mallet ; the wooden hammer used by stonecutters. 


!«l 


II 


'll 






826 


GLOSSARY OF TERMS 


Mandrel; an iron rod used as a core around which a flat piece may be bent into a cylindrical shape 

Also the shaft that carries the chuck of a lathe. 

Manhole; an opening by which a man can enter a boiler, culvert, 4c. to clean or repair it. 

Mattock; a kind of pick with broad edges for diggiug. 

Maul; a heavy wooden hammer. 

Mean, arithmetical: half the sum of two numbers. 

“ , geometrical; the sq rt of the product of two numbers. 

Mean-proportional; the same as the geometrical mean. 

Meridian; a north and south line. Noon. 

Mitre-joint; a joint formed along the diagonal line where the ends of two pieces are united at an 
angle with each other. 

Mitre-sill; the sill against which the lock gates of a canal shut. 

Modulus ; a datum serving as a means of cotnparisoh. Same as constant or coefficient. 

Modulus of elasticity ; seep 434. Modulus of Rupture, p 485. 

Moment; tendency of force acting with leverage. See p 335. 

Moment of inertia. See p 486. 

Moment of rupture, or of bending; the tendeney which any load or foree exerts to break or bend a 
body by the aid of leverage. Its amount is found in foot-pounds by multiplying the force in lbs, by 
the length of leverage in feet between it and that part of the body upon which the tendeucy is exerted. 
Moment of stability. See Art 69. of Force in Sigid Bodies. 

Momentum; moving foree. See page 310, Force in Bigid Bodies. 

Monkey; the hammer or ram of a pile-driver. 

Monkey-wrench. or screw-wrench; a spanner, the gripping end of which can be adjusted by means 
®f a screw to fit objects of different sizes. 

Moorings; fixtures to which ships, &c, can make fast. 

Mortise; a hole cut in one piece, for receiving the tenon which projects from another piece. 

Muck; soft surface soil containing much vegetable matter. 

Muntins, or mullions: the vertical pieces which separate the panes in a window-sash. 
Nailing-blocks; block* of wood inserted in walls of stone or brick, for nailiug washboards, &c, to. 
Nave; the main body of a building, having connecting wings or aisles on each side of it. The hub 
of a wheel. 

Newel; the open space surrounded by a stairway. 

Newel-post; a vertical post sometimes used for sustaining the outer ends of steps. Also the large 
baluster often placed at the foot of a stairway. 

Nippers ; pincers. An arrangement of two curved arms for catching hold of anything. 

Normal; perpendicular to. According to rule, or to correct principles. 

Nosing ; the slight projection often given to the front edge of the tread of a step ; usually rounded. 
Nut, or burr; the short piece with a central female screw, used on the end of a screw-bolt, &e, for 
keeping it in plaee. 

Ogee; a moulding in shape of an S, the same as a cima. 

Ordinate; a line drawn at right angles from the axis of a curve, and extending to the curve. 
Oscillate ; to swing backward and forward like a pendulum. 

Out of wind, pronounced wynd; perfectly straight or flat. 

Ovolo; a projecting convex moulding of quarter of a circle; when it is concave it is a eavetto, or 
hollow. 


th< 

i 

S' 


it 

II 

bi 

(o 

lb 

n 

ci 

a 


P 

l 


Packing ; the material placed in a stuffing-box, &c, to prevent leak*. 

Packing-pieces; short pieces inserted between two others which are to be riveted or bolted together, 
to prevent their coming in contact with each other. 

Pull, or pawl. See Ratchet. 

Parapet; a wall or any kind of fence or railing to prevent persons from falling off. 

Parcel; to wrap canvas or rags round a rope. 

Purge ; to make the inside of a flue smooth by plastering it. 

Patent hammer; a hammer with several parallel sharp edges for dressing stone. 

Pay. To cover a surface with tar. pitch. Ac. A ship word. 

Pay out. To slacken, or let out rope. 

Pediment: the triangular space in the face of a wall that is Included between the two stoping side* 
•f the roof and a line joining the eaves. 

Penstock. 8ee Forebay.. 

Pier; the support of two adjacent arches. The wall space between windows, &e. A structure built 
•ut iuto the water, 

Pierre perdue; lost itone; random stone, or rough stones thrown into the water, and let find their 
own slope. 


Pilaster; a thin flat projection from the face of a wall, as a kind of ornamental substitute for a 
column. 

Pile-planks; planks driven like piles. 

Pillow-block, or plummer-block; a kind of metal chair or support, upon which the journals of hor¬ 
izontal shafts are generally made to rest, and on which they revolve. 

Pinion; a small cog-wheel which gives motion to a larger one. 

Pintle; a vertical projecting pin like that often placed at the tops of crane-posts, and over which 
the holding rings at the tops of the wooden guys At. Also, such as is used for the hinges of rudders, 
or of window-shutters to turn around. 




i 

















GLOSSARY OF TERMS 


827 


thfi^ th 7!° Pe ° f a *°- The distance from center to center of the teeth of a cog-wheel, or 
the threads of a screw. Boiled tar. Also the dist apart of rivets, &c. 

moved by ’it a connectin t5‘ rod fo1 ' transmitting motion from a prime mover to machinery at a distance, 

PW-a&w ; a large saw worked vertically by two men, one of whom (the pitman; stands in a pit. 

„ i, k k « r end a vertical revolving shaft, whether a part of the shaft itself, or attached to 
it. it should be tiat; and both it and the step or socket upon which it rests should be of hard steel, 
it a steel pivot has to revolve rapidly and continuously, it is well to proportion its diam, so as not to 
nave to sustain more than 250 lbs per sq inch; otherwise it will wear quickly. Dust and grit should 
for the same reason be carefully guarded against. Pivots which revolve but seldom, and slowly, as 
those of a railroad turntable, may be trusted with half a ton, or even a whole ton per sq inch. As a 
rude rule, cast-iron pivots should not be loaded with more than half as much as steel ones. A steel 
one may be welded to the foot of its cast-iron shaft; or may be inserted part way into it; and the 
whole strengthened by irou bands shrunk on. 

Planish; to polish metals by rubbing with a hard smooth tool. 

Plant; the outfit of machinery, &c, necessary for carrying on any kind of work. 

Plaster-bead; a small vertical strip of iron or wood nailed along projecting angles in rooms, to 
protect the plaster at those parts. ° * ’ 

Platband ; a plain, fiat, wide, slightly projecting strip, generally for ornament. When narrow, It 
Is called a fillet. 

Pliers ; a kind of pincers. 

Plinth ; the square lowest member of the base of a column or pillar. 

Plug; a piece iuserted to stop a hole. Screw-plug, a plug that is screwed into a hole. 

Plumb; vertical. 

Plummet, or plumb-bob; a weight at the lower end of a string, for testing verticality. 

Plunger; a kind of solid piston, or one without a valve. 

Point; a kind of pointed chisel for dressing stone. To put a finish to masonry by touching up the 
outer mortar joints. To dress stone with a point and mallet. 

Pole-plate; a longitudinal timber resting on the ends of tie-beams of roofs; and for supporting 
the feet of the common or jack rafters, when such are used. 

Port; the opening or passage controlled by a valve. 

Prime ; to put on the first coat of paint. Priming also is when water passes into a steam cylinder 
along with the steam. 

Projection. If parallel straight lines be imagined to be drawn in any one given direction, from 
every point in any surface s. whether flat, curved, or irregular, then if all these lines be supposed to 
be intersected or cut by a plane, either at right angles to their direction, or obliquely, the figure 
which their cross-section thus made would form upon said plane, is called the projection of the sur¬ 
face s. If such lines be supposed to be drawn from a person's face, in a direction in front of him, 
and to be cut by a plane at right angles to their direction, tbeir projection on the plane would be the 
person’s full-face portrait. If the lines be drawn sideways from his face, the projection will be his 
profile. The projection of a globe upon a flat plane, will evidently be a circle if the plane cuts the 
lines at right angles ; and an ellipse if it cuts them obliquely. Shadows cast by the sun are projec¬ 
tions. 

Puddle; earth well rammed into a trench, &c, to prevent leaking. A process for converting cast 
iron into wrought by a puddling furnace. 

Pug-mill; a mill for tempering clay for bricks or pottery. &c. 

Pulley ; a circular hoop which carries a belt in machinery. 

Puppet; in machinery, a small short pedestal or stand. Puppet-valve. See Valves. 

Purlins; the horizontal pieces placed on rafters, for supporting the roof covering. 

Put-logs, or put-locks; horizontal pieces supporting the floor of a scaffold; one end being inserted 
into put-log holes left for that purpose in the masonry. 

Quay; a wharf. 

Quoin; the hollow into which a quoin-post of a canal lock-gate fits. Stones, usually dressed, placed 
along the vertical angles of buildings, chiefly for ornament. 

Quoin-post; the vertical post on which a lock-gate turns. The heel-post. 

Rabbet, or rebate ; a half groove along the edge of a board, &c. See 16, p 613, of Trusses, where 
two rabbets are shown overlapping each other. 

Race; the channel which conducts water either to or from a water-wheel: the first is a head race; 
the last a tail race. The waves produced by the meeting of strong opposing currents; also, a rapid 
tideway, or roost. 

Rack and pinion; the rack is a straight row of cogs on a bar, and called a rack-bar; and the pinion 
is a small cog-wheel working into it. . _ /moment of inertia 

-Radius of gyration. See Center of gyration. Rad of gyr is = I - 

Rag-bolt. ‘See Jag-spike. N weight of body 

Rag-wheel, or Sprocketwheel; one with teeth or pins which catch into the links of a chain. 

Ram ; the hammer of a pile-driver. 

Random stone, 1 rip rap, or rough stones thrown promiscuously into the water, to form a founda¬ 
tion. Ac. 


Rasp ; a coarse file. 

Ratchet and pall; the former Is sometimes a straight bar, at others a wheel: in either ease it is 
furnished with teeth between which the pall drops and prevents backward motion. Used for safety 
in hoisting machinery, &c. The pall is sometimes called a click. 

Ream. A bole, wider at top than at bottom, (see 19, p 613, of Trusses,) through which a screw, 
bolt, <kc, is to be inserted, so that its head shall not project above the general surface, is said to be 
reamed, or reamed out; or to be countersunk. 

Rebate. See Rabbit, above. 

Reciprocal of a number is the quotient found by dividing 1 by that number. 

Reciprocating motion. See Alternating motion. 












828 


GLOSSARY OF TERMS 


Re-entering angle ; an angle or corner projecting inward. See Salient, below. 

Revetment; steep facing of stone to the sides of a ditch or parapet in fortification. A retaining-wall. 

Rib ; the curved pieces which form the arches of iron or wooden bridges, Ac. Also, those to which 
the outer plaukiug of a sailing vessel, Ate, are fastened. 

Ridge of a roof; its i>eak, or the sharp edge along its very top. Has various similar applications. 

Ridge-pole, ridge-piece, or ridge-plate; the highest horizontal timber in a roof, extending from top 
to top of the several pairs of rafters of the trusses; for supporting the heads of the jack-rafters. 

Right and left; a lock which in its proper position suits one flap of a pair of folding doors, will 
not suit if fastened to the other flap ; nor even to the same flap if required to open to the right in¬ 
stead of to the left, or vice versa, according to whether it is a right or a left-hand lock. And so with 
many other things, as, for iustauce, certain arrangements for working railway switches, &c. Right 
aud left boots and shoes are a familiar illustration ; also, right and left screws. Therefore, in order¬ 
ing several of anything, it is necessary to consider whether they may all be of the same pattern, or 
whether some must be right-hand, and others left-haud ones. 

Right shore of a river; that which is on the light hand when descending the river. 

Right-solid body ; cue which hag its axis at right augles to its base; w hen not so, it is oblique. 

Ring-bolt; a bolt with an eye and a ring at one end. 

Rip-rap. See Random stone. 

Roadstead; anchorage at some distance from shore. 

Rock-shaft; a shaft which only rocks or makes part of a revolution each way, instead of revolving 
entirely around. 

Rockwork; squared masonry in which the face is left rough to give a rustic appearance. 

Rubble ; masonry of rough, undressed stones. Scabblcd rubble has ouly the roughest irregularities 
knocked otf by a hammer. Ranged rubble has the stones in each course rudely dressed to nearly a 
uniform height. 

Rundle, or round; the step of a ladder. 

Rustic; much the same as rockwork. 

Saddle; the rollers aud lixtures on top of the piers of a suspension bridge, to accommodate ex¬ 
pansion aud contraction of the cables. The top piece of a stone cornice of a pediment. Has many 
other applications. 

Sag ; to bend downward. 

Salient; projecting outward. See Re-entering, above. 

Sandbag; a bag filled with sand for stopping leaks. 

Scabble; to dress off the rougher projections of stones for rubble masonry, with a stone-axe, or 
acabbliug hammer. 

Scantling ; the depth and breadth of pieces of timber ; thus we say, a scantling of 8 by 10 ins, &c. 

Scarf; the uuiting of two pieces by a long joint, aided by bolts, Ac. 

Scarp i a steep slope, fn fortification the inner slope of a ditch. 

Scotia ; a receding moulding consisting of a semi-circle or semi-ellipse, or similar figure. 

Screeds ; long narrow strips of plaster put on horizontally along a wall, and carefully faced out of 
wind, to serve as guides for afterward plastering the wide intervals between them. 

Screw-bolt, a bolt with a screw cut on oue end of it. 

Screw-jack. See Jack. 

Screw-tvrench. See wrench. 

Scribe; to trim off the edge of a board, Ac, so as to make it fit closely at all points, to an irregular 
surface. The lower edges of an open caisson are scribed to fit the irregularities of a rocky river bottom. 

Scroll; an ornamental form consisting of volutes or spirals arrauged somewhat in the shape of S. 

Scupper nails ; nails with broad heads for nailing down canvas, Ac. 

Scuppers ; on shipboard, holes for allowing water to flow off from the deck into the sea. 

Scuttle ; a small hatchway. To make holes iu a vessel to cause sinking. 

Sea-wall; a wall built to prevent encroachment of the sea. 

Secret nailing ; so nailing down a floor by nails along the edges of the boards, that the nail-heads 
do not show. 

Serve ; to wrap twine or yarn, Ac, closely round a rope to keep it from rubbing. 

Set screw, or tightening-screw; a screw for merely pressing one thing tightly against another at 
will; such as that which confiues the movable leg of a pair of dividers in its socket. 

Shackle, or clevis; a link In a chain shaped like a U, and so arranged that by drawing out a bolt 
or pin, which fits into two holes at the ends of the U, the chain can he separated at that point. 

Shaft; a vertical pit like a well. The body of a column. A large axle. 

Shank; the body of a bolt exclusive of its head. The long straight part of many things, as of an 
anchor, a key, Ac. 

Shears, or sheers; two tall timbers or poles, with their feet some distance apart, and their tops 
fastened together; and supporting hoisting tackle. 

Sheave; a wheel or round block with a groove around its circumference for guiding a rope. 

Sheeting, or sheathing; covering a surface with boards, sheet iron, felt, Ac. 

Shingle; the pebbles on a seashore. 

Shoes; certain fittings at the ends of pieces; as the pointed iron shoes for piles. The wall shoe* 
into which the lower ends of iron rafters generally fit, Ac. 

Shore; a prop. 

Shot; the edge of a board is said to be shot when it is planed perfectly straight. 

Shrink. When an iron hoop or band is first heated, and then at once placed upon the body which 
it is intended to surround, it shrinks or contracts as it cools, and therefore clasps the body more firmly. 
This is called shrinking on the hoop. 

Shuttle ; a small gate for admitting water to a water-wheel, or out of a canal lock, Ac. 

Siding; a short piece of railroad track, parallel to the main one, to serve as a passing-place. 

Silt; soft fine mud deposited bv rivers. Ac. 

Siphon culvert; a culvert built in shape of a U, for carrying a stream under an ohstacle. and allow¬ 
ing it afterward to rise again to its natural level. The term is improper, inasmuch as the principle 
of the siphon is not involved. 

Skewback; the inclined stone from which an arch springs. 

Skids; vertical fenders, on a ship's sides. Two parallel timbers for rolling things upon. 

Skirting; narrow boards nailed along a wall, as the washboards in dwellings. 

Sledge; a heavy hammer. 

Sleeper; any low-er nr foundation piece in contact with the ground. 

Sleeve; a hollow cylinder slid over two pieces to hold them together. 


GLOSSARY OF TERMS, 


829 


Slide-bars, or slides; bars for anything to slide along; as those for the cross-heads of piston-rods 
I &c. Often called guides. ’ 

Slings ; pieces of rope or chain to be put around stones, &c, for raising them by. 

Slip; the slidiug down of the sides of earth-cuts or bauks. A long narrow water space or dock 
i between two wharf-piers. 

Slope-wall; a wall, generally thin and of rubble stone, used to preserve slopes from the action of 
water in the banks of cauals, rivers, reservoirs. &c ; or from the action of rain. 

Slot; a long narrow hole cut through anything. 

Sluice; a water-chanuel of wood, masonry, &c; or a mere trench. The flow is usually regulated 
by a sluice-gate. 

Smoke-box; in locomotives, that space in front of the boiler, through which the smoke passes to 
the chimney. 

Snag ; a lug with a hole through it, for a bolt. 

Socket; a cavity made in oue piece for receiving a projection from, or the end of, another piece; as 
that into which the movable leg of a pair of dividers fits. 

Soffit; the lower or uuderneath surface of an arch, cornice, window, or door-opening. &c. 

Solder; a compound of different metals, which when melted is used for uniting pieces of metal also 
heated. Soft solder is a compound of lead and tin. and is used for uniting lead or tin. There are 
various hard solders, such as spelter solder, composed of copper and ziuc, for uniting iron, copper, or 
brass. 

Sole; that lining around a water-wheel which forms the bottoms of the buckets. 

Spandrel; the space, or the masonry, &c, between the back or extrados of an arch and the roadway. 

Spanner; a kind of wrench, consisting of a handle or lever with a square eye at one end of it; much 
used for tightening up the nuts upon screw-bolts, &c. The eye fits over or surrounds the nut. 

Spar; a beam; but generally applied to round pieces like masts, &c. 

Spelter; ziuc. 

Spigot; the pin or stopper of a faucet. The smaller end of a common cast-iron water or gas pipe. 

Spindle; a thin delicate shaft or axle. 

Splay ; to widen or flare, like the jambs of a common fireplace, or those of many windows ; or like 
the wing-walls of most culverts. 

Splice; to unite two pieces firmly together. 

Springer; the lowest stone of an arch. 

Sprocket-wheel, or rag-wheel ; one with teeth or pins which catch in the links of a chain. 

Spur-wheel;a. common cog-wheel, in which the teeth radiate from a common cen, like those of a spur. 

Square; in roofing; 100 square feet. 

Square-head; a square termination like that upon which a watch-key fits for winding; or that 
upon which the eye of the handle of a common grindstoue fits for turning it, &c. 

Staging ; the temporary flooring of a scaffold, platform, &c. 

Stanchion ; a vertical prop or strut. 

Standing-bolt, or stud-bolt; a bolt with a screw cut upon each end ; one end to be screwed perma¬ 
nently into something, and the other end to hold by means of a nut something else that may be re¬ 
quired to be removed at times. 

Stand-pipe. See p. 2D8. 

Staple; a kind of double pin in shape of a U; its two sharp points are driven into timber, and 
curved part is left projecting, to receive a hoop, pin, or hasp, &c. 

Starlings ; the projecting up and down-stream ends or cutwaters of a bridge pier. 

Stay; variously applied to props, struts, and ties, for staying anything or keeping it in place. 
Stay-bolts; long bolts placed across the inside of a boiler, &c, to give it greater strength. 
Steam-chest; the iron box in locomotive engines and others, through which the steam is admitted 
to the cylinders. 

Steam-pipe ; the one which leads steam from a boiler to the steam-chest. 

Step ; a cavity in a piece for receiving the pivot of an upright shaft; or the end of any upright piece. 
Stiles ; the flat vertical pieces between and at the sides of the panels in doors. &c. 

Stock; the eye with haudles for turning it, in which the dies for the cutting of screws are held. 
Stove-up, or stored, or upset; when a rod of iron is heated at one end, and then hammered end¬ 
wise so that that part becomes of greater diameter or stouter than the remainder. The heads of bolts 
are frequently made in one piece with the shank in this way; and the screw ends of long screw-rods 
are often upset, so that the cutting of the threads of the screw may not reduce the strength of the bar. 

Strap; a long thin narrow piece of metal bolted to two bodies to hold them together. A strap- 
hinge is a strap fastened to a shutter, &c, and having an eye or a pin at one end for fitting it to the 
other part of the hinge which is attached to the wall. 

Stratum ; a layer, or bed; as the natural ones in rocks, &c. 

Stretcher; a brick, or a block of masonry laid lengthwise of a wall. A frame for stretching any 
thing upon. 

Stretcher-course; a course of masonry all of stretchers, without any headers. 

Strike; an imaginary horizontal line drawn upon the inclined face of a stratum of rocks. Thus, 
if the slates or shingles on a roof represent inclined strata of rocks, then either the ridge or the eaves 
of the roof, or any horizontal line between them, will represent their strike. The inclination is 
called the dip of the strata; and the strike is always at right angles to it by compass. 

String ; variously applied to longitudinal pieces. 

String-board; the boarding (often ornamented) at the outer ends of steps in staircases. It hides 
the horses, as the inclined timbers which carry the steps are called. 

String-course; a long horizontal course of brick or masonry projecting a little beyond the others; 
and often introduced for ornament. 

Stringer; any longitudinal timber or beam, &c. 





830 


GLOSSARY OF TERMS 


Strut; a prop. A piece that sustains compression, whether vertical or inclined. 

Strut-tie, or tie-strut; a piece adapted to sustaiu both teusiou aud compression. 

Stub-end; a blunt end. ... . . , , 

Stud; a short stout projecting pin. A prop. The vertical pieces in a stud partition. 

Stud-bolt. See Standing-bolt. .... . ... 

Stuffing-box; a small boxing on the end of a steam cylinder, and surrounding the piston-rod nk« 
a collar; or in other positions where a rod is required to move backward aud forward, or to revolve, 
In an opening through auy kind of partition, without allowing the escape of steam, air, or water, Ac, 
as the case may he. The box is tilled with greased hemp or other packing, which is kept pressed close 
around the moving rod by meaus of a top-piece or kind of cover called the gland, which may be 
screwed down more or less tightly upon it at pleasure. The rod passes through the gland also. 

Sumpt, or sump; a draining well into which rain or other water may be led by little ditches from 
different parts of a work to which it would do injury. 

Surhase ; the inside horizontal mouldings just uuder a window-sill. Also those around the top of a 
pedestal, or of wainscoting, Ac. 

Swage, or swedge; a kind of hammer, on the face of which is a semi-cylindrical, or other shaped 
groove or indentation ; and which, being held upon a piece of hot iron and struck by a heavy hammer, 
leaves the shape of the indentation upon the iron. 

Switch ; the movable tongue or rail by which a train is directed from one track to another. 

Swivels; devices for permitting one piece to turn readily in various directions upon auother, with¬ 
out danger of entanglement or separation. At 13. p 583, of Trusses, is a tightening swivel; the 
castors under the legs of heavy furniture are swivelled rollers. 

Synclinal axis; in geology, a valley axis, or one toward which the strata of rocks slope downward 
from opposite directions. The line of’the gutter in a valley roof may represent such an axis. 

Ts; pieces of metal in that shape, whether to serve as straps, or for other purposes. So also with 
L's. S's, W's, -ps. &c. See figs to Welded Iron Tubes, p 405. 

Tackle; a combination of ropes and pulleys. 

Talus; the same as batter. 

Tamp ; to fill up with sand or earth, Ac, the remainder of the hole in which the powder has been 
poured for blasting rock. To compact earth generally, as under cross-tics, &c. 

Tap; a kind of screw made of hard steel, and having a square head which may be grasped by a 
wrench for turning it around, and thus forcing it through a hole arouud the inside of which it cuts an 
interior screw. To strike with moderate force. To make an opening in the side of any vessel. 

Tappet; a pin or short arm projecting from a revolving shaft; or from an alternating bar, and in¬ 
tended to come into contact with, or tap, something at each revolution or stroke. 

Teeth; or cogs of wheels. 

Temper; to change the hardness of metals by first heating, and then plunging them into water, oil, 
Ac. To mix mortar, or to prepare clay for bricks, Ac. 

Templet ; the outline of a moulding or other article, cut out of sheet metal or thin wood, to serve 
as a pattern for stonecutters, carpenters. &c. 

Tenon; a projecting tongue fitting into a corresponding cavity called a mortise. 

Terra cotta; baked clay. Brick is a coarse kind. 

Thimble; an iron ring with its outer face curved into a continuous groove. A rope being doubled 
around this and tied, the thimble acts as an eye for it, and prevents that part of the rope from wear¬ 
ing. Also, a short piece of tube slid over another piece, or over a rod, Ac, to strengthen a joint, Ac. 

Thread : the continuous spiral projection or worm of a screw. 

Through-stone; a stone that extends entirely through a wall. 

Throw; the radius, or distance to which a crank “throws out” its arm. Applies in the same way 
to lathes. Some use it to express the diameter instead of the radius. To avoid mistakes, the terms 
“single" and “double” throw might be used. 

Tie ; any piece that sustains tension or pull. 

Tie-strut; a piece adapted to sustain either tension or compression. 

Tightning-ring. See 14, of Figs 21*4, of Trusses. 

Tightning-screw. See Set-screw. 

Tire ; the iron ring placed around the outer circumference of the felloe of a wheel. 

Toggle joint. In Fig 16, of Force in page 323, suppose a m and a n to be two stiff bars, hinged 
together at a. It is plain that if we press downward at a, the result will be a great pushing force 
against any bodies placed at the ends m and n. Such an arrangemeut of two bars for producing such 
pressure is a toggle-joint. 

Tongue; a long slightly projecting strip to be inserted into acorresponding groove, as in tongued 
and grooved floors. 

Tooling ; dressing stone by means of a tool and mallet; the tool being a chisel with a cutting edge 
of l to 2 inches wide. Tooling is generally done in parallel stripes across the stone. 

Torus; a projecting semi-circular, or semi-elliptic moulding; often used in the bases of columns. 
It is the reverse of a scotia. 

Trailing-wheels ; in a locomotive, those sometimes placed behind the driving-wheels. 

Train; a number of cog-wheels working into each other. 

Tramway ; any two smooth parallel tracks upon which wheels without flanges may run. In rail 


GLOSSARY OF TERMS 


831 


tramways the rails themselves have flanges; but in wide stone ones tor common vehicles, none are 
required. 

Transom; a beam across the opening for a door, Ac. Also, a horizontal piece dividing a high 
window into two stories, Ac, Ac. Also, an opening above a door, for ventilation or light. 

Tread; the horizontal part of a step. 

Treadle ; a kind of foot-lever, for turning a lathe, grindstone, Ac, by the foot. 

Treenail; a long wooden pin. 

Trimmer; a short cross-timber framed into two joists so as to sustain the ends of intermediate 
joists, to prevent the latter from entering a chimuey-tiue, or interfering with a wiudow, Ac. 

Trip-hammer, or tilt-hammer; a large hammer worked by camb machinery, and used for heavy 
iron work, especially for hammering irregular masses into the shape of bars, Ac. 

Truck; a kind of small wagon consisting of a platform on two or more low wheels. Also, those 
frames and wheels usually placed under railroad cars and engines, and which, by means of a pintle 
connecting the two, allow them to vibrate or move laterally to some extent independently of each other. 

Trundle, lantern-wheel, or wallower; used instead of a cog-wheel, and consisting of two parallel 
circular pieces some distance apart, and united by a central axis, and by cylindrical rods placed 
around and parallel to the axis, to serve instead of cogs or teeth. 

Trunk; a long woodeu boxing forming a water channel. 

Trunnions ; Cylindrical projections, as at the sides of a cannon, forming as it were an interrupted 
- axle or shaft for supporting the cannon on its carriage; and allowing it to revolve vertically through 
" some distance. 

Tumbler; a kind of spring catch, which at the proper moment falls or tumbles into a notch or 

{ bole prepared for it in a piece; thus holding the piece in position until the tumbler is lifted out of the 
notch. 

Tumbling-bay; see “ waste-weir.” 

Tumbling-shaft; in locomotives, a shaft used in the “ link motion.” 

Turnbuckle ; variously applied, as to the ordinary fastenings at the outer face of a wall, for bold 
ing window-shutters back when opened; also, to the tightening-swivel at 13, page 583, of Trusses, Ac. 
Turntable; the well-known arrangement for turning locomotives at rest. See page 791. 
Undermine ; to excavate beneath anything. 

Underpin; to add to the height of a wall already constructed, by excavating and building beneath 
it. Also, to introduce additional support of any kind beneath anything already completed. 

Upset. See Stove-up. 

Valves ; various devices for permitting or stopping at pleasure the flow of water, steam, gas, Ac. 
A safety valve is one so balanced as to open of itself when the pressure becomes too great for 
safety. A slide valve is one that slides backward and forward over the opening through which the 
flow takes place. A ball valve, or spherical valve, is a sphere, which in any position fits the open¬ 
ing. When the pressure below it raises it off from its seat, it is prevented from rolling away by 
means of a kind of open caging which surrounds it. A conical or puppet valve is a horizontal slice 
of a cone, which fits into a corresponding conical seat made in the opening. In rising and falling it 
is kept in position by a vertical valve-stem or spindle, which passes through its center, and which 
plays through guide-holes in bridge-pieces placed above and below the valve. A trap, clack, flap, 
or door valve, is a plate with hinges like a door. When two such valves are used, with their hinged 
edges adjacent to each other, so that in opening and shutting they flap like The wings of a butterfly, 
they constitute a butterfly valve. A throttle valve is one which when closed forms a partition 
across a pipe; and opens by partially revolving upon an axis placed along its diameter. A rotary 
valve works like a common stopcock. A snifting valve is one which lets out steam under water ; and 
is so called from the sniffing noise thereby produced. The port valve is the sliding one which ad¬ 
mits steam from the steam-chest into the cylinders. A double seat, or double beat valve is a pe¬ 
culiar one with two seats, one above the other: and so arranged that the pressure of steam or water 
against it when shut, does not oppose its being opened. A cup valve is in shape of au inverted 
cylindrical cup, with a length somewhat greater than its diameter. Its lower or open edge is ground 
to fit the seat over which it rests. As this cup rises and falls, it is kept in place by a cylindrical 
eaging closed at top, and having for its sides four or more vertical pieces, against the inner sides of 
which the sides of the cup play. A check valve is any kind so placed as to check or prevent the 
return of the fluid after its passage through the valve into the pipe or vessel beyond it. 

:7 Vault; an arch long in comparison with its span. The space covered by such an arch. 

■ Veneer; a very thin sheet of ornamental wood glued over a more common variety. 

Wainscot; a wooden facing to walls in rooms, instead of piaster, or over a facing of plaster; usually 
not more than 3 or 4 feet high above the floor. 

Wales ; long longitudinal timbers in the sides of a ship, coffer-dam, caisson, &c. 

Wallow; a water-wheel, &c, is said to wallow when it does not revolve evenly on its journals. 
Wallower. See Trundle, 

Wall-plate, or raising-plate ; a timber laid along the tops of walls for the roof trusses or rafters to 
rest on, so as to distribute their weight more equally upon the wall. 

Warped; twisted, as a board, or the face of a stone, &c, which is not perfectly flat. To warp ; to 
haul a vessel ahead by means of an anchor dropped some distance ahead. To flood an extent of 
ground with water for a short time to increase its fertility. / 

Washboards; boards nailed around the walls of rooms at the floor, so as to prevent injury to the 
plaster when washing the floors. 

Washers; broad pieces of metal surrounding a bolt, and placed between the faces of the timber 
through which the bolt passes, and the head and nut of the bolt, so as to distribute the pressure over 
a larger surface, and prevent the timber from being crushed when the bolt is tightly screwed up. 

Waste-weir; an overfall provided along a canal, &c, at which the water may discharge itself in 
case of becoming too high by rain, &c. Sometimes called a tumbling-bay. 

Watch-tackle; ropes running in different directions from a boat, and used in bringing it into a 
desired position. 







832 


GLOSSARY OF TERMS 


Watershed; the sloping ground from which rain-water descends into a stream. * 

Water-table; a slight projection of the lower masonry or hrickwork on the outside of a wall, an 
reaching to a few feet above the ground surface, as a partial protection against rain, or as ornamen 

Ways: the inclined timbers along which a vessel glides when being launched. 

Weather-boards ; boards used instead of bricks or masonry for the outsides of a building, or bridge 
Ac. They are nailed to vertical and inclined indoor timbers; and may be either vertical or hoi 
When hor, they are so placed that the lower edge of one overlaps the upper edge of the one below 
When vert, their edges should be tongued and grooved ; and narrow slips be nailed over the vert joint: 
to keep out rain, Ac. 

Weir, or wier; a dam. or an overfall. 

Weld; to join two pieces of metal together by first softening them by heat, and then hamtnerin 
them in contact with each other. In this operation fluxes are used. 

Welt; see “ butt-joint." 

Wharf; a level space upon which vessels lying along its sides can discharge their cargoes ; or fron 
which they can receive them 

Wheel-base; the distance from center to center from the extreme front wheels, to the extreme bin 
ones in a locomotive, car, Ac. 

Wicket; a small door or gate made in a larger one; as the shuttle or valve in a lock-gate, for lettin 
out the water. 

Winch ; a handle bent at right angles, and used for turning an axis; that of a common grindstone 

Wind. See Out of wind. 

Winders; those steps (often triangular) in a staircase by which we wind, or turn angles. 

Windlass; the wheel and axle, or winch and drum, as often used in common wells. Also, a hori 
sontal shaft on shipboard, by which the auchor is raised; the windlass beiug revolved by means ol 
wooden levers called handspikes. 

Wing-dam; a projection carried out part way across a shallow stream, so as to force all the watei 
to flow deeper through the channel thus contracted. 

Wing*; applied in mauy ways to projections. The flauges which radiate out from a gudgeon; ant 
by which it is fastened to the shaft. Small buildings projecting from a main one. The wings 01 
flaring wing-walls of a culvert or bridge. 

Wing-walls; the retaining-walls which flare out from the ends of bridges, culverts, Ac. 

Wiper. See Carnb. 

Working-beam, or walking-beam ; a beam vibrating vertically on a rock-shaft at its center, as seen 
in some steam-engines; one end of it having a connection with the piston-rod; and the other end with 
a crank, or with a pump-rod. Ac. 

Worm; the so-called endless screw, which by revolving without advancing gives motion to a cog 
wheel (worm-wheel), the teeth of which catch in the thread of the screw. 

Wrench; a long handle having at one end an eye or jaw which may catch hold of anything to b« 
twisted or turned around, as a screw-nut. Ac. When it has a jaw which by means of a screw is 
adaptable to nuts, Ac, of different sixes, it is a monkey-wrench, or screw-wrench. 


ill 

1st 

ta 

o» 

ill 


INDEX. 


The numbers refer to the pages. In the alphabetical arrangement, minor 
vords, as “and,” “between,” “in,” “on,” “ through,” Ac, are neglected, 
ee also Glossary, p 819, Ac, and Table of Contents, p xxiii. 


irt 

of 

•t# 


Abrasion—Angle. 


ti 


Offl 

iii 


A. 

# » 

Lbrasion 
of cement, 678. 
by streams, 279/, 633, Ac. 
ibsorbent bodies, specific gravity of, 
381. 


■'* 

It 

u 


.bsorbents for nitro-glycerine, 662, 664. 
tbsorption by bricks, 671. 

.hutment, Abutments, 
of arch, 697. 
batter of, 699. 

courses in, inclination of, 359, 700. 
of dams, 285. 
foundations for, 633, Ac. 
line of thrust in, 359, 700. 
masonry, quantity in, 703. 
piers, 699. 
to proportion, 697. 
rubble, dimensions, rule, 698. 
thickness of, rule, 697. 
xceleration 

of gravity, 258, 311, 317, 362-364. 
on inclined planes, 363. 

,cid fumes, effect of, on roofs, 428. 

.ere, Acres, 
area of, 389. 

required for railroads, 722. 
surveying, 168. 
iCtion and re-actiou, 311. 


idhesion 
of cement, 677. 
of glue, 466, 824. 
of locomotives, 374a, 808, 809. 
of mortar, 670. 

of nails and spikes, 4256, 762. 
adjustment 
of box sextant, 195. 
of clinometer, 206. 
of compass, 195. 
of erroir in survey, 168, Ac. 
of hand level, 205. 
of level, 202. 
of plumb level, 206. 
of slope instrument, 206. 
of theodolite, 193. 
of transit, 191. 

Ldjutages, flow through, 259. 


54 


Admiralty knot, 387. 

Air, 215. 

buoyancy of, 234. 
compressed, 215, 648, 658. 
breathing, 215, 648, Ac. 
iu diving bells, 215. 
in foundations, 647. 
in rock-drills, 658. 
compressors, 658. 
lock, 648. 
in pipes, 297. 
pressure, 215. 

barometer, leveling by, 207. 
of compressed aii*, 215, 648, A«. 
slacking, 669. 
in tunnels, 754. 
valves, 297. 

ventilation, quantity required for, 
215. 

vessel, 298. 
weight, 215, 381. 
wind, 216. 

Alcohol, weight, 381. 

Alioth, 178,179. 

Alligation, 36. 

Altitude. See Height. 

Aluminium, weight, 381. 

Anchorages of suspension bridges, 620. 
Aneroid barometer, 207. 

Angle, Angles, 54. 
acute, 54. 
adjacent, 54. 
alternate, 55. 

arc, angle subtended by, 141. 
to bisect, 56. 

blocks, of Howe trusi^ 594. 

in a circle, 56. 

chords subtending, 105. 

complement and supplement of, 54. 

contiguous, 54. 

co-secants of, 59. 

cosines of, 59, 60. 

co-versed sines of, 59. 

defined, 54. 

deflection, 726-731. 

degree, decimals of, mius Ac in, 5T. 

of direction, 615. 

to draw a given angle, 56. 

833 











834 


INDEX. 

Angle—Arch. 


Angle, Angles—continued, 
to draw a right angle, 55. 
exterior, interior, Ac, 55. 
of friction, 315,355,371. 
frog, 781,785, Ac. 
given, to draw, 56. 
interior, exterior, Ac, 55. 
iron, 441, 442, 525. 
limiting, of resistance, 355. 
of maximum pressure, 687. 
to measure 

with the hand, Ac, 58. 
with the sextant, 114. 
with the tape line, 114. 
with the two-foot rule, Ac, 58. 
minutes and seconds in decimals of a 
degree, 57. 
ohtnse, 54. 
opposite, 55. 
in a parallelogram, 57. 
in pipes, 256. 
plates for rail-joints, 764. 
in polygons, 110. 
of reflexion, 255. 
of repose, 353. 

of resistance, limiting, 353, 355. 
right, to draw, 55. 
rule, 2-ft, to measure by, 58. 
secant, 59. 

seconds in decimals of a deg, 57. 
in a segment, 56. 
in a semicircle, 56. 
sines, 59. 
table, 60. 

of sliding. 353, 356, 369. 

of slope, 724. 

on sloping gronnd, 113. 

subtended by arc, 141. 

supplement and complement of, 56. 

switch, 773. 

tangent of, 59, 60. 

tangential, tables of, 726-728. 

in triangles, 110, Ac. 

versed sines of, 59. 

vertical, 55. 

Angular velocity, 365. 

Animal power, 377. 

Anthracite 
heat from, 212. 
for locomotives, 809, Ac. 
weight of, 381,389. 

Annual earnings, expenses, Ac. See Earn¬ 
ings, Expenses, Ac. 

Anti-friction rollers, 374e, 792, Ac. 

Antimony 
strength, 464. 
weight, 381. 

Anti-resnltant, 322, Ac. 

Apertnres 

contiguous, flow through, 261. 
flow through, 257, Ac. 
shape of, effect on flow, 260. 
in thin partition, 260. 

Apothecaries’ weight, 387. 

Application of force, 309, 314. 
direction of, 309, 314. 
line of, 314, 
oblique, 314. 


Application of force—continued 
perpendicular, 314. 
point of, 309, 314, Ac. 

Aqueduct, Aqueducts, 
flow in, 268. 

Kutter’s formula, 271. 

Pittsburgh, 624. 

Arc, Arcs, 
circular, 141. 
angles subtended by, 141 
center of gravity of, 348 
chords of, 105, 141. 
co-secant, 59. 
cosine, 59. 
table, 60. 

co-versed sine, 59. 
graduated, 189. 
large, to draw, 142. 
lengths of, 141. 
ordinates of, 141, 726,761, 786 
radii of, 141. 
rise of, 141. 
secant, 59. 
sine, 59. 
table, 60. 

tables of, 143-145. 
tangent, 59. 

table, 60. 
versed sine, 59. 
elliptic, 149. 
circumference of, 149. 
ordinates of, 149. 
table, 150. 

tangent to, to draw, 15C 
parabolic, 152. 
semi-elliptic, 149. 
circumference of, 149. 
ordinates of, 149. 
table, 150. 

Arch, Arches, 693. 
abutments of, 697. 
back of, defined, 693. 
braced, strains in, 592, 598 
brick, 709, 712. 
bridge, 693. 
cast-iron, 599. 
centers for, 711. 
concrete, 681, 695, 696. 
crown of, defined, 693. 
derangement of, 713. 
elliptic. 696, Ac. 

joint in, to draw, 150. 
existing, dimensions of, C35. 
extrados, 693. 
face, 693. 
foot, 693. 
intrados, 693. 
keystone 
defined, 693. 

depth of, rules, 693, Ac, 695. 
lever, principle of, applied to, 341. 
line of pressure in, 350, &c, 700. 
line of resistance in, 359, Ac, 700. 
line of thrust in, 359, Ac, 700. 
pressure in, 694. 
pressure, line of, 359, Ac, 700. 
radius, to find, 694. 
rise of, 693. 










INDEX. 

Arch-Beam 


835 


Arch, Arches—continued, 
roof, 600. 
rubble, 681, 696. 
skewback, 693. 
soffit, 693. 
span, 693. 
spring, 693. 
stability of, 359, Ac. 
stones, 693. 
chamfering, 714. 
pressure in, 694. 
pressure of, on centers, 713. 
voussoir, defined, 693. 

Archimedes screw, 379. 

Area 

of a circle, to find, 123. 

tables, 125-140. 
crippling, of rivet, 471. 
sectional, of flange in beams, 518, 521, 
537. 

sectional, of tunnels, 754. 
sectional, of web, in beams, 518, 521, 
539. 540. 

of surfaces. See the surface in ques¬ 
tion. 

Arithmetic, 33-35, Ac. 

decimal, 34, 35. 

Arithmetical progression, 36. 

Artesian wells, 627. 

Artificial 
horizon, 195. 
islands, 651. 
stones, 466, 678, 681. 

Ascent. See Grade, Height, Slope, 
effect of, 

on power of horse, 375. 
on power of locomotive, 808. 

Ash wood 

strength, 434, 436, 463, 493. 
weight, 381. 

Ashlar masonry, cost of, 667, 668. 
Asphaltum, weight, 381. 

Atlas powder, 664. 

Atmosphere, 215. See Air. 
buoyancy of, 234. 
in tunnels, 754. 
ventilation, 

quantity required for, 215. 
weight of, 215, 381. 

Augers for earth and sand, 626. 
Automatic 
frogs, 784, 785. 
switch-stand, 775. 

Average pressure of steam, 809. 
Avoirdupois weight, 387. 

Axis. See the given surface or solid 
of buoyancy,235. 
of equilibrium, 235. 
of flotation, 235. 
neutral, 479, 485, 487. 

to find position of, 487. 
of symmetry, 235. 

Axle 
car. 813. 

driving, of locomotive, 808. 
friction, 374d, 374e. 

standard dimensions, master mechan¬ 
ics’, 813. 


Back, Backs, 
of arch, defined, 693. 
of retaining wall, 683, Ac. 

Backing of walls, 683. 

Back-stays of suspension bridges, 616, 
Ac. 

Bag-scoop or spoon, 632. 

Baggage cars, 811. 

Bailing by bucket, day’s work at, 378. 
Baldwin locomotives, 805-810. 

Ballast for railroads, 759. 
cost of, 804. 

Balloon, principle of, 234. 

Balls, weight of, 398, 400, 416. 

Bar, Bars, 
brass, 415. 
copper, 415. 
dredging, 631. 
iron, weight of, 400,401. 
lead, 415. 
sand, 669. 

Barbed fence, 803. 

Barometer, 
aneroid, 207. 
effect of latitude on, 207. 
leveling by, 207. 
pressure of air, 215. 

Barrel, contents of, 390. 

Barrow. See Wheelbarrow. 

Base, wheel-, of locomotives, Ac, 546,805, 
Ac. 

Batter 

of abutments, 699; 
of retaining walls, 685, Ac. 

Battered walls, 685, Ac. 

Beam, Beams, 
angle iron, 525. 
box,537. 

breaking loads, constants, 491-493. 

formula, 488. 
cast-iron 

modifications in sections of, 518. 
strength of, 519, Ac. 
channel, 521, Ac. 

as pillars, 441, 442, 456. 
closed, 485. 

concrete, strength of, 682. 
constant 

for breaking loads, 491-493. 
for deflection, 506, 507, 509. 
continuous, 515. 
curved beams, 484. 
curved flanges, 530. 
cylindrical, 492, 497, 503, 511, 516. 

hollow, 516. 
dock 

deflections of, 499, 505-513. 
constant for, 506-509. 
dimensions for a given, 508, 510. 
load for a given, 506, Ac, 510. 
rules for, 505-511. 
under sudden loading, 499. 
table, 499, 500. 
dimensions of 
to find, 497. 

for a given deflection, 508, 510. 
elastic limit in, 504. 











836 


INDEX. 

Beam—Bolts, 


Beam, Beams—continued, 
flanges of 
curved, 530. 
oblique, 530. 
strains in, 529, 537. 
granite, 501. 

Hodgkinson’s, 518. 
hollow, strength of, 516. 

I and channel, 520, Ac. 
deflections of, 521. 
as pillars, strength of, 441, 442, 454. 
for railroad bridges, 524. 
inclined, 340,496. 
iron 

angle, 525. 

breaking loads, table, 502. 
cast, 

modifications in sections of, 518. 
strength of, 519. 
flanges, strains in, 529,537. 
loads, table of, 502. 
rolled, 520, Ac. See also Beams I, Ac. 
sate loads and deflections, 500. 
and steel, loads, table, 512, 513. 

T, 525. 

wrought, 520, Ac. See Beams I. 
of irregular shape, breaking loads for, 
495, 496. 

irregularly loaded, 496. 
lever, principle of, applied to, 339. 
limit of elasticity in, 504. 
loads on, 488. See also Beams, Strength, 
applied irregularly, 496. 
applied suddenly, 499. 
constants for, 491—193. 
formula, 488. 

for a given deflection, 506, Ac, 610. 
within limit of elasticity, 504. 
longitudinal sections of, 495. 
moments in, 478-489, 528, Ac, 537, Ac. 
open,528. 
plate, 537. 
resistance of, 484. 
riveted, 537. 

rolled. See Beams I, Beams Channel, 
Ac. 

shearing strains in, 532. 
of solid cross-section, 484. 
to splice, 610. 
square, on edge, 494. 
steel and iron, loads, table, 512, 513. 
stone, 493, 501. 
strains in flanges of, 529, 537. 
strains in, shearing or vertical, 532. 
strains in, vertical, 532. 
strength of, 478, Ac., 493. See also 
Beams, Loads on. 

affected by methods of supporting 
and loading, 494, 496. 
when loaded irregularly, 496. 
practical methods for finding, 491, 

Ac. 

suddenly loaded, 499. 

T-iron, 525. 

with thin webs, 528. 

tie-, 551, Ac. 

timber. See Beams, Wooden, 
tubular, strength of, 516. 


Beam, Beams—continued, 
vertical strains in, 532. 
wooden 

deflections, table, 499. 
loads, tables, 499, 502, 512, 613. 
for railroad bridges, 514. 

Bearing piles, 641. 

Bearing power of soils, 634, 644. 

Bearing and reverse bearing, 171. 
Beech-wood, strength, 434,436, 463, 493. 
Bell 

diving, pressure in, 215, 648, Ac. 
joint for pipes, 295. 

Belts, leather, strength of, 466. 

Bending 

of beams, rules for, 505-513. See 
Beams, Deflections of. 
rails, ordinates for, 761. 

Bends in pipes, 255. 

Bents in trestles, 756. 

Bessemer steel ties, 759. 

Beton concrete, 678, 681. 

Beveled joints for rails, 764. 

Birch, strength, 434, 436, 463, 493. 
Birmingham gauges, 410, 411. 

Bismuth, 
strength, 464. 
weight, 381. 

Bitumen, weight, 381. 

Bituminous coal, 

for locomotives, 809, Ac. 
weight of, 381, 389. 

Blake stone-crusher, 680. 

Blasting, 651-666, 754. 

Blocks, concrete, 679, 681. 

Blow, defined, 310. 

Board measure, table, 420. 

Boat, canal, 
cost of, 376, 
traction of, 376. 

Bodies, Body. See the body iu question, 
absorbent, specific gravity of, 381, 384. 
center of gravity, 349. 
defined, 306. 

expansion of, by heat, 212. 
falling, 258, 362. 
floating, 234, Ac. 
mass of, 318. 

regular, volumes, Ac, of, 154. 
rigid, force in, 305. 
rotating, 365. 
specific gravity of, 380. 
strength of, 434-525. 
vibrating, 364. 
weight of, 380. 

Boiler, 

incrustation of, top of 218. 
iron, 402, 464. 
pressure, 809, Ac. 
thickness for, 233. 
tubes, 405. 

Boiling-point, 213, 217. 

leveling by, 209. 

Boll man truss, 586, 603. 

Bolster plates, 514, 524, 544. 

Bolts, 406. 

copper, strength of, 408. 
iron, table, 408, 409. 



INDEX 


837 


Boltless—Brokerage. 


Boltless rail-joint, 767. 

Boring 

Artesian wells, 627. 

! augers for earth, 626, 627. 
test, 626, 633. 
wells, 626. 

Borrow-pits, to measure, 155,156. 
Bottom, 

heading, 754. 

of stream, scouring action on, 279/. 
velocity, 268. 

Bowstring 
centers, 717. 

truss, strains in, 588, 597. 

Box 

beams, 537. 
cars, 811. 

center, of turntable, 792. 
drains, 707. 
girders, 537. 
sextant, 194. 

Braced arch, strains in, 592, 598. 
Bracing, 

! counter, 549, 564, 714. 
lateral, 542, 610. 
sway, 543. 

Brad spikes, 762. 

Brake friction, 374, &c. 

Branches in pipes, 296. 

Brandt drill, 652. 

Brass 

balls, weight of, 416. 
bars, 415. 

compressibility of, 434. 
ductility of, 434. 

effect of cement, mortar, Ac, on, 670, 
673. 

effect of water on, 218. 

[ elastic limit of, 434. 

expansion by heat, 212. 
friction of, 373. 
modulus of elasticity, 434. 
pipes and tubes, seamless, 417. 
sheets, 415. 

strength of, 438, 464, 493. 

stretch of, 434. 

tubes, seamless, 417. 

weight of, 381, 398-400, 401, 410, 415. 

wire, 410, 412. 

Breaker, stone, 680. 

Breaking. See also Strength of Mate¬ 
rials, Ac. 

loads for beams, constants for, 491. See 
Beams. 

moment, 479-484. 

Breathing, 
air consumed in, 215. 
in diving-bells, 215, 648. 

Brick, Bricks, 670, 671. 
absorption of water by, 671. 
adhesion to cement, 677. 
adhesion to mortar, 670. 
arches, 709, 712. 
buildings, cost of, 668. 
cylinders, sinking of, 650. 
dust, 669. 
friction of, 373. 
laying, 671. 


Brick, Bricks—continued, 
number of, in a sq ft of wall, 671. 
strength of, 437, 466, 493. 
weight of, 381, Ac. 
work, 671, Ac. 
cylinders, sinking of, 650. 
incrustations on, 673, 678. 
rod of, English, 672. 
strength of, compressive, 437. 
water, to render impervious, 672. 
weight of (under Masonry), 383. 
Bridge, Bridges. See also Arch, Beam, 
Girder, Trestle, Truss, Ac. 
arch, 693. 
cast-iron, 599. 
centers for, 711. 
brick, 709, 712. 

centers for, 711. 

Brooklyn, foundations, 649. 
camber of, 607. 
cast-iron, 599. 

Chestnut St, Phila, 599. 

East River, foundations, 649. 
factor of safety, 607. 
false works, 608. 
floor girders for, 610. 
girder, 537-546. 
iron, cast, 599. 

iron, wrought. See Trusses, Ac. 

joints, 470, 611, 613. 

loads on, greatest probable, 606, 623. 

loads on, moving, 546, 564, 805, Ac. 

plate girders, 537-546. 

repairs, annual, 815. 

riveted girders, 537-546. 

roadways, drainage of, 708. 

rollers, expansion, 614. 

safety, factor of, 607. 

Severn Valley, 599. 
stone, 693. See also Arch. 

centers for, 711. 
suspension, 615-625. 

cables of, 412, 413, 615, Ac. 
suspension links, 614. 
swing, 593. 

friction rollers for, 798. 
truss, 547. See Truss. 

Whipple, 599. 
widths of, 542, 609. 

Wissahickon, Phila, 720. 
wooden, 514. See also Truss, Trestle, 
Ac. 

British 

Imperial measure, 391. 
measures, to reduce U S measures to, 
and vice versa, 390. 
railroads, miles, Ac, 818. 
rod of brickwork, 389, 672. 

Broach channeling, 658. 

Broken 

bubble-tube, to replace, 193. 
cross-hairs, to replace, 193. 
joints, 764. 
pipes, 296. 

stone (see also Rubble), 
foundations, 634. 
voids in, 380, 678, 751. 

Brokerage, 37. 








838 


INDEX. 

Broil ze—i’liortl. 


Bronze 
weight, 381. 

phosphor, wire, strength, 464. 
Brooklyn bridge, foundations, 649. 
Brunlee’s iron piles, 647. 

Bubble-tube, to replace, 193. 

Buckled plates, 409. 

Builder’s level, to adjust, 206. 
Building, Buildings, 
cost of, per cubic foot, 668. 
repairs, railroad, 815. 

Buoyancy 

of air, 234. « 

of liquids, 234-236. 

Burleigh rock-drill, 657. 

Burnettizing, 425a. 

Burnham’s frost-proof tank, 801. 

Burr truss, 601. 

Bursting 

of pipes, 234, 298, 303. 
pressure in pipes, 239. 

Bushel, volume of, 390. 

Butt-joint, 469. 

Buttresses, 692. 

c. 

Cable, Cables, 
number of wires in, 412. 
stays, 616. 

of suspension bridges, 
dimensions of, 616. 
strains in, 616. 

Caisson, 636. 

Brooklyn bridge, 649. 

Calking, 295, Ac. 

Camber of trusses, 607. 

Cambria nut-lock, 765. 

Canal, Canals, 
boats, cost of, 376. 
boats, traction of, 376. 
flow in, 268, Ac. 

Kutter’S formula, 244, 271. 
leakage from, 222, 269. 
traction on, 375. 

Cant bevel's, 479-482, 494-496, 535, 593. 
Caps, blasting, 665. 

Car, Cars, 811, 812. 
axles of, 812. 
derrick, 750. 
earthwork, 749. 
friction of, 374e, 808, 812. 
pile-driver, 642. 
repair, cost of, 815. 
resistance of, 374e, 808, 812. 
wheels, 765, 812. 
wrecking, 750. 

Carat, 385. 

Cart, Carts. 

earthwork, 742, Ac, 747. 
excavating (wheeled Bcrapers), 747. 
road, repail's of, 743. 
rock, removing, 752. 
traction, 375. 

Cartridge, dynamite, 663. 

rack-a-rock, 664. 

Cast-iron. See Iron, Cast. 

Castelli's quadraut^.269. 


Casting, 
rough, 674. 
safety, 770, 779. 

weight of, by size of pattern, 398. 

Cattle-cars, 811. 

Cedar, strength, 436, 463, 493. 

Ceiling, weight of, 553. 

Cellar walls, cost of, 668. 

Cement, 673. 
abrasion of, 678. 
brick-dust, 669. 
concrete, 678. 
expansion of, 678. 
and iron, pipes of, 294. 
for leaks, 429, 431. 
moisture, effect on, 673. 
mortar, 676. 
setting of, 674. 
in stone bridges, 696. 
strength of, 437, 466, 493. 
weight of, 382, 674. 

Center, Centers, 
for arches, 711. 
of buoyancy, 2:35. 
of circle, to find, 123. 
of force, 350. 
of gravity, 347, 348. 
of gyration, 440. 
of oscillation, 365. 
of percussion, 365. 
of pressure, 227,235, 350, 700. 

Centigrade thermometer, 213. 

Centimetre, length of, 392. 
cubic, weights per, 381. 

Central forces, 368. 

Centrifugal force, 368. , 

Centripetal force, 368. 

Chain, Chains, 

Gunter’s, 176. 

iron, 414. 

pump, 379. 

riveting, 470. 

surveying, 168, 176. 

of suspension bridges, 615. 

Chaining, slope, allowance for, 113, 176. 

Chair, railroad, 767. 

Chalk, 

strength, 437. 
weight, 381. 

Chamfering arch-stones, 714. 

Channel, Channels, 
flow in, 268. 

Kutter’s formula, 244, 271. 
iron, 521. 

as pillars, 441, 442, 456. 

Channeling in rock, 658. 

Charcoal, weight, 381. 

Cherry-wood, weight, 381. 

Chestnut St bridge, Phila, 599, 636. 

Chestnut-wood, 
strength, 434, 436, 463, 493. 
weight, 382. 

Chord, Chords, 
of arcs, to find, 141. 
in circles, 124, 141. 
long, table, 729. 

natural (to radius 1), table, 105. 
of trusses, 550, 588, 610, 612, Ac. 





839 


INDEX. 

Ch u r n-d r i 11 in g —Con leal. 


Churn-drilling, 651. 

Circle, Circles, 123. See also Circular, 
angles in, 56. 
areas of, to find, 123-140. 
center of, to find, 123. 
center of gravity of, 348. 
chords in, 124, 141. 
circumference, to find, 123, 133, 140. 

tables of, 125-140. 
diameter, to find, 123, 133,140. 
to draw, 123. 
great, earth’s, 144. 
mensuration of, 123. 
radius of, 123,141. 
tables, 125-140. 
tangeuts to, to draw, 124. 

Circular 

arc, 141. See also Arc. 

tables of, 143-145. 
curves for railroads, 726. 
inch,389. 
lune, 146. 
motion, 365. 
ordinates, 141. 

tables, 726, 728, 761, 786. 
rings, 146, 167. 
sector, 146. 

center of gravity of, 348. 
segment, 146. 
center of gravity of, 348. 
table of, 147. 
spindle, 167. 
zone, 146. 

Circumference 

of circle, to find, 123,133,140. 

tables, 125-140. 
of ellipse, 149. 

Cisterns, 233, 800-803. 

City 

water-supply, 287. 

Civil time, 395. 

Clamp, Clamps, 

pouring, for pipe-joints, 294, 295. 
rod, switch, 771. 

Clay, 

effect on mortar. 670. 
in foundations, 634. 
loosening of, 742. 
swelling of, by absorption, 634. 
Clearing, cost of, 804. 

Clinometer, 206,724. 

Clock, 

to regulate by star, 395. 
time, 395. 

Close piles, 641. 

Cloth, tracing, 433. 

Coal 

cars, 811. 

consumption of, by steam-engines, 

805, etc. 

corrosive fumes from, 403. 418. 
for locomotives, 805-810. 
oil, weight (Petroleum), 383. 
ton of, volume of, 389. 
weight of, 381, 389. 

Cocks, corporation, 299. 

Coefficient, Coefficients. See Strength, Ac. 
for beams, 491. 


Coefficient, Coefficients—continued, 
of contraction, 261. 
for deflection, 506, 507, 509. 
of friction, 371. 
for loads on beams, 491. 
for loads within elastic limit, 505. 
of resistance, 485. 
of roughness, 244, 272, 273. 
of rupture, 485. 

of safety. See Safety, Factor of. 
of torsion, 476. 
for transverse strength, 491. 
Coffer-dam, 636, 637, Ac. 

Cog-wheels, 342. 

Cohesion, 463. 

Cohesive strength, 463. 

Coignet beton, 681. 

Coin, Coins, 386, 387. 

Coke, weight, 381. 

Cold, 

effect on cement, 675. 
on explosives, 661, 663. 
on iron, 466. 
on mortar, 672. 
on trusses, 614. 

Collision, 310. 

Color, Colors, 
of cement, 674, 678. 
draughtsmen’s, 433. 

Columns (pillars). See Pillars. 

water, 801. 

Combination, 36. 

Commercial 

measures, size of, by weight of water, 
391. 

weight, 387. 

Commission, 37. 

Compass, 
to adjust, 195. 
variation of, 196, 197. 

Compensating reservoir, 290. 
Compensation water, 290. 

Complement and supplement, 56. 
Component, 319, &c. 

Composition of forces, 319, Ac. 

Compound 
interest, 37. 
levers, 342. 
proportion, 35. 

Compressed 
air, 215, 648, Ac, 658. 
gun-cotton, 664. 

Compressibility, 434. 
of air, 215. 
of liquids, 236. 

Compressive strength, 436, Ac. 

Concrete, 678. 

beams of, strength of, 682. 
beton Coignet, 681. 
strength of, compressive, 437. 
under water, 680. 
w’eight, 681. 

Concretions in pipes, to prevent, 292. 
Cone, Cones, 160. 
center of gravity of, 349. 
frustum, 160. 
center of gravity of, 349. 

Conical rollers, 792. 







840 


INDEX. 

Conoid—Curve«I. 


Conoid, 
parabolic, 167. 
frustum of, 167. 

Consolidation locomotives, 546, 805-810. 
Constants. See Coefficients. 
Construction, 
railroad, 722. 
cost of, 804. 

Consumption 

of coal by steam-engines, 805, etc. 
of fuel, effect of grades on, 810. 
by locomotives, 808, Ac. 

Contiguous openings, flow through, 261. 
Continuous 
beams, 515. 
girders, 515. 

Contour lines, 197. ■ 

Contracted vein, 258, 260. 

Contraction, 
coefficients of, 261. 
by cold, 212. 

and expansion in trusses, allowance 
for, 614. 

incomplete, 259, Ac, 263, Ac. 

on weirs, 263, Ac. 
of outflow, 258, 260. 
of rails, 764. 
of water-way, 703. 

Contractor’s profit, 743. 

Contrary flexure, point of, 515. 

Converse pipe-joint, 293. 

Copper 
bars, 415. 

balls, weight of, 416. 
compressibility of, 434. 
cost of, 415-417. 
ductility of, 434. 

effect of cement, mortar, Ac, on, 670, 
673. 

of water on, 218. 
elastic limit of, 434. 
expansion by heat, 212. 
modulus of elasticity, 434. 
pipes, seamless, 417. 
roofs, 416. 
sheets, 415, 416. 

strength of, 408, 438, 464, 476, 477. 
stretch of, 434. 
tubes, seamless, 417. 
weight of. 382, 398, 399, 400, 401, 410, 
415,416. 

Cord (funicular machine), 325, 344. 

of wood, volume of, 389. 

Cork, weight, 382. 

Corporation cocks or stops, 299. 
Corrosion, 

by acid fumes, 428. 
by coal fumes, 403, 418. 
of timber, prevention, 425. 
by water, 218, 645. 

Corrugated sheet-iron, 403. 

Co-secants, 59. 

Cosines, 59, 60. 

Cost of articles. See article in question. 
Co-tangent, 59. 

Counter-bracing, 549, 564. 

of centers, 714. 

Counter-forts, 692. 


Counter-scarp revetment, 692. 
Counter-sloping revetment, 692. 

Couples, 351. 

Couplings for pipes and tubes, 294, Ac, 
405. 

Courses of masonry, inclination of, 683, 
700. 

Cover in a butt-joint, 469. 

Cover-plate, 545, Ac. 

Co-versed sines, 59. 

Cracks in pipes, 296. 

Creeping of rails, 764. 

Creosote, 425, 769. 

Crescent truss, strains in, 588. 

Crib, Cribs, 635. 
coffer-dam, 638. 
dams, cost of, 285. 
foundations, 635. 

Crippling, 
of beams, 478, Ac. 
of riveted joints, 471, Ac, 539. 
Cross-girts, turntables, 792. 

Cross-hairs, to replace, 193. 

Cross-section paper, 433. > 

Cross-ties, 759. 
cost of, 804, 815. 

Crossings, railroad, repair, annual, 815. 
Crowds, weight of, 606, 623. 

Crown 

of arch, defined, 693. 

(coin) value of, 386. 

Crumlin viaduct, 756. 

Crusher, stone, 680. 

Crushing 
loads, 436, Ac. 
of stone, 680. 

Cube, Cubes, 41,154, 155. 
center of gravity of, 349. 
roots, 40. 

of decimals, to find, 53. 
of large numbers, to find, 52. 
tables, 40. 
tables, 41. 

Cubic 

centimetre, weights per, 381. 
foot, equivalents of, 389. 

weights per, 381. 
inch, equivalents of, 3S9. 
measure, 389. 
metric, 392. 

yards, earthwork, 732, Ac. 
yard, equivalents of, 389. 

Culvert, Culverts, 
arches for, 693. 
box,707. 

foundations of, 707. 
lengths of, 702. 

quantity of masonry in, 702, Ac. 
Curvature of the earth, table, 115. 

Curve, Curves. See Arc, Circle, Ellipse, 
Parabola, Ac. 
in pipes, 255. 
railroad, 726. 
tables of, 726-731. 
in tunnels, 754. 
in turnouts, 786. 

Curved 
beams, 484. 




841 


INDEX. 

Curved—Dodecagon. 


i Curved—continued, 
flanges, 530. 

! profiles, retaining walls, 692. 
Curvilinear motion, 365. 

Cuttings, level, 732. 

Cycloid, 154. 

Cylinder, Cylinders, 156. 
brickwork, hollow, in foundations, 650. 
center of gravity of, 349. 
contents, table, *157, 246, Ac. 
in foundations, 647, 650. See also 
Foundations. 

frustum of, center of gravity of, 349. 
iron, foundations, 645. 

friction of, 644. 
of locomotives, 805, Ac. 
masonry, hollow, in foundations, 650. 
one ft diam, 1 ft high, contents of, 390. 
one in. diam, 1 ft high, contents of, 390. 
plenum process, 648. 
pneumatic, 647. 
pressure in, 232. 
steam, 809, Ac. 

sinking, for foundations, Ac, 647-650. 
strength of, 232, 516. 
vacuum process, 647. 

Cylindrical 

beams, 492, 497 , 503, 511, 516. 

riveted sheet-iron, 516. 
pillars, 441, 442, 443, Ac. 
ungula, 159. 

Cyma, to draw, 151. 

33. 

I Dam, Dams, 279e, 282, 287. 
coffer, 636, 637, Ac. 
construction of, 282, 636. 
discharge over, 264. 
height of water on, 286. 
leakage through, 222, 288. 
pressure on, 222, Ac. 
stability of, 229. 
trembling in, 285. 
walls, 229, Ac, 691. 

Day, 395. 
j Dead 

load, cars, 811. 

bridges, 564, etc. 
oil, 425. 

| Decagon, 110. 

Decimal, Decimals, 34, 35. See Metric, 
of a degree, mins and secs in, 57. 
of a foot, inches in, table, 388. 
fractions reduced to, 34. 
roots of, 53. 

Deck beams, 521. 

Deflection, Deflections, 
angle, 726-731, 785. 

of beams, 505, &c. See Beams, Defls of. 

distances, tables, 726-728. 

of shafting, 505. 

of suspension bridges, 615. 

of trusses (camber), 607. 

of turnouts, 785. 

of turntables, 791, Ac. 

Degree 

of latitude, length of, 144, 388. 


Degree—continued, 
of longitude, length of, 387, 388. 
mins and secs in decimals of, 57. 
Dekametre, 392. 

chord of 2 dekams, curve, table, 728. 
Demi-revetment, 692. 

Departures and latitudes, 168. 

Depot, railroad, cost of, 804. 

Depth, Depths, 
on dams, 286. 

flow at different, 237, Ac, 268. 
hydraulic mean, 272. 
of keystone, 693, Ac. 
pressures at different, 223, Ac, 286. 
of rain-fall, 220. 

Derrick car, 750. 

De Vout’s switch-stand, 776. 

Dew-point, 215. 

Diagonal 

bracing, 542, Ac, 608. 
of parallelogram, 57, 119. 
of trapezoid, Ac, 120. 
of truss, 547, 608. 

Diagram, Diagrams, 
of forces, 320, Ac. 
for Kutter’s formula, 278. 
of loads, moving, 546. 
of moments, 482, 483. 
of pressure of water, 240. 
of trusses, 551, Ac. 

Dial, to make, 397. 

Diameter, Diameters, 
of bolts, 406. 

of circle, to find, 123, 133, 134, 140. 
of pipes, 245, Ac, 248, 405, 416. 
of rivets, for safety, 471, Ac, 539. 
square roots of, 247. 
of wire, 410, Ac. 

Diamond drill, 652. 

Diffusion of force, 231a, 308. 

through liquids, 224, 227. 

Dimensions. See the article in question. 
Direction, angle of, 615. 

Discharge 

through adjutages, 259. 
through apertures, 257, Ac. 
through channels, 268, Ac. 
through contiguous openings, 261. 
over dams, 264, Ac. 
head for a given, to find, 248. 
through notches, 267. 
through pipes, 236, Ac. 
through sewers, 279c. 
through short tubes, 259. 
through thin partition, 260. 
over weirs, 264, Ac. 

Discount, 37. 

Distance, Distances, 
frog, 785, Ac. 
polar, of North star, 177. 
by sound, 211. 

Distributing reservoirs, 290. 

Distribution of pressure 
in plane surfaces, 231a. 

Diving-bell, pressure in, 215. 

Diving dress, 651. 

Docks, concrete for, 679. 

Dodecagon, 110. 







842 


INDEX. 

Dodecahedron—End-wheels. 


Dodecahedron, 154. 

Dollar, 387. 

U S, weight, Ac, of, 387. 
value of, in different countries. 386. 
Double 
float, 269. 
riveting, 468. 
rule of three, 35. 
shear, 470, 476. 

Draft 

of horse, 375, 377, Ac. 
of locomotive, 808. 
of vessels, 236. 

Drag scrapers, 747. 

Drain, Drains, 
area drained by, 279c. 
box drains, 707. 
foundations of, 707. 
pipe, 279dL 
Drainage 

of roadways of bridges, Ac, 708. 
sewers, 279c. 
of tunnels, 754. 

Draw-bridge, strains in, 593. 

Drawing materials, 433. 

Drawn pipes and tubes. 417. 

Dredge, 631. 

land, 750. 

Dredging, 631. 

by screw-pan, 647. 

Dress, diving, 651. 

Dressing of stone, 667. 

Drill. Drills, 
churn, 651. 
jumper, 651. 
machine, 652. 
rock, 651, Ac. 
steam, 652. 

Drilling, 

Artesian wells, 627. 
rock, 651-658. 
tunnel, 754. 

Driving 
axles, 808. 
tires, 806. 
wheels, 805, Ac. 
weights on, 546, 564, 805, Ac. 
Drop timbers, 284. 

Dry 

drains, 707. 
measure, 390. 
rot, 425. 

Dual in, 664. 

Ducat, value of, 386. 

Ductility, 434. 

Dump-cars, 811. 

Duodecimals, 35. 

Dynamic rock-drill, 657. 

Dynamics, 306. 

Dynamite, 662. 

E. 

E and W line, to run. 171. 

Earnings, railroad, 814, Ac. 

Earth, 

augers for, 626, Ac. 
bearing power of, 634. 


Earth—continued, 
blasting of, 663. 
boring of, 626. 
cars (dump-cars), 811. 
curvature of, table, 116. 
friction of, 375, 692. 
hauling of, 743. 
heat of, 215. 

leakage through, 222, 288. 
leveling of, 743. 
loosening of, 742. 
natural slope of, 687, 690. 
pressure of, 687. 
shoveling, 742. 
shrinkage of, 741. 
slope of, natural, 687, 690. 
spreading of, 743. 
supporting power of, 634. 
weight of, 382. 
work, 732-753. 
cost of, 742, 804. 
cubic yards of, 732. 
in tunnels, 754. 
volume of, 732. 

East River bridge, foundations of, 649. 

East and west line, to run, 171. 

Easting, 168. 

Eclipse rock-drill, 654. 

Economizer rock-drill, 656. 

Edge Moor turntable, 793. 

Effective cross-section, 538. 

Efflorescence, 673, 678. 

Elastic frog, 783. 

Elastic limit, 434, 504. 

Elasticity, 
limit of, 434, 504. 

in beams, 504. 
modulus of, 434. 

Electric blasting machines, 665. 

Elevation of outer rail in curves, 729. 

Ellipse, 149, 150. 
area of, 150. 

center of gravity of, 348. 
to draw, 150. ** 

false, to draw, 151. 
ordinates of, 149. 
tangent to, to draw, 150. 

Ellipsoid, 166. 
center of gravity of, 349. 

Elliptic 
arc, 149. 

ordinates of, 149. 
table of, 150. 
arch, 696. 

joints in, to draw, 150. 
ordinates, 149. 

Elm wood, 

strength, 434, 436, 463, 493. 
weight, 382. 

Elongation 
by heat, 212. 
of North star, 177, 178. 
under tension, 434. 

Embankment, 732-753. 
cost of, 742, 804. 
shrinkage of, 741. 
volume of, 732. 

End-wheels of turntables, 792. 




INDEX. 843 

Energ-y—Flow. 


Energy, 318. 
kinetic, 318. 
potential, 318. 

Engine 

locomotive, 805-810. 
dimensions, 805, Ac. 
weight, 805, Ac. 
performance, 808. 
pumping, 801. 

; English cement, 673. 

English rod of brick-work, 389, 672. 

Enlargement in pipes, effect of, 257. 

;Entry head, 237. 

Equality of moments, 338. 

Equation of payments, 37. 

Equilibrium 

of floating bodies, axis of, 235. 

of forces, 338. 

indifferent, 235. 347. 

of moments, 338. 

stable, 235, 347. 

unstable, 235, 347. 

vertical of, 235. 

Equipment, railroad, cost of, 804, 814, Ac. 

Erection of trusses, 608. 

Establishment of a port, 219. 

Europe, railroad, miles of, 818. 

Evaporation, 222, 269. 
by locomotives, 803, 809. 

Even joints, 764. 

Excavating carts (wheeled scrapers), 747 

Excavation, 732-753. 
cost of, 742, 804. 
cubic yards, 732. 
in tunnels, 754. 
volume of, 732. 

; Excavator, steam (land dredge), 750. 

Expansion 
of cements, 678. 
by heat, 212. See Heat, 
of rails, 764. 

rollers, to provide for, 614. 
in trusses, allowance for, 614. 
steam, 809. 

Expense, Expenses, 
fuel, 815. 
locomotive, 810. 
railroad, 814, Ac. 
telegraph, 815. 
train, 815. 

i Exploders for blasting, 665. 

Explosive, Explosives, 660, Ac. 
foreign, 664. 
freezing of, 661, 663. 
gelatine, 665. 
modern, 661. 

Express cars, 811. 

Extrados, 693. 

Eye bars and pins, 612. 


F. 

Face 

of arch, 693. 
wall, 683. 

Facing switch, 770. 

Factor of safety. See Safety, factor of. 
for piles, 644. 


Factor of safety—continued, 
for pillars, 442, 446. 
for truss bridges, 607. 

Fahrenheit thermometer, 213. 

Fall, Falls, 
rain, 220. 

required for a given discharge, 274,2796. 
in sewers, 279c, Ac. 

Falling bodies, 258, 862. 

Falling water, horse-power of, 280. 

False ellipse, to draw, 151. 

False works, 608. 

Fascines, 650. 

Fatigue of materials, 435. 

Faucet in pipe-joint, 295. 

Fellowship (partnership), 37. 

Fence, 803, 815. 

Ferrule for water-oipe, 299. 

Fifth 

powers, 251. 

square roots of, 253. 
roots, 251. 

Figure, Figures, 110, Ac. 
center of gravity of, 348, 349. 
defined, 54. 
to draw, 121, Ac. 
to enlarge, 122. 

irregular, area of, to find, 122. 
to reduce, 122. 

two or more, center of gravity of, 348. 
Filling, spandrel, 693. 

Filtration, 222. 

Finish, hard, 426. 

Fink truss, 574, 578-580, 584, 003. 

Fir, strength, 460, 463. 

Fire, Fires, 
heat of, 212. 

hydrant (fire-plug), 304. 

Firing, simultaneous, of blasts, 665. 
Fish-plates, 764. 

Fisher rail-joints, 766, 767. 

Fittings for pipes, 293, Ac, 405. 

Flagging, strength, constants for, 493. 
Flange, Flanges, 
curved, 530. 
oblique, 530. 
strains in, 529, 537. 
strains in, in riveted girders, 537. 
Flexible joints for pipes, 296. 

Flexure, contrary, point of, 515. 

Floating 
bodies, 234, Ac. 
mills, 280. 

Floats, 268, 269. 

Floor 

beams. See Beam, 
buckled plates for, 409. 
girders, 545, Ac, 610. 
glass, 432. 
loads on, 606, 623. 
weight of, 553. 

Florin, value of, 386 
Flotation, 234, Ac. 

Flow 

through adjutages, 259. 
through apertures, 257, Ac. 
in channels, 268, Ac. 
through contiguous openings, 261. 







844 


INDEX. 

Flow—Fractions. 


Flow—continued, 
full, 259. 

Kutter’s formula, 244, 271. 
obstructions to, 279e, Ac. 
in pipes, 236, Ac. 
in sewers, 279c. 
in streams, 268, Ac. 
through short tubes, 259. 
in syphons, 240, Ac. 
through thin partition, 260. 
in troughs, 263. 

Fluid, Fluids, 
friction of, 374c. 

Follower, in pile driving, 645. 

Foot, 

cubic, equivalents of, 389. 

of substances, weight of, 381. 
decimals of, inches in, table, 388. 
pound, 311. 

spherical, equivalents of, 389. 

Force, Forces, 

application of, point of, 309, 314, Ac. 
center of, 227, 235, 350, 700. 
centrifugal, 368. 
centripetal, 368. 
composition of, 319, Ac. 
defined, 307. 

in different planes, 332, Ac. 
diffusion of, 308. 

through liquids, 227. 
equilibrium of, 338. 
imparted, 309. 
gradually, 315. 

on inclined planes, 352, Ac, 363. 

living, 308, 317. 

measure of, 310. 

moving, 310. 

obliquely applied, 314. 

parallel, 347. 

parallelogram of, 320, Ac. 

parallelopiped of, 333. 

point of application of, 309, 314, Ac. 

polygon of, 329. 

resolution of, 319, Ac. 

in rigid bodies, 305. 

“single,” “triple,” Ac, caps, 665. 
in trusses, 551. 
working, 310. 

Forcite, 664. 

Foreign coins, 386. 

Foreign explosives, 664. 

Formula. See also the given problem. 
Gordon’s, 439. 

Kutter’s, 244, 271. 
prismoidal, 161. 

Foundations, 633. 
of arches, 693. 
artificial islands, 651. 
brick cylinders, 650. 

Brooklyn bridge, 649. 
caissons, 636. 
car pile-driver, 642. 
for centers, 711. 
in clay, 634. 
close piles, 641. 
coffer-dams, 636, 637, Ac. 
concrete, 680, 681. 
crib, 635. 


Foundations—continued, 
of culverts, 707. 
cylinders, 645-647. 
brick, 650. 

plenum process, 648. 
vacuum process, 647. 
with concrete, 679. 
friction of, 644, 645. 
masoury, 650. 
with piles inside, 661. 
diving dress, 651. 
of drains, 707. 

East River bridge, 649. 
fascines, 650. 
on gravel, 634. 
grillage, 641. 

iron cylinders, 645-647. See Founds 
tions, Cylinders, 
iron piles, 645. 
islands, artificial, 651. 
leveling by concrete, 680. 
loads for, 634. 
masonry cylinders, 650. 

Nasmyth pile-driver, 642. 

Pierre perdue, 634. 
pile, piles, 626, 640. 
adhesion of ice to, 645. 
bearing, 641. 
blasting of, 663. 
in cylinders, 651. 
drivers, 379, 641, 642. 
car, 642. 
guupowder, 641. 

Nasmyth, 642. 
driving, by jets, 646. 
factor of safety, 644. 
friction of, 644. 
grillage, 641. 
heads for, 645. 
hollow, 647. 
ice, adhesion to, 645. 
iron, 645. 
loads for, 643. 

Mitchell's screw, 645. 
sand, 626, 650. 
screw, 645. 
sheet, 641. 
shoes for, 644. 
sustaining power of, 643. 
water-jet for driving, 646. 
withdrawal of, 645. 
plenum process, 648. 
pneumatic, 647, Ac. 
random stone, 634. 
of retaining walls, 692. 
rip-rap, 634. 
on sand, 634. 
sand piles, 626,650. 
sand pump, 650. 
screw piles, 645. 
sheet piles, 641. 
sustaining power, 634, 643. 


for trestles, 756. 
for turntables, 791. 
vacuum process, 647. 
Four-way stop-valve, 302. 
Fractions, 33. 
addition, Ac, of, 33. 






INDEX. 

Fractions—Glass, 


845 


Fractions—continued, 
decimal, 34. 

greatest common divisor, 33. 
lowest terms, to reduce to, 33. 

Franc, value of, 386. 

Franklin Inst standard dimensions of 
bolts, Ac., 406. 

Freezing 
of cement, 675. 
of dynamite, 663. 
of explosives, 661, Ac. 
of mercury. 213. 
of mortar, 672. 
of nitro-glycerine, 661. 
in pipes, 293. 

], behind retaining-walls, 684. 
in stand-pipes, 298. 
in track tank, how prevented, 802. 
of water, 217. 

Freight 
cars, 811, 812. 
earnings, 814. 
expenses, 814, Ac. 
locomotives, 805-810. 
ton-mile, 809, Ac, 814, Ac. 
train expense, 815. 

French 

measures, 391, 393. 
weights, 393. 

Freyburg suspension bridge, 622. 
Friction, 353, 370. 
angle of, 315, 355, 371. 
axle, 374<t. 
of cars, 374e, 808. 
coefficients of, 371, Ac. 
of earth, 375, 692. 
at feet of rafters, 355. 
head, 237, 248. 

on inclined planes, 352-361, 364. 
of iron cylinders, 644. 
journal, 374d. 

kinetic, 370. , . ,L 

launching, 374c. 

Morin’s laws, 372. 
of masonry, 373, 375, 692. 
of piles, 644. 
in pipes, 257. 

' of pivots, 371. 

in pumping mains, 257. 
rollers, 374e, 792, Ac. 
rolling, 3745. 
static, 370. 
of w T alls, 688. 
of water, 257, 374c. 

Frictional stability, 352-361. 

Fritzsche turntable, 794. 

Frog, Frogs, 780-786. 
angle of, 781, 785, Ac. 
distance, 785. 
length, 781. 
number, 781, 786, Ac. 
point, 781. 

Frost-jacket in fire-hydrant, 304. 
Frost-proof tank, 801. 

Frozen. See Freezing. 

Frustum, 

center of gravity of, 349. 
of cone, 160. 


Frustum—continued, 
parabolic, 152. 
of paraboloid, 167. 
of prism, 155. 
of pyramid, 160. 

Fuel, 

coal, 809, Ac. 

consumption by locomotive, 809. 

effect of grade on, 810, Ac. 
expense, of locomotives, 810, 815. 
grades, effect on consumption, 810, Ac. 
wood, 810, Ac. 

Fulcrum, 335. 

Full flow, 259. 

Fumes, 

acid, effect on roofs, 428. 
coal, effect on iron, 403, 418. 
Funicular machine, 325, 344. 

C>. 

Gallon, 390, 391. 

Galton’s experiments, 374. 

Galvanic action in water-pipes, 293. 
Galvanized 
iron, 403. 
pipes, 299. 

Gas, weight of, 381, Ac. 

Gasket, 295, Ac. 

to prevent washing into pipe, 295, 296. 
Gates for water-pipes, 301. 

Gauge, Gauges, 

Birmingham, 410, 411. 
narrow 

cars for, 811. 
car-wheels, 812. 
locomotives, 805, Ac. 
statistics, 818. 
railroad, 773, 814. 

Stub’s, 411. 
stuff, 426. 
wire, 410, 411, 412. 

Gauging of streams, 268, Ac. 

Gauthey’s pressure plate, 269. 

Gearing, 342. 

Gelatine, explosive, 665. 

Genesee viaducts, 756, 757. 

Geographical mile, 387. 

Geometrical 
progression, 36. 
similarity, 54. 

Geometry, 54. 

German cement, 673. 

Giant powder, 664. 

Gibbon boltless rail-joint, 767. 

Gin. 378. 

Girder, Girders. See also Beam, 
box and plate, 537. 
continuous, 515. 
floor, 545, Ac, 610. 
riveted, 537. 
transverse, 545, Ac, 610. 
of turntables, 791, Ac. 

Girts, cross-, of turntables, 792. 

Glass, 431. 
compressibility, 434. 
dimensions, Ac, 431. 
ductility, 434. 








846 INDEX. 

Glass—Hemisphere. 


Glass—continued, 
elastic limit, 434. 
expansion by heat, 212. 
friction of, 373, Ac. 
modulus of elasticity, 434. 
prices of, 432. 

strength, 432, 437, 466, 493. 
stretch of, 434. 
weight, 431. 

Glazing, 431. 

Globe, 162, 163. 

Glossary of terms, 819. 

Glue, adhesion of, 466. 

Glycerine, nitro-, 661. 

Gneiss, weight, 382. 

Gold, 

strength. 464. 
value, 387. 
weight, 382, 387. 

Gondola cars, 811. 

Gordon’s formula, 439. 

Grade, Grades, 

allowance for, in chaining, 176. 
contour lines, 197. 
effect on fuel consumption, 810. 
on horse, 375. 
ou locomotive, 808. 
hydraulic, 240. 
resistance of, 808. 
of roads, 375,723. 
of sewers, 279c, d. 
tables of, 176, 354, 723, 724, 725. 
traction on, 375, 808. 
in tunnels, 754. 
on turnpikes, 723. 
of water-pipes, 290. 

Grading, cost of, 742, Ac, 804, 

Granite, 
beams, 501. 
cost of blocks, 667. 
expansion by heat, 212. 
rubble, cost of, 668. 
strength of, 437,493. 
weight, 383. 

Granular bodies, specific grav of, 381,384. 

Gravel, 

boring in, 626. 
dredging in, 631. 
for foundations, 634. 
natural slope of, 690. 
weight, 382. 

Gravity, 

acceleration of, 258, 311, 317, 362, 363. 
center of, 347, 348. 
on inclined planes, 363. 
specific, 380. 

Great Bear, 179. 

Great Britain, railroads in, miles of, 818. 

Greenleaf turntable, 795. 

Greenwood switch-staud, 772. 

Grillage, 641. 

Ground lever, 772. 

Grout, 670. 

Grubbing, cost of, 804. 

Guard rails, 774, 779,781. 

Gudgeon, 
diameter, 824. 
friction, 374d. 


Guide-rails, 774, 779, 781. 
Gun-cotton, compressed, 664. 

Gun metal, strength, 464. 
Gunpowder, 660. 
pile-driver, 641. 

weight of (under Pow’der), 384. 
Gunter’s chain, 176, 387. 
Gutta-percha 
pipe, 294. 
weight of, 382. 

Gypsum process, 425a. 

weight, 382. 

Gyration, 

center of, 440. 

radius of, 366, 440, 538, 540. 


H. 

Hairs, cross, to replace, 193. 

stadia, 190. 

Hand level, 205. 

Hard finish, 426. 

Haul, mean, 743. 

Hauling, 375, 377, Ac, 743, 747. 

Head, Heads, 
block, 771. 
of bolts, 406. 
entry, 237. 
friction, 237, 248. 

for a given discharge, to find, 248. 
for a given velocity, to find, 245, 248. 
for piles, 644. 
plate, 771. 
pressure, 239. 

required for bends, Ac, 255. 

theoretical, 258. 

tripod, 189. 

velocity, 237. 

virtual, 258,280. 

of water, 223, 237, Ac. 

for water supply, 290, Ac. 

Heading in tunnel, 754. 

Headway in bridges, 609. 

Heat, 

of the air, 215. 
conduction of, by air, 215. 
effect of, on cement, 675. 
expansion of air by, 215. 
of rails by, 212, 764. 
of solids by, 212. 
of surveying chains ,by, 168. 
of trusses by, 614. 
of fires, 212. 
subterranean, 215. 
thermometer, 213. 

Ilecla powder, 664. 

Heel 

of frog, 781. 
of switch, 771, 774, 785. 

Height, 

effect on temperature, 215. 
effect on weight, 317. 
to find, by barometer, 207. 
by boiling point, 209. 
by trigonometry, 113. 
of locomotive smoke-stack, 805, Ac. 
of water. See Head. 

Hemisphere, center of gravity of, 349. 




INDEX. 

Hemlock—Iron 


847 


Hemlock, 

strength, 436, 460, 476, 493. 
weight, 382. 

Hemp ropes, 414. 

Heptagon, 110. 

Hercules powder, 664. 

Hexagon, 110,121. 

Hickory, 

strength, 436, 463, 493. 
weight, 382. 

High explosives, 661. 

Hodgkinson beams, 518. 

Holes 

for blasting, 651. 
boring, in earth, 626. 
boring, in rock, 651. 
for rivets, 470. 

Hook-head spikes, 762. 

Horizon, artificial, 195. 

Horizontal 
defined, 115. 

Horse, Horses, 
power of, 375, 801. 

-power, 377. 
of falling water, 280. 
of running streams, 280. 
pumping, day’s work, 801. 

-walk, diameter of, 377. 
weight, 377. 

House, engine, cost, 799. 

Howe truss, 594. 

Hydrant, fire (fire-plug), 804. 

Hydraulic, Hydraulics, 236. See also 
Water, Flow, Velocity, Discharge, 
Ac. 

cements, 673. See also Cement. 

dams, 229, 264, 279e, 282. 

grade-line, 240. 

lime, 673. 

mean depth, 272. 

radius, 244, 272. 

ram, 280. 

weirs, 229, 264, 279«?, 282. 

Hydrogen, specific gravity of, 382. 
Hydrometric pendulum, 269. 

Hydrostatic, Hydrostatics, 222. 
press, 227. 

I. 

I beams, 520, Ac. 
as pillars, 441, 442, 454. 
for railroad bridges, 524. 

Ice, 217, (fee. 
adhesion to piles, 645. 
blasting of, 663. 
in stand-pipes, 298. 
strength of, compressive, 437. 
weight, 217, 383. 

Icosahedron,154. 

Impact, 310. 

Impartation 
of force, 314, Ac. 
of velocity, gradual, 315. 

Imperial measure, British, 391. 

Impulse, 310. 

Inch, Inches, 
circular, 389. 


Inch, Inches—continued, 
cubic, equivalents of, 389. 
in decimals of a foot, 388. 
fractions of, common and decimal, 34. 
of rain, 221. 

spherical, equivalents of, 389. 
Inclination. See Grade, 
of courses in masonry, 683, 700. 
tables of, 176, 354, 723-725. 
in tunnels, 754. 

Inclined 

beams, 340, 480, 496. 
plane, 352, Ac, 363. 
descent on, 363. 
ropes for, 413, 414. 
stability on, 352-361. 
tables, 176, 354, 723-725. 
velocity on, 363. 

Incomplete contraction, 263. 

Incrustation 
of boilers, top of, 218. 
of walls, 673, 678. 

India-rubber, weight, 383. 

Indifferent equilibrium, 235, foot of 347. 
Inertia, 308. 

moment of, 365, 486, 487. 

Ingersoll rock-drill, 654. 

Initial pressure of steam, 809. 
Instability, 235, 347, 356. 

Insurance, 37. 

Interest, 37. 

Intrados, defined, 693. 

Iron 

angle, 525. 
beams of, 525. 
pillars of, 441-442. 
strength of, transverse, 525. 
balls, weight, 416. 
bars, weight, 400, 401. 
beams. See Beams, Iron, 
blasting of, 663. 
bolts, 406-409. 
bridges, cast, 599. 
bridges. See Trusses, Bridge, Ac. 
buckled plates, 409. 
cars, 811. 
cast 

beams of, modification in sections of, 
518. 

beams of, strength of, 519. 
bridges, 599. 
cohesive strength, 464. 
compressive strength, 438. 
crushing strength, 438. 
expansion by heat, 212. 
friction of, 373, &c. 
pillars, 439, Ac. 
pipes, flow in, 243, Ac. 
pipes, weight of, 293, 297, 399. 

6alt water, effect on, 218, 645. 
shearing strength, 476. 

Sterling’s toughened, 520. 
strength, 438,464, 476, 477, 493. 
tensile strength, 464. 
torsional strength, 477. 
transverse strength, constants for, 
493. 

turntables, 792. 





848 


INDEX. 

I roil—Keystone. 


Iron—continued, 
cast 

water, salt, effect on, 218, 645. 
weight, 382, 398, 401. 
casting, weight of, by size of pattern, 
398. 

and cement, pipes of, 294. 
chains, 414. 
channel, 521. 

as pillars, 441, 442, 456. 
cohesive strength of, 409, 464. 
cold, effect on, 168, 466, 764. 
columns, 439, Ac. See also Pillars, Iron, 
compressibility of, 414. 
compressive strength of, 438. 
contraction of, by cold, 168, 764. 
corrosion of, by coal fumes, 403. 
corrugated sheet, 403. 
cost of, 402. 

crushing strength, 438. 
cylinders, bursting pressure in, 232,233. 
cylinders, in foundation, 644, Ac. See 
also Foundations, 
ductility of, 434. 
effect of cement on, 673. 
of cold on, 168, 466, 764. 
of heat on, 168, 212, 764. 
of mortar on, 670, 673. 
of water on, 218, 645. 
elastic limit, 434. 

expansion of, by heat, 168, 212, 764. 

-frame cars, 811. 

friction of, 373, Ac. 

galvanized, 403. 

heat, effect on, 168, 212, 764. 

limit of elasticity, 434. 

modulus of elasticity, 434. 

net, 470. 

paints for preserving, 430. 
piles, 645. See also Foundations, 
pillars, 439, Ac. See also Pillars, 
pipes, 

cast, weight, 293. 
fittings for, 405. 
flow in, 243, Ac. 
galvanized, 299. 
joints for, 293, 295. 
kalameined, 293. 
thickness of, 233, 293. 
wrought, 293. 
plates, 

buckled, 409. 
prices of, 402. 
porosity of, 233. 
prices of, 402. 
re-rolled, 402, 464. 
rolled. See Iron, Wrought, 
roofs, 403, Ac, 582, Ac. See Roofs, 
salt water, effect on, 218, 645. 
shearing strength, 476. 
sheet, 402, 403. 
corrugated, 403. 
galvanized, 403. 
specific gravity of, 382. 
spikes, 762. 

strength. 409, 438, 464, 476, 477, 493. 
stretch of, 434. 

T, 441, 442, 525. 


Iron—continued, 
tensile strength, 409, 464. 
torsional strength, 477. 
transverse strength, 477, 493. 
trestles, 756. 
tubes, 405. 

water, effect on, 218, 645. 
weight of, 382, 400, Ac. See also Iron, 
Cast; Iron, Wrought, 
wire, 412. 
rope, 413. 

-wood (Canadian), strength, 493. 
wrought 

bars, weight, 400, 401. 
beams, 520, Ac. See Beams, 
cohesive strength, 464. 
compressive strength, 438. 
crushing strength, 438. 
expansion by heat, 168, 212, 764. 
friction of, 373, Ac. 
pillars, 439, Ac. See also Pillars, 
pipes, 293, 294. 
fittings for, 405. 
joints for, 293, 295. 
weights, prices, Ac, 293, 405. 
prices, 402. 

shearing strength, 476. 
strength, 438, 464, 476, 477, 493. 
tensile strength, 464. 
torsional strength, 477. 
transverse strength, 493. 
tubes, weight, 405. 
water-pipes, 293. 
weight, 383, 400, 401. 

Islands, artificial, for foundations / 651. 

J. 

Jag-spikes, 762. 

Jet, pile-driving, 646. 

Joint, Joints, 
in arches, 709. 
bell, for pipes, 295. 
butt, lap, 469. 

in chimneys, Ac, cement for, 429, 431. 
Converse, for wrought-iron pipes, 293. 
distribution of pressure in, 23la. 
flexible, for pipes, 296. 
iron, 583. 
lap, butt, 469. 

masonry, inclination of, 683, 700. 

distribution of pressure in, 231a. 
net, 470. 

for pipes, 293, 295, Ac, 405. 

pressure in, distribution of, 231a. 

rail, 763. 

riveted, 468, 539. 

roof, 418, 427, 429. 

timber, 610, 612. 

in trusses, 610, 612, Ac. 

Journal friction, 374d. 

Judson dynamite, Ac, 664. 

Jumper drill, 651. 

K. 

Keyed frog, 782. 

Keystone, 693, 695. 
pressures on, 330, 359, 694. 




INDEX. 849 

Kieselguhr-Link. 


Kieselguhr, 662. 

Kinetic energy, 318. 

friction, 370. 

King 
-post, 554. 

-rod, 553, &c. 
truss, 551, &c, 578. 

Kinzua viaduct, 758. 

Knees in pipes, 256. 

Knife-edge, strength of, 438. 

Knot (nautical), length of, 387. 

Kutter’s formula, 244, 271. 

Kyanizing, 425a. 

L. 

Lagging for centers, 711, 712, 719. 
Laitance, 681. 

Land 

dredge, 750. 
measure, 389. 
metric, 392. 

required for railroads, 722. 
i section of, 

area of, 389. 
surveying, 168. 
ties, 692. 

Lap-joint, 469. 

Lap-welded boiler-tubes, 405. 

water-pipe, 293. 

Lard 

as a lubricant, 374c. 
weight of, 383. 

Lateral bracing, 542, 610. 

Laths, 426, 427. 

Latitude, Latitudes, 
degree of, length of, 387. 
and departures, 168. 
effect of, on barometer, 207, 209. 

on gravity, 317, 362. 
secants of, 177. 

Lattice truss, 596. 

Launching, friction of, 374c. 

Laying 

bricks, 670. 
out of turnouts, 785. 
f pipe, cost of, 297. 
track, cost of, 804. 

Lead 

balls, weight of, 416. 
bars. 415. 

compressibility, 434. 
ductility, 434. 
elastic limit, 434. 

effect of cement, mortar, &c, on, 670, 
673. 

expansion by heat, 212. 
in masonry joints, 438. 
modulus of elasticity, 434. 
paint, 429. 
pencils, 433. 
pipe, 234, 416. 

for pipe-joints, 293, 295, 297. 
roofs, 415, 416. 
sheets, 415. 416. 
strength, 438, 464. 
stretch of, 434. 
tensile strength of, 464. 

55 


Lead —co n ti n ued. 
weight, 383, 398-401, 410, 415, 416. 
white, cement, 
for leaks, 429. 
white, paint, 429. 

Leaded tin, 418. 

Leak in roof, to stop, 429, 431, 
Leakage, 222, 269, 282, 288. 

Leather, 
friction, 374c. 
strength, 466. 

Length, 
frog, 781. 
switch-rail, 776. 
switch, 786, &c. 

Lengthening scarfs, 610. 

Level, 201. 

builder’s, to adjust, 206. 
cuttings, 732. 
hand, Locke’s, 205. 
lines, defiued, 115. 
note-book, form of, 204. 

Y, 201. 

Leveling 

by barometer, 207. 
by boiling-point, 209. 
by concrete, 680. 
of earth, on embankments, 743. 
screws, 189, 202. 

Lever, Levers, 335. 
compound, 342. 

principle of, applied to arches, 341. 

beams, 339. 
switch. 772, 776, 779. 
tumbling, 772, 776, 779. 

Leverage, 335. 

Life, average, 
of cars, 811. 
of locomotives, 810. 
of rails, 760. 
of shingles, 429. 
of ties, 759. 
of wire ropes, 414. 

Lignum vitae, 
strength, 463, 493. 
weight, 383. 

Lime, 669. 

effect on cement, 680. 
hydraulic, 673. 
paste, 670. 

to preserve timber, 425a. 
weight, 383. 

Limestone, 383, 437. 

Limit of elasticity, 434, 504. 

in beams, 504. 

Limnoria, 425. 

Line, Lines, 54. 
contour, 197. 
hydraulic grade, 239. 
of no variation, 197. 
parallel, to draw, 56. 
of pressure, 359, 700. 
of resistance, 359,700. 
of thrust, 359, 700. 

Lining of tunnels, 754. 

Link. See Chain, 
expansion, 614. 
suspension, 614. 






850 


INDEX. 

Liquid—Members. 


Liquid, Liquids. See Water, 
buoyancy of, 234, 235. 
compressibility, 217, 236. 
flow of, 236, 4c. 
friction of, 374c. 
measure, 390. 

pressure of, 222, 4c, 239, 4c. 
pressure, transmission through, 227. 
specific gravity of, 381. 

Li thofracteur, 664. 

Little Giant rock-drill, 656. 

Live load, 546, 564, 4c, 805-S07. 

Living force, 308, 317. 

Load. See also Beam, Truss, Pillar, 
Bridge, 4c. 

on beams, constants for, 491. 

on bridges, greatest probable, 606, 623. 

cart load, of earth, 4c, 742, 4c. 

on driving-wheels, 546, 564, 805, 4c. 

on earth, safe, 634. 

for a given deflection, 506,508, 510. 

for I beams, 521. 

live, 546, 564, 805-807. 

for locomotives, 808. 

moving, 546, 564, 805-807. 

for piles, 643. 

on roof, 216, 221, 580. 

of sand, 427. 

on turntables, 794, 795. 

Lock, air, 648. 

Lock-nut, 408, 765, 768. 

Lock-Ken viaduct, 599, 647. 

Locke level, 205. 

Locomotive, Locomotives, 805-810. 
“adhesion ” of, 374a, 808, 809. 
driving-wheels, 805, 4c. 

weights on, 546, 564, 805-807. 
evaporation by, 803. 
house, cost, 799. 
repairs, 810, 815. 
statistics, 814, 4c. 
turntables for, 791. 
water for, 218, 800. 
weights, 546, 805, 4c. 

Locust, strength, 436, 463, 493. 
Logarithms, 38, 39. 

to find roots by, 39. 

Long 

chords, table, 729. 
measure, 887, 392. 

Longitude, degree of, length of, 387, 388. 
Lorenz switch, 775. 

Lowering of centers, 711-713, 720. 
Lubricants, 374c, 

Lubrication of turntables, 792, 4c. 
Lumber, 420-425. See also Wood, Timber. 

price, 4255. 

Lune, circular, 146. 

M. 

Machine, 
drill, 652,754. 
funicular, 325, 344. 
riveting. 471. 

for tapping pipes, 294, 299. 
Magneto-electric blasting, 665. 
Mahogany, strength, 434,436, 463, 493. 
weight, 383. 


Mail 

cars, 811. 
earnings, 814. 

Maintenance of road and real estate, 815 
Man, 

power of, 378. 
weight, 606, 623. 

Manilla rope, 414. 

Mansfield frog, 783. 

Map, to reduce or enlarge, 122. 
Maple-wood, 
strength, 463, 493. 
weight, 383. 

Marble, 

expansion by heat, 212. 
strength, 437, 493. 
weight, 383. 

Masonry, 

in abutments, quantity, 703. 
adhesion of cement to, 677. 

of mortar to, 670. 
in arch bridges, quantity, 702-708. 
backed by concrete, 679. 
cost, 667. 

courses, inclination of, 683, 700. 

lead between, 438. 
friction of, 373, 4c, 683, 700. 
incrustation of, 673, 678. 
in piers, quantity, 708. 
joints. See Joints, 
quantity in arches, 702-708. 
in piers, 708. 
in retaining walls, 690. 
in walls of wells, 158. 
in wing-walls, 704. 
railroad, cost, 804. 
in retaining walls, 683, 4c. 
strength of, compressive, 437. 
weight, 229, 381, 4c. 

Mass, 317, foot-note. 

Materials, 
fatigue of, 435. 
strength of, 434. 
weight of, 381. 

Matter, defined, 306. 

Maximum 

pressure, angle of, 687. 
prism of, 687. 
slope of, 687. 
velocity, 268. 

Mean 

depth, hydraulic, 244, 272. 
haul, 743. 
radius, 244, 272. 
velocity, 243, 268, 4c. 

Measure, Measures, 385. 
commercial, size of, by weight of water, 
391. 

cubic, 389. 

French, 391-393. 
long, 387. 
metric, 391, Ac. 

Russian, 394. 

Spanish, 394. 
square, 389. 

Mechanics, 305. 

Melting points, 212. 

Members, web, of trusses, 529. 














INDEX. 


851 


Mercu ry—N atn r al 


Mercury, 

barometer, 207, 215. 
freezing-point, 213. 
thermometer, 213, 215. 
weight, 383. 

Meridian, to find by North star, 177. 
longitude, 387, 388. 
variation of compass, 193, 196. 

Metacenter, 235. 

Metal, Metals. See also the names of the 
several metals, 
blasting of, 663. 
cohesive strength, 464. 
compressibility, 434. 
compressive strength, 438. 
ductility, 434. 

effect of cement on, 670, 673. 
of heat on, 212. 
of lime on, 670, 673. 
of mortar on, 670. 
of water on, 218, 645. 
elastic limits of, 434. 
expansion by heat, 212. 
friction of, 370, &c. 
limit of elasticity, 434. 
modulus of elasticity, 434. 
shearing strength, 476. 
sheet, 402-404, 410, 415, 416, 418, 419. 
strength, 434, 438, 464, 476, 477, 493. 
stretch of, 434. 
tensile strength, 464. 
torsional strength, 477. 
transverse strength, 493. 
weight of, 381, Ac. 

Meters, wheel, 270. 

Metre, Metres, 
length of, 391. 
radii, &c, of curves in. 728. 

Metric 

measures, 391. 
railroad curves, table,. 728. 
weights, 381, Ac, 393. 

Mica, weight, 383. 

Middle ordinates, 141,726,728,730,761,786. 

Mile, Miles, 

freight-ton-mile, 809, Ac, 814, &c. 

geographical, 387. 

land and sea, 387. 

nautical, 387. 

passenger, 814, Ac. 

scale of, 187.. 

sea-mile, 387. 

square (section), 3S9. 

ton-mile, 809, Ac, 814, Ac. 

train-mile, 809, &c. 

Mills, floating, 280. 

Miner’s friend powder, 661. 

Minutes in decimals of a degree, 57. 
of time, 395. 

Mitchell’s screw pile, 645. 

Mitred joints for rails, 764. 

Models, caution, 490. 

Modern explosives. 661. 

Modulus. See Coefficient, Strength, Ac. 
of elasticity, 434. 
of flow, 259, 266. 
of friction, 371. 
of resistance (coef of res), 48';. 
of rupture, 485. 


Mogul locomotives, 805, Ac. 

Moisture, effect on cement, 673. 
on sound, 211. 
on zinc, 419. 

Moment, Moments, 335. 
in beams, 478-489, 528, 537, Ac. 
breaking, 479-484. 
equilibrium of, 338. 
of inertia, 365, 486, 487. 
of resistance, 484, 486, 488. 
of rupture, 479-484. 
of stability, 229, 235, 337, 357, 688. 

Momentum, 310. 
of water, 234. 

Money, 386, 387. 

Monkey-switch, 772. 

Mont Cenis tunnel, 754. 

Morin’s laws of friction, 372. 

Mortar, 

adhesion of, 670. 

in arches, 690, 709, 713. 

bricks, Ac, 669. 

cement, 676. 

clay, effect on, 670. 

effect on iron, 620, 670, 673. 

on wood, 670. 
frozen, 675. 
grout, 670. 
pointing, 674. 
in retaining walls, 684. 
rubble, cost of, 668. 

weight of, 383. 
salt, effect on, 670, 678. 
sand for, 669, 677. 
strength of, tensile, 466, 676, 678. 
in water tanks, 801. 
weight, 383, 670. 

Moseley roof, 600. 

Motion, 306. 
accelerated, 311. 
circular, 365. 
defined, 306. 
quantity of, 310. 
retarded, 311. 
uniform, 311. 

Moving 
force, 310. 

load, 546, 564, 805-807. 

Mud 

penetrability, 644. 
in reservoirs, 288. 
weight of, 383, 632. 

Muskrats, 288. 

N. 

Nails, 4256. 
shingling, 429. 
slating, 428. 

Narrow-gauge railroad 
cars, 811, 812. 
locomotives, 806, &c. 
statistics, 818. 

Nasmyth pile-driver, 642. 

Natural 
chords, 105. 

Portland cement, 673. 
sines, Ac, 59, 60. 
slope, 684, 686, 690. 

| Nautical mile, 387. 









852 


INDEX. 


Needle—Paste. 


Needle, compass, 190, 196. 

Net 

earnings of railroads, 814, Ac. 
irou, net plate, net joint, 470, 638. 
Neutral axis, 479, 485, 487. 

Niagara suspension bridge, 622. 
Nitro-glycerine, 661. 

Nonagon, 110. 

North and south line, to find, 177. 
North star, 177. 

Northing, 168. 

Note-book, level, form of, 204. 
Number 

of frog, 781, 786. 
by wire-gauge, 410, 411, 412. 
Nuts, 406. 

Nut-locks, 408, 765, 768. 

Cambria, 765. 


o. 

Oak, 

strength, 434, 436, 463, 476, 493. 
weight, 383. 

Oblique, Obliques, 
beams, 340, 480, 496. 
flanges, 530. 
lines, 54. 
pillars, 457. 

pressure, 225, 314, 352, 687. 
in trusses, best inclination for, 548. 
length of, to find, 122, 608. 
Obstacles in surveying, to pass, 175. 
Obstruction, Obstructions, 
to flow, 279d, Ac. 
by piers, 279d, Ac. 
in pipes, to prevent, 292. 

Octagon, 
area, 110. 
to draw, 121. 

Octahedron,154. 

Offset, 683. 

Oil, Oils, 

coal, weight (Petroleum), 383. 
dead, 425. 

for locomotives, cost, 810, 815. 
olive, 374c. 
weight, 383. 

wells, nitro-glycerine in, 661. 

Olive oil, lubricating power of, 374c. 
Open channels, flow in, 268, &c. 
Openings, 

contiguous, flow through, 261. 
flow through, 257, Ac. 
with short tubes, flow through, 259. 
in thin partition, flow through, 260. 
Ordinate, Ordinates, 
elliptic, 149. 
to find, 141. 

middle, 141, 726-731, 761, 786. 

parabolic, 152. 

tables, 726-731, 761, 786. 

Oscillation, center of, 365. 

Osgood excavator, 750. 

Otis excavator, 751. 

Outer rail, elevation of, 729. 

Outflow, velocity of, theoretical, 258. 
Outlet valves, 290. 


Oval, to draw, 151. 
Overfall, 
dams, 282. 
discharge over, 264. 
for reservoir, 289. 


Pa 

Pa 


?( 


P. 

Packing piece, 471, 545, Ac. 

Paint, Paints, 429. 
brushes, to clean, 430. 
for iron, 430. 
on zinc, 403. 

Painting, 429. 

Panel, 

defined, length of, 548. 

diagonal of, to find length of, 122, 608. 

Paper, 433. 
car-wheels, 812. 
pipes, 294. 
wheels, 812. 

Parabola, 152, 153. See Parabolic, 
arc of, 152. 

center of gravity of, 348. 
to draw, 153. 
frustum of, 152. 
ordinates, 152. 

semi-, center of gravity of, 348. 
tangent to, to draw, 153. 
zone of, 152. 

Parabolic 
arc, 152, 153. 
conoid. 167. 

frustum of, 167. 
curve, 152, 153. 
frustum, 152. 
ordinates, 152. 
zone, 152. 

Paraboloid, 167. 
center of gravity of, 349. 
frustum of, 167. 

Parallel 
forces, 235, 347. 
lines, to draw, 56. 
plates, 189. 

Parallelogram, Parallelograms, 57, 119. 

angles iu, 57. 

center of gravity of, 348. 

of forces, 320, Ac. 

Parallelopiped, 155. 
center of gravity of, 349. 
of forces, 333. 

Parapets of suspension bridges, 621, Ac. 
Parlor cars, 811, 812. 

Partial 

contraction, 259, Ac, 263, Ac. 
on weirs, 263, Ac. 

payments (Equation of Payments), 37. 
Partition, thin, flow through, 260. 
Partnership, 37. 

Passage-way in tunnel, 754. 

Passenger 
cars, 811, 812. 

earnings and expenses, 811, 814, Ac. 
locomotives, 805-810. 
mile, 814, Ac. 
train expenses, 815. 

Paste, lime, 670. 








853 


INDEX. 

Patterns-Pipe. 


’atterns, weight of castings by size of, 
398. 

’aving, 

j Belgian, 668. 
brick, 671. 

(Payments, 
equation of, 37. 

partial (Equation of Payments), 37. 

Pencils, lead, 433. 

Pendulums, 364. 
hydrometric, 269. 
seconds, 385. 

Pennsylvania R R, 
bridges 

of I beams, 524. 
of riveted girders, 543, Ac. 
standard rolling loads, 546. 
locomotives, 546, 806, 807. 
track-tank, 802. 

Pentagon, 110. 

Perch 389. 

Percussion, 
center of, 365. 
drills, 653. 

Perimeter. See also Circumference, 
wet, 244, 271. 

Permanent way, 759. 

Permutation, 36. 

Perpendicular, to draw, 55. 

Persian wheel, 379. 

Petroleum, weight, 383. 

Philadelphia, 

Chestnut St bridge, 599. 

foundations, 636. 

South St bridge, 648. 

Wissahickon bridge, 720. 

Phoenix segment-columns,441,442,443,449 
dimensions, weights, Ac, 449. 
strength, 442, 443, 449. 
in trestles, 758. 

Phosphor-bronze wire, strength, 464. 

Picks, wear of, 743. 

Pier, Pi ere, 

■ abutment, 699. 

[ foundations for, 633, Ac. 
masonry, quantity in, 708. 
obstructions by, 2794, Ac. 
of suspension bridges, 618, &c. 

Pierre perdue, 634. 

Piezometer, 239. 

Piles, foundation, 640. See also Founda¬ 
tions. 

Pillar, Pillars, 439, Ac. 
of angle-iron, 440-442. 
capitals of, shapes of, 457. 
of channel-iron, 441, 442, 456. 
ends of, shapes of, 439, 457. 
factor of safety, 442, 446. 

Gordon’s formula, 439. 
hinged ends, 439. 
of I beams, 441, 442, 454. 
iron, 439, Ac. 

coefficients of safety, 442, 446. 
strains usually allowed, 457. 
strength of, 439, Ac. 
masonry, strength of, 437. 
oblique, 457. 

Phoenix segment, 441-443, 449, 75 ; . 


Pillar, Pillars—continued, 
pin-ended, 439. 
radius of gyration, 440. 
with rounded ends, 439. 
safety, factor of, 442, 446. 
segment, Phoenix, 

dimensions, weights, Ac, 449. 
strength, 441, 442, 443, 449. 
in trestles, 758. 
steel, 442, 458. 
strength of, 439, Ac. 

T and + iron, 441, 442. 
wooden, 458, Ac. 

Pin, Pins 

and eye-bare, 439, 612. 
surveying, 176. 

Pine, 

pillars of, 459. 

strength, 434, 436, 458, 463,476,477,493. 
weight, 383, 384. 

Pinions and wheels, 342. 

Pipe, Pipes, 
air-valves for, 297. 
angles in, 256. 

areas and contents of, 157, 247. 
bends in,255. 
branches in, 296. 
brass, seamless, 417. 
bursting of, 234, 298, 303. 

thickness required to prevent, 232- 
234, 293. , 

bursting pressure in, 239. 
cast-iron, 290-293, 297, 399. 

weight, 293, 297, 399. 
cement and iron, 294. 
concretions in, to prevent, 292. 
contents and areas of, 157, 247. 
copper, seamless, 417. 
cost of, 293, 297. 

of laying, 297. 
couplings for, 293, 295, 405. 
cracks in, 296. 
curves in, 255. 

diameters of, 245, 290, 291, 293, 297. 
square roots of, 247. 
for water-supply, 290, Ac. 
discharge from, 
formulae, 243. 
principles of, 236, Ac. 
drain, 2794. 

terra-cotta, 2794. 
drawn brass and copper, 417. 
enlargements in, 257. 
ferrules for, 299. 
flexible joints for, 296. 
flow in, 236, Ac. 

Kutter’s formula, 214, 271. 
friction in, 257. 
galvanic action in, 293. 
galvanized, 299. 
gates for, .‘101. 
gutta-percha, 294. 
iron, 

cast, 290-293, 297, 399. 

weight, 293, 297, 399. 
and cement, 294. 

♦fittings for, 293, Ac, 405. 
joints for, 293, 295, 407. 






854 


INDEX. 

Pipe—Potential. 


Pipe, pipes, iron—continued, 
thicknesses of, 233, 293. 
wrought, 293, 405. 
weight, 293, 405. 
joints for, 293, 295, 407. 

flexible, 296. 
kalameined, 293. 
knees in, 256. 
laying, 297. 

lead for, 293.295, 297. 
leaden, 234, 299.416. 

thicknesses of, 234. 
long, pressure of water in, 257. 
material of, effect on velocity, 244. 
to mend, 296. 

obstructions in, to prevent, 292. 
paper, 294. 

pressure of water in, 232, Ac, 239. 

resistance to pumping, 257. 

seamless, 417. 

service, 294, 299, 416. 

sleeves for, 296. 

stand, 298. 

steam, 405. 

stop-valves for, 301. 

street, 290. 

swellings in, effect of, 257. 
tapping of, 294, 299. 
terra-cotta, 279d. 
thickness required, 232-234, 293. 
valves for, 301. 

of varying diameter,disch through, 254. 
velocity in, 243, 245, &c. 
water, 290, Ac, 293, 297, 399. 
weight, 293, 297, 399,405. 
wooden, 294. 

wrought-iron, 293, 294, 405. 
weights, prices, Ac, 293, 405. 

Pit of turntable, 791. 

Pitch 

of rivets, 472, Ac, 539. 
of roofs, 428,581. 
weight of, 384. 

Pitot’s tube, 269. 

Pittsburgh suspension bridge, 624. 

Pivot 

of turntables, 792, Ac. 

Plan, to reduce or enlarge, 122. 

Plane, Planes, 110. 
of flotation, 235. 
forces in different, 332, Ac. 
forces in one, 319, Ac. 
inclined, 352, Ac, 363. See also In¬ 
clined Plane, 
surfaces, 110. 
trigonometry, 112. 

Plank, 

board measure, table, 420. 
in foundations, 633. 
sheet piling, 641. 
thickness for a given pres, 637. 

Plant, railroad, 814, Ac. 

Plaster, 426. 
of Paris 

effect on metals, 673. 
price of. 673. 

strength, 437, 466. , 

weight, 384. 


Plate, Plates. See Sheet. ,, 

angle, 764. I. 

beams, 537. 
bolster, 524, 544, 614. 
buckled, 409. 
fish, 764. 
frog, 783. 
girders, 537. 
glass, 432. 
iron 

prices, 402. 
turntables, 793. 
net, 470. 

openings in, flow through, 257, Ac, 

parallel, 189. 

steel, tinned, 418. 

terne, 418. 

tin, 418. 

tinned steel, 418. 
wall, 524, 544, 614. 

Platform 
cars, 811. 
revolving, 799. 

Platinum, 384, 464. 

blasting caps, 665. 

Plenum process, 648. 

Plug, fire (fire-hydrant), 304. 

Plumb level, to adjust, 206. 

Plumbago, as a lubricant, 374c. 

Pneumatic . 

foundations, 647, Ac. 

Pocket sextant, 194. 

Point, Points, 

of application of force, 309, 314, Ac. 
boiling, 210, 217. 

leveling by. 209. 
of contrary flexure, 515. 
freezing, 212, 217. 
frog, 781-789. 
melting, 212. 
position of, to find, 118. 
switch, 774-776. 

Pointing mortar, 674. 

Pole star, 177. 

Polygon, Polygons, 110. 
center of gravity of, 348. 
of forces, 329. 

irregular, to find area of, 122. 
to reduce to a triangle, 121. 
regular, to draw, 121. 

Polyhedron, Polyhedrons, regular, 154. 
Pond, discharge of, time required for, 264 
Poplar, strength, 436, 463, 493. 

Pound (coin), value of, 386. 
weight, 387. 

Porous bodies, specific gravity of, 381,384. 
Port, establishment of, 219. 

Portage viaduct, 756, 757. 

Portland cement, 673. 

Post, Posts. See also Pillars, 
fence, 803. 
king, 554. 

pivot, in turntables, 792, Ac. 
queen, 555. 

Samson, 630. 
in trestles, 756. 
in trusses, 547,549. 

Potential energy, 318. 








855 


INDEX. 


1*4X1 rin 

Pouring-clamps for pipe-joints, 2!) 4, 295. 
Powder, 384, 660. 

Power, Powers. See Steam, Water, Wind, 

Ac. 

animal, 377. 
defined, 311. 
fifth, 251. 

square roots of, 253. 
gain of, 336. 
of horse, 375, 377, 801. 
of locomotives, 808. 
man, 378. 

second and third, tables, 41. 
tractive, 375. See also Traction. 

Pratt truss, 595. 

; Prejudicial work, 316. 

Press, Presses, 
hydrostatic, 227. 

Pressed brick, 670. 

Pressure. See Load, 
of air, 215, 648, Ac. 

barometer, leveling by, 207. 
in arches, 342,359, 694. 
boiler, 809. 

center of, 227, 235, 350, 700. 
on centers for arches, 713. 
cylinder, of steam, 809, Ac. 
in dams, 286. 
distribution of, 
in plane surfaces, 231a. 
of earth, 683, Ac, 687. 
effect of, on friction, 371. 
on foundations, 634. 
on inclined plates, 352, Ac, 363. 
initial, of steam, 809. 
line of, 359, 700. 
maximum, angle of, 687. 
prism of, 687. 
slope of, 687. 
in pipes, 232, 239. 
plate, Gauthey’s, 269. 

( in reservoirs, 288. 

on retaining walls, 683, Ac. 
of running streams, 279/, Ac. 
of running water in pipes, 239. 
steam, 809. 

strength, compressive, 436. 
transmission of, through liquids, 227. 
of water, 222, Ac, 2:39, Ac. 
in cylinders, 232. 
in long pipes, 257. 
in pipes, 232, Ac, 239. 
running, 239, 279/, Ac. 
still, 222. 

walls to resist, 229, Ac. 
of wind, 216. 

Prices. See the article in question. 
Prism, Prisms, 155. 
center of gravity of, 349. 
frustums of, 155. 

center of gravity of. 349. 
of maximum pressure, 687. 

Prismoid, 161. 

Prismoidal formula, 161. 

Profile, Profiles, 199, Ac. 
curved, 692. 
paper, 433. 

transformation of, 691. 


g— Rail. 

Progression, 36. 
arithmetical, 36. 
geometrical, 36. 

Proportion, 35. 
compound, 35. 
simple, 35. 

Protracting by chords, 105. 

Puddle-walls, 288. 

Pulley, 342. 

Pump, Pumps, 801. 
chain, 379. 

day’s work at, 378,801. 
sand, 626, 650. 

Pumping. See Pump, 
engine, 801. 
mains, friction in, 257. 

Purlins, 551, 5S2. 

Pyramid, Pyramids, 160. 
center of gravity of, 349. 
frustum of, 160. 
oenter of gravity of, 349. 

Q* 

Quadrant, Castellfs, 269. 

center of gravity of, 348. 

Quarrying, 651-667. 

Quartz, weight, 384. 

Queen truss, 555, 578. 

Quicklime, 669. 
to preserve timber, 425a. 
weight of, 383. 

It. 

Rack-a-rock, 664. 

Radii, Radius, 
to find, 123, 141. 
of gyration, 366, 367, 439, 440. 

square of, 440, 538, 540. 
mean, 244, 272. 

of railroad curves, tables, 726-728. 
of turnouts, 786. 

Rafter, Rafters, 551, Ac, 582. 

feet of, friction at, 355. 

Rail, Rails, 760, 763. 
bending, ordinates for, 761. 
creeping of. 764. 
elevation of outer, 729. 
expansion by heat, 212, 764. 
fence, 803. 
frog, 781. 

guard or guide, 774, 779, 781. 
joints, 763. 

ordinates for bending, 761. 
outer, elevation of, 729. 
renewals, 815. 
roads, 722. 
ballast, 759. 

bridges. See Bridge, Truss, Arch, Ac, 

cars, 811, 812, 814, Ac. 

construction, 722, 804. 

cost of, 804. 

cross-ties, 759. 

gauge, in U. S., 814. 

pile-driver, 642. 

resistance on, 374e. 

roadway, 759. 







856 INDEX. 

Kail—Rock. 


Rail, Rails—continued, 
roads, 

shops, cost of, 799. 
spikes, 762. 
statistics, 814 to 818. 
switch, 770. 
ties, 759. 

time, standard, 396. 
track-tank, 802. 
traction on, 377, 808, 810. 
turnout, 770. 
water stations, 800. 
safety, 779. 
stock, 774. 

switch-rail, length of, 776. 
way, see Rail-road. 

Rain, 220. 

reaching sewer, rate of , 279c. 
water, 218, 385. 

Ram, hydraulic, 280. 

water, 234, 298, 303. 

Ramming 
of cement, 675. 
of concrete, 680. 

Ramsbottom’s track tank, 802. 

Rand rock-drill, 656. 

Random stone, 634. 

Range of stress, 435. 

Ratio, 35. 

Reaction, 311. 
of soils, elastic, 644. 

Real estate, maintenance, railroad, 815. 
Reaumur thermometer, 213. 

Re-burning of cement, 674. 

Rectangle, Rectangles, 119. 

center of gravity of, 348. 

Reflection, to measure heights by, 117. 
Reflexion, angle of, 255. 

Refraction and curvature, table, 115. 
Regular figures, 110. 

Regular solids, 154. 

Repair, Repairs, 
of bubble-tube, 193. 
of cars, 811, 815. 
of cross-hairs, 193. 
of locomotives, 810, 815. 
of pipes, 296. 
in reservoirs, 289. 
of road, 743. 

of road-bed, railroad, 815. 
of rolling-stock, 810,811, 815. 
of track, 815. 

Repeated stress, 435. 

Repose, angle of, 355, 371. 

Reservoir, Reservoirs, 287, etc. 
discharge from and into, 262. 
evaporation from, 222. 
for railroads, 801. 

Resistance. See also Loads, Strength, Ac. 
angle of, limiting, 355, 371. 
of beams, 484. 
of cars, 374c, 808. 
coefficient of, 485. 
to flow, 244, 255, 257, 271, Ac. 
of friction, 370. 
on grades, 808. 
limiting angle of, 355, 371. 
line of, 359, Ac, 700. 


Resistance—continued, 
modulus of, 485. 
moment of, 484. 486, 488. 
on railroads, 374e. 

Resolution of forces, 319, &c. 

Resultants, 319, Ac. 

Retaining walls, 683. 
clay backing, <534. 
curved profiles, 692. 
masonry in, quantity of, COO, 692. 
surcharged, 685, Ac. 
theory of, 686. 

transformation of profile, 691. 

Retarded velocity, 311, 

Reverse bearing, 171. 

Revetment, 692. 

Revolving bodies 365. 

Rhomb, 119, 155. 

Rhombic prism, 155. 

Rhombohedron, 155. 

Rhomboid, 119. 
center of gravity of, 348. 

Rhombus, 119. 
center of gravity of, 348. 

Rhumb-line, 171. 

Right angle, to draw, 55. 

Rigid bodies, force ill, 305. 

Ring, Rings. See Circle, Ellipse, Ac. 
circular. 146,167. 
joint, 768. 
tightening, 583. 

Rip-rap, 634. 

Rise of arch, 693. 
of roof, effect on weight, 581. 

Rivers. See Water, Rain, Ac. 
dams, 282. 
flow in, 268, Ac. 
reservoirs, 287. 
scour of, 279/. 

Rivet, Rivets, 468, 539. 

Riveted 
beams, 537. 
girders, 537. 
joints, 468, 539. ,, 

Road, Roads, 

-bed, repairs, 815. 
cart, repairs, 743. 
grade, 375, 723. 

tables, 176, 354, 723-725. 
maintenance, 743, 815. 
rail-. Sec Railroad, 
traction on, 375. 

-way, acres required for, 722. 
drainage of, in arches, 708. 
rail mid, items of, 759. 
width of, in bridges, 604. 

Rock, Rocks, 
blasting, 660. 

broken, voids in, 380, 678, 751. 
channeling, 658. 
drill, 651, Ac. 
hand, 658. 
machine, 652. 
steam, 652. 
removal, 751, Ac. 
strength of, 434, Ac. 
weight of, 381, Ac. 
work in tunnels, 754. 







INDEX. 

Rockers—Sand 


857 


Rockers, expansion, 614. 

Rod, Rods, 

of brickwork, 389, 672. 
iron, 402. 
king, 553. Ac. 
queen, 555. 
suspending, 620. 
tie, 551, &c, 572. 
upset, 408. 

Rolled 

iron. See Iron, Wrought. 

Roller, Rollers, 
anti-friction, 374e. 792, Ac. 
expansion, 614. 
friction, 374e, 792, Ac. 

Rolling 
friction, 3745. 
load, 546, 564, 805-807. 
stock, 805, Ac, 814, 815, Ac. 

repairs, 815. 
resistance of, 374e. 

Roof, Roofs, 
arched, COO. 
copper, 416. 
cost of, 580. 

coverings, weight of, 551, Ac, 581. 
details, 582. 

effect of acid fumes on, 418,428. 

of rise on weight of, 581. 

Fink, 574, 578-580. 
frost-proof, Burnham’s, 801. 
iron for, 403. 
arch, 600. 
details, 582, 583. 
lead, 415, 416. 
leak in, to stop, 429, 431. 

Moseley, 600. 

painting of, 403, 430. 

pitch of, 428,581. 

purlin, 551, 582. 

rafter, 355, 551, Ac, 582. 

rise of, effect on weight of, 581. 

sheet-iron, 403. 

shingle, 429. 

81 ate, weight of, 428. 
snow on, 221. 
to stop leak, 429, 431. 
tin, 418. 

truss. See Truss. 

weight of, affected by rise, 581. 

with load, 580. 
wind on, 216. 
wooden, details, 613. 
zinc, 418. 

Root, Roots, 40. 
cube and square, tables, 40. 
of decimals, to find, 53. 
fifth, 251. 

of large numbers, to calculate, 52. 
square and cube, tables, 40. 
square, of diameters, 247. 
of fifth powers, 253. 

Rope, Ropes, 414. 
strength of, 414, 466. 
weight of, 414. 
wire, 413. 

Rosendale cement, 673. 

Rosin, weight, 384. 


Rot of timber, 425. 

Rotary drills, 652. 

Rotary motion, 365. 

Rotating bodies, 365. 

Rough-casting, 431, 674. 

Roughness, 

coefficient of, 244, 272, 273. 

Rubble, 

adhesion to mortar, 670, 677. 
arches, 681,696. 
cost, 668. 
foundations, 634. 
proportion of mortar in, 383. 
quarry, loose, 380, 751. 
retaining walls, 690. 
strength, 437. 
voids in, 669, 741. 
weight of, 383. 

Rule, Rules, 

{ of three, 35. 

two-loot, to measure angles by, 58. 
Rupture. See Strength, 
coefficient of, 485. 
constant of, 485. 
modulus of, 485. 
moment of, 479-484. 

Russian weights and measures, 394. 

s. 

Safety, 

allowance for. See Safety, Factor of. 
castings, 770, 779. 

coefficient of. See Safety, Factor of. 
factor of, 

for beams, 499, 521, 540. 
for piles, 644. 
for pillars, 442, 446. 
for retaining walls, 685. 
for suspension bridges, 617. 
for truss bridges, 607. 
rail, 779. 

switch, 770, 775, 778. 

Salmon brick, 671. 

Sal t, 

effect on mortar, 670, 678. 
water, effect on iron, 218, 645. 

weight of, 217. 
weight of, 384. 

Samson 
J joint, 765. 

1 post, 630. 

Sand 

augers, 626, Ac. 
bar sand, 669. 
blasting of, 663. 
in cement, 674. 676, 678, 679. 
for centers, striking, 713. 
in concrete, 678. 
cost of, 669. 
dredging, 631. 

effect on cement, 674, 676,678. 
excavating in, 742, Ac. 
for foundations, 634. 
load of, 427. 

for mortar, 669, 677, 678. 
natural slope of, 690. 
penetrability of, 644. 
piles, 626, 650. 









858 INDEX. 

Sand-Single. 


Sand—continued, 
in plaster, 42(5. 
pressure of, t>83, &c. 
price of, G69. 
pump, G26, 6.00. 
retaining walls for, 683, &c. 
slope, natural, 690. 
specific gravity of, 381, 384. 

-stone, 

expansion by heat, ‘212. 
strength of, 431, 466, 493. 
weight, 384. 

sustaining power of, 634,644. 
voids in, 384,677. 
weight of, 381, 384. 

Sap of timber, 425. 

Saylor’s Portland cement. 673. 

Scales, track, 803. 

Scarfs, lengthening, 610. 

Scarp revetment, 692. 

Scoop, tender, 802. 

Scour of streams, 279/. 

Scrapers, earthwork, 747. 

Sc reeding, 426. 

Screw, Screws, 342. 

Archimedes, 379. 
for centers, striking, 713. 
cylinders, 645. 
leveling, 189, 202. 
piles, 645. 

standard dimensions, 406. 

Seamless pipes and tubes, 417. 

Sea 

mile, 387. 
tides, 219. 

water, 217, 219, 645. 
worms, 425. 

Secant, 59. 
of latitudes, 177. 

Seconds in decimals of a degree, 57. 
Seconds pendulum, 385. 

Section 

effective, in riveted girders, 538. 
of land, area of, 389. 

Sector, circular, 146. 

center of gravity of, 348. 

Sediment in reservoirs, 288. 

Segment, Segments, 
circular, 146, 147. 

center of gravity of, 348. 

-columns, Phoenix, 441, 442, 449. 

in trestles, 758. 
spherical, 166. 
center of gravity of, 349. 

Self-acting frog, 785. 
switch stand, 775. 

Sellers standard dimensions of bolts, &c, 
406. 

Sellers turntable, 792. 

Semi-circle, center of gravity of, 348. 
Semi-parabola, center of gravity of, 348. 
Service pipe, 294, 299, 416. 

insertion of, 294, 299. 

Setting of cement, 674. 

Settlement 
of backing, 684. 
of centers, 713, &c, 720. 
of embankment, 741. 


Sewer, Sewers, 
flow in, 279c. 

Rutter’s formula, 244, 271. 
grade of, 279c. 

rain-water, rate of reaching, 279c. 
velocities in, 279c. 

Sextant, 

angles measured by, 114. 
box or pocket, 194. 
to adjust, 194. 

Shaft of tunnel, 754. 

Shafting, 

deflection of, 505. 
friction of, 374d. 
strength of, 477. 

Shale, weight, 384. 

Sharpening tools, cost, 743. 

Shear, 476, 532. 

double and single, 470, 476. 

Shearing 
of beams, 532. 
of nails, 4255. 
of rivets, 470, &c. 
strains, 532. 
strength, 476. 

Sheet, Sheets, 
brass, 415. 
copper, 415, 416. 
iron, 403. 
corrugated, 403. 
galvanized, 403. 
roof, 403. 
lead, 415, 416. 

metals, thickness of, 410, 411. 
piles, 641. 
zinc, 418. 

Sheeting of centers, 711, 719. 

Shell 

-lime, 670. 
spherical, 166. 

weight of. 398, 400. 

Shilling, value of, 386. 

Shingles, 429. 

Shoes for piles, 644. 

Shops, railroad, cost, 799. 

Shoveling earth, 742. 

Shovels, wear of, 743. 

Shrinkage of embankment, 741. 

Siemens’ electrical blasting machine, 665, 
Sieves for cement, 678. 

Signal target, 772, &e, 775, 779. 

Silver, 

coins, &c, 387. 
strength, 466. 
weight, 384, 387. 

Similarity, geometrical, 54. 

Simple 

proportion, 35. 
interest, 37. 

Simultaneous firing of blasts, 665. 

Sine, Sines, 59. 
natural, defined, 59. 
table, 60. 

of polar distances of Polaris, 177. 
Single 

riveting, 468. 
rule of three, 35. 
shear, 470. 









859 


INDEX. 

Siphon—Statics. 


Siphon, 240. 

Skew-back, 693. 

Skidding of wheels, 374a. 

Slacking of lime, 669, 670. 

Slate, 427. 

compressibility, 434. 
expansion by heat, 212. 
roofs, weight of, 428. 
strength, 

compressive, 437. 
tensile, 466. 
transverse, 493. 
weight, 384. 

Slating, 427. 

Sleeping cars, 811, 812. 

Sleeves for pipes, 296. 

Sliding, 370. 
angle of, 355, 371. 
friction, 370, &c. 
of retaining walls, 692. 

Slope, Slopes. See Grade, 
allowance for, in chaining, 176. 
angle of, 176, 354, 723, 724. 
hydraulic, 244, 272. 
instrument, 206, 724. 
of maximum pressure, 687. 
natural, 684, 686, 690 
tables, 176, 354, 723-725. 
in tunnels, 754. 

Slugger rock-drill, 656. 

Sluices in dams, 285. 

Snow, 221, 384. 

Soakage, loss by, 269. 

Soap as a lubricant, 374c. 

Soapstone, weight, 384. 

Soap-wash for walls, 672. 

Soffit, defined, 693. 

Soil, Soils, 
boring in, 626. 
dredging, 631. 
excavation of, 742. 
leakage through, 222. 
penetrability of, 644. 
pressure of, 683, &c. 
reaction of. elastic, 644. 
scour of, 279/. 

sustaining power of, 634, 644. 
weight, 382, under “ earth.” 

Solid, Solids, 154. 

center of gravity of, 349. 
defined, 54. 

expansion by heat, 212. 
measure, 389. 

metric, 392. 
mensuration of, 154. 
specific gravity of, 380, Ac. 

Sound, 211. 

South St bridge, Phila, 648, 

Southing, 168. 

Sovereign, 386. 

Span, defined, 693. 

Spandrel, 693. 
walls, 693, 698. 

Spanish weights and measures, 394. 

Specific gravity, 380. 

Speed, Speeds. See Velocity, 
of locomotives, 809. 
of teams, 743, 747. 


Spelter. See Zinc. 

Sphere, Spheres, 162, 163. See Spherical, 
center of gravity of, 349. 
one foot diameter, volume of, 389. 
one inch diameter, volume of, 389. 
table, 163. 

Spherical 

segment, 166. 

center of gravity of, 349. 
shell, 166. 

weight of, 398, 400. 
zone, 166. 

Spheroid, 166. 

center of gravity of, 349. 

Spigot in pipe-joint, 295. 

Spindle 
circular, 167. 
torsional strain in, 477. 

Spikes, 762. 

Splices, timber. 610, 612. 

Split switch, 774-776. 

Spreading of earth, 743. 

Spring, Springs, 
of arch, 693. 

effect of, in easing blows, 319. 
in foundations, 634. 
frog, 784. 

Spruce, 

strength, 434, 436, 463, 476, 493. 
weight, 384. 

Spudding, 628. 

Square, Squares, 
area, 119. 

equivalents of, in circles, 123. 
center of gravity of, 348. 
measure, 389. 

metric, 392. 
mensuration of, 119. 
of numbers, table, 41. 
of radius of gyration, 440, 538, 540. 
roots, 40. 

of decimals, to fiud, 53. 
of diameters, 247. 
of fifth powers, 253. 
of large numbers, to calculate, 52. 
tables, 40. 
sides of, 119, 123. 
tables of, 41. 

Stability, 235, 356. 
of arches, 358, &c, 700. 
frictional, 352-361. 
on inclined planes, 352, Ac. 
moment of, 357. 
of retaining walls, 683. 

Stable equilibrium, 235, 347, 358. 

Stadia hairs, 190. 

Stand-pipes, 298. 
for railroad water-stations, 801. 
for water-works, 298. 

Stand, switch, 772. 

Standard railway time, 396. 

Star, Stars, 

Alioth, 177. 

iron, sizes and prices, 402. 

North, 177. 

Pole, 177. 

to regulate a watch, Ac, by, 395. 
Statics, 306. 






860 


INDEX 


Station—Strength. 


Station, Stations, 
expense, 815. 
in surveys, 197, Ac, 204. 
water, 800. 
way, cost, 803. 

Statistics, 
arches, 095. 
railroad, 814-818. 
rainfall, 220. 

Stays, cable, 616. 

Steam, 

average pressure, 809. 
boiler pressure, 809. 
cylinder pressure, 809. 
dredges, 631. 
engine, 

locomotive, 805-810. 
pumping, 801. 
excavator, 750. 
expansion of, 809. 
initial pressure of, 809. 
pile drivers, 641, 642. 
pipes, 405. 
pressure, 809. 
rock-drill, 652. 

warming by, surface required, 399. 
Steel 

beams, 493, 512. 
buckled plates, 409. 
chains, strength. 415. 
cohesive strength of, 464, 465. 
columns, 442, 458. 
compressibility, 434. 
compressive strength, 438. 
cost of, 402. 
ductility, 434. 
elastic limit, 434. 
expansion by heat, 212. 
friction of, 373, Ac. 
modulus of elasticity, 434. 
nails, 4256. 
pillars, 442, 458. 
plates, 

buckled, 409. 
tinned, 418. 
price, 402. 
rails, 760. 

frogs of, 781. 
rope, 413. 

shearing strength, 476. 

strength, 438, 464, 465, 476, 477, 493. 

stretch of, 434. 

tensile strength, 464, 465. 

ties, 759. 

tires, 807, 812, 813. 
torsional strength, 477. 
transverse strength, 493. 
weight, 384, 400, 401. 
wire, 412. 
rope, 413. 

Sterling’s toughened cast-iron, 520. 
Stiffeuers for plate-girders, 539. 

Stock rails, 774. 

Stone, Stones, 
adhesion to cement, 677. 

mortar, 670. 
arch-, 693. 

arches, quantity in, 702. 


Stone, Stones—continued, 
artificial, 466, 678, 681. 
ballast, 759, 804. 
beams, 493, 501. 

-breaker, 680. 

bridges, 693. See also Arch. 

centers for, 711. 
broken, voids in, 380, 678, 751. 
buildings, cost of, 668. 
cohesive strength, 466. 
compressive strength, 437. 
for concrete, 678, Ac. 
crusher, 680. 

-cutter, day’s work, 667. 
dams, 229, 231. 
dressing, 667. 
drilling, 651, Ac. 
excavating, 751. 
expansion by heat, 212. 
friction of, 373, Ac. 
key-, 693. 

quantity in arches, Ac, 702, Ac. 
quarrying, 651-667. 
random, 634. 

Kansome’s, 466. 
strength, 437, 466, 493. 
tensile strength, 466. 
transverse strength, 493. 
weight, 381, Ac. 

-work, 651, 751. 
strength, 437. 
weight, 229, 383. 

Stop, Stops, 

corporation, for pipes, 294, 299. 
leak in roof, 429, 431. 

-valves for water-pipes, 301. 

Storage reservoirs, 289. 

Strain, Strains, 306-311, 434, Ac. 
in beams, vertical, 532. 
flange, 529, 537. 
repeated, 435. 
shearing, 532. 

in suspension bridges, 616, Ac. 
in trusses, 551, Ac. See also Trusses, 
vertical, in beams, 532. 

Stream, Streams, 
abrasion by, 279/. 
flow in, 268. 
to gauge, 268. 
horse-power of, 280. 
pressure of running, 279/, Ac. 
scour of, 279/. 
virtual head, 280. 

Street pipes, 290, Ac. See also Pipes. 

Strength, Strengths. See also the article 
in question, 
of arches, 693, Ac. 
of beams, 478,493. 
of bridges. See Arch, Truss, Ac. 
cohesive, 463. 
compressive, 436, Ac. 
of cylinders, 232, 516. 
of materials, 434. 
of piles, 643. 
of pillars, 439, Ac. 
of retaining walls, 683, Ac. 
of riveted joints, 468, Ac, 472. 
of shafting, 477. 






INDEX. 861 

Strength—Tie. 


Strength, Strengths—continued, 
shearing, 476. 
tensile, 463. 
torsional, 476. 
transverse, 478. 

Stress,defined, 307. See Strain, Strength, 
range of, 435. 
repeated, 435. 

Stretch 

of materials, 434, 463, Ac. 
of riveted joints, 471. 

Striking of centers, 711, 713, 720. 
Stringers, track, 545, Ac. 

Strut, Struts, 
defined, 325, 547, 554. 

-tie, defined, 325, 547. 

and tie, to distinguish, 325, 590. 

Stub’s gauge, 411. 

Stub switch, 770, 771. 

Stucco, 426, 674. 

Stumps, blasting of, 663. 

Sub-delivery, cost, 804. 

Subterranean temperature, 215. 

Sulphur, weight, 384. 

Sun-dial, to make, 397. 

Supplement of angle, 56. 

Supported joints, 763. 

Surcharge, 685, Ac. 

Surface velocity, 268. 

Surveying, 168. 

Suspended joints, 763. 

Suspenders of suspension bridges, 620. 
Suspension 

bridges, 615, Ac. 

cables of, 412, 615, Ac. 
links, 614. 
trusses, 548. 

Sway-bracing, 543. 

Swellings in pipes, effect of, 257. 

Swing bridges, strains in, 593. 

Switch, Switches, 770, Ac. 

Swivel, 583. 

Sycamore, 

strength, 434, 436, 463, 493. 
weight, 384. 

Symmetry, axis of, 235. 

Syphon, 240, Ac. 

flow in, 240. 

System, metric, 391. 

Systeme 
ancien, 393. 
usuel, 393. 

T. 

T iron, 442, 525. 

rails, 760, 763. 

Table, turning, 790. 

Tables. See the article in question. 
Tallow, 

as a lubricant, 374c. 
weight of, 384. 

Talus, 692. 

Tamping, 661, Ac. 

Tangential 
angle, table, 726-728. 
distance, table, 726-728. 

Tangent, Tangents. 59, 60. 
to circles, to draw, 121. 
to an ellipse, to draw, 150. 


Tangent, Tangents—continued, 
natural, 59. 
table, 60. 

to a parabola, to draw, 153. 
screw, 190, 202. 

Tank, 

frost-proof, 801. 

of tender, capacity of, 805, Ac, 808. 
thickness, 227, 803. 
track, 802. 
water, 800, Ac. 

Tapping of pipes, 294, 299. 

Tar, weight, 384. 

Target, signal, 772, Ac, 775, 779. 
Tarpaulin, 680. 

Taxes, railroad, 815. 

Teams, speed of, 743, 747. 

Telegraph expense, annual, 815. 
Tejnperatme, 212. See Heat, 
of air, 215. 

altitude, effect on, 215. 
effect on 
cement, 674, 675. 
evaporation, 222. 
metals, Ac, 212. 
rails, 212, 764. 
rainfall, 220. 
strength of iron, 466. 
survpying chains, 168. 
trusses, 614. 
velocity of sound, 211. 
weight of water, 217, 385. 
thermometers, 213. 

Tender, Tenders, 546, 805-810. 
capacity, 805, Ac, 808. 
dimensions, 805, Ac. 

-scoop, 802. 
weights, 546, 805. Ac. 

Tensile strength, 463. 

of riveted joints, 472. 

Teredo, 425. 

Terne plates, 418. 

Terra-cotta pipes, 279d. 

Test 

borings, 626, 633. 
of cements, 674, 675. 
of instruments, 191-206. 
Tetrahedron, 154. 

Theodolite, 193. 

Thermometers, 213. 

Thilmany process, 425a. 

Thin partition, flow through, 260. 
Three, rules of, 35. 

Three-way valves, 302. 

Throat of frog, 781. 

Throw of switch, 773. 

Thrust, line of, 359, Ac, 700. 

Tides, 219. 

Tie, Ties, 

-beam, 551, Ac. 
cross, 759, 804, 815. 
defined, 325, 547. 
land-, 692. 
rod, 551, Ac. 

raised, 572. 
steel, 759. 

-strut, defined, 325, 547. 

and strut, to distinguish, 325, 590. 








862 


INDEX. 

Timber—Truss, 


Timber. See also Wood, Wooden, 
beams, deflections, 499. 
loads, 499, 502, 512, 513. 
for railroad bridges, 514. 
board measure, table, 420. 
cohesive strength, 403. 
compressibility, 434. 
compressive strength, 430. 
cost, 425. 
creosoting, 425. 
crushing strength, 436. 
dams, 282. 
ductility, 434. 
durability, 425. 
elastic limit, 434. 
friction of, 373, &c. 
joints, 610, 612. 
limit, elastic, 434. 
modulus of elasticity, 434. 
preservation of, 425. 
shearing strength, 476. 
splices, 610, 612. 
strength, 436, 463, 476, 477, 493. 
stretch of, 434. 
tensile strength, 463. 
for ties, 759. 
torsional strength, 477. 
transverse strength, 493. 
trestles, 755. 
turntables, 797. 
weight, 381, &c. 

Time, 395. 

-piece, to regulate by star, 395. 
standard railway, 396. 

Tin, 418. 

compressibility, 434. 
ductility, 434. 
elastic limit, 434. 
expansion by heat, 212. 
leaded, 418. 

modulus of elasticity, 434. 
roofing, 418. 
strength, 438, 466. 
stretch of, 434. 
weight, 385, 400. 

Tinned steel plates, 418. 

Tire, 

car-wheel, 811, 812. 
locomotive, 807. 
wagon, 380. 

Toe of switch, 771, 774, 785. 

Ton, x, 387, 

of coal, volume of, 389. 

-mile, S09, &c, 814, &c. 

Tongue 
of frog, 780. 

-switch, 774. 

Tonite, 664. 

Tools, wear of, 743. 

Top heading, 754. 

Torpedoes, nitro-glycerine, 661. 
Torsion, 476. 

Toughened cast-iron, Sterling’s, 520. 
Towers of suspension bridges, 618, 620. 

valve towers, 289. 

Town’s truss, 596. 

Tracing-cloth and paper, 433. 

Track. See Rail, 
gauge of, 773, 814. 


Track—continued, 
laying, cost, 804. 
repairs, 815. 

-scales, 803. 
stringers, 545, &c. 
tank, 802. 
trough, 802. 

Traction, 375. 
of cars, 808. 
on grades, 808. 
of horses, 375, 377. 
of locomotives, 808, 809. 

Trailing switch, 770. 

Train, 

earthwork, 749. 
expenses, 815. 

-mile, 809, Ac. 
weight, 546, 564. 
of wheels, 342. 

Transformation of profile, 691. 

Transit, the engineer’s, 188. 

Transmission of pressure in liquids, 227. 

Transverse 
girders. 545, Ac. 
strength, 478. 

Trap rock, weight, 384. 

Trapezoid, 120. 
center of gravity of, 348. 

Trapezium, 120. 
center of gravity of, 348. 

Traverse table, 180-187. 

Tread, 765. 
of car-wheel, 765. 

-wheel, 378, 641. 
of wheel, 380,765. 

Trees, blasting of, 663. 

Trembling of dams, 285. 

Tremie, 680. 

Trenton wire gauge, 412. 

Trestles, 755. 

Triangle, Triangles, 110. 
center of gravity of, 348. 
in or'about a circle, 123. 
of forces, 330, 588. 
mensuration of, 110. 
right-angled, 112. 

Triangular truss, 558. 

Trigonometry, plane, 112. 

Tripod, 189. 

Trough, 

flow through, 263. 
track, 802. 

Troy weight, 385. 

Trunnion, friction of, 374d. 

Truss, Trusses, 547. 

Bollman, strains in, 586. 
bow-string, strains in, 588, 597. 
braced arch, strains in, 592, 598. 
bracing, lateral, 610. 
bridge, arrangement of, 603, Ac. 

Burr, 601. 
camber, 607. 
cantilever, 593. 
for centers, 716, Ac. 
chords of, 550, 612. 
contraction by cold, 614. 
cost, 580. 

counter-bracing, 564. 
crescent, strains in, 588. 


til noli 
! tOMUftfU 









863 


INDEX. 

Truss—Velocity. 


Truss, Trusses—continued, 
diagrams, 551, Ac. 
dimensions, for bridges, 595-605. 
distance apart in bridges, 609. 
erection of, 60S. 

expansion by heat, allowance for, 
614. 

rollers, 614. 
eye-bars and pins, 612. 
factor of safety, 607. 
false-works, 608. 

Fink 

bridge, arrangement of, 603. 
strains in, 584. 

roof compared with king and queen, 
578. 

strains in, 574. 
floor girders, 610. 
forces acting on, 551. 
headway, 609. 

horizontal bracing, 542, 610. 

horizontal strains in, 562, 591, Ac. 

Howe, 594. 

joints, 583, 610, 612. 

king and queen, 551. Ac. 

compared with Fink, 578. 
lateral bracing, 610. 
lattice, 596. 

loads on, greatest probable, for bridges, 
606, 623. 

moving, 546, 564, 805-807. 

Moseley, 600. 

moving loads on, 546, 564, 805-807. 
obliques, 

best inclination for, 548. 
to find length of, 122, 608. 
overturning tendency, 609. 
panel, defined, 548. 

length of, to find, 548. 
pins and eye-bars, 612. 
posts in, 549. 

< Pratt, 595. 

purlins, 551, 582. 
queen compared with Fink, 578. 
rafters of. 355, 551, Ac, 582. 
raising of, 608. 
rise of, effect on weight, 581. 
rollers, expansion, 614. 
roof, details, 582. 
forms of, 551, Ac, 570, Ac. 
strains in, 551, Ac, 570, Ac. 
safety, factor of, 607. 
splices, 610, 612. 
strains in, 551, Ac. 
suspension, 548. 
in suspension bridges, 615. 
suspension links, 614. 
of swing bridges, strains in, 593. 
tendency to overturn, 609. 
tie-beams in, 551, Ac. 

Town’s lattice, 596. 
triangular, 558. 
verticals in, 549. 

Warren, 558. 

counter-bracing, 568. 
weight of, affected by rise, 581. 
weights of, for bridges, 605. 
for roofs, 573, 578-580. 


Tube. Tubes. See also Pipes, 
boiler, 405. 

brass, seamless drawn, 417. 
bubble, to replace, 193. 
copper, seamless drawn, 417. 
flow in, 236, Ac. See Pipes, Flow, Ac. 
iron, 405. 

Pitot’s, 269. 

pressure of water in, 232, Ac, 239. 
seamless, 417. 
short, flow through, 259. 
welded, 405. 

Tumbling lever, 772, 776,779. 

Tunnel, 754. 

Turf, weight, 385. 

Turn-buckle, 583. 

Turnouts, 770, 785. 

Turnpike, grades on, 723. 

Turntables, 790. 
cast-iron, 792. 
wooden, 797. 
wrought-iron, 793. 

Turpentine, 430. 

Tyler switch, 770. 

Tympa.n, 379. 

u. 

Undecagon, 110. 

Ungula, cylindric, 159. 

Uniform velocity, 311. 

Unit 

of rat^of w r ork, 311. 
of work, 311. 

United States 
cements, 673, 679. 

measures, to reduce to British, and 
vice versa, 390. 
railroad statistics, 814, Ac. 
standard dimensions of bolts, Ac, 406. 
Unstable equilibrium, 235, 357. 

Upright switch-stand, 772. 

Upset rods, 408. 

Useful work, 316. 

V. 

Vacuum process for sinking cylinders, 
647. 

Valve, Valves, 
air, 297. 
four-way, 302. 
outlet, 290. 
stop, 301. 
three-way, 302. 

-tower, 289. 

for water-pipes, 301. 

Variation 
of compass, 196. 
line of no, 197. 
vernier, 193. 

Vegetation in reservoirs, 289. 

Vehicles, friction of, 3746. 

Vein, contracted, 258, 260. 

Velocity, Velocities, 
of abrasion, 279/. 
affected by material of pipe, 244 
accelerated, 311. 
through adjutages, 259. 
augular, 365. 








864 INDEX. 

Velocity—Water. 


Velocity, Velocities—continued, 
through apertures, 257, Ac. 
in channels, 268, Ac. 

Kutter’s formula, 271. 
defined, 309. 
effect of, on friction, 374. 

of material of pipe on, 244. 
of falling bodies, 258, 362. 
head, 237. 

for a given velocity, to find, 248. 
imparted gradually, 315, 319. 
on inclined planes, 363. 

Kutter’s formula, 244, 271. 

material of pipe, effect on, 244. 

mean, 243, 268, Ac. 

of outflow, 258. 

in pipes, 243, Ac, 245, Ac. 

retarded, 311. 

in rivers, 268, Ac. 

in sewers, 279c. 

in short tubes, 259. 

of sound, 211. 

theoretical, of outflow, 258. 

through thin partition, 260. 

of trains, 809. 

uniform, 311. 

virtual, 339. 

of water, 236, Ac. 

of wind, 216. 

Vena contracta, 258, 260. 

Ventilation. # 

air, quantity required, 215. 
of tunnels, 754. 

Vernier, 190. 

variation, 193. 

Verrugas viaduct, 758. 

Versed sines, 59. 

Vertical, Verticals, 
defined, 54. 
of buoyancy, 235. 
of equilibrium, 235. 
of flotation, 235. 
strains in beams, 532. 
in a truss, 547. 

Vessel, Vessels, 
air, 298. 

contents of, 154, Ac, 390, Ac. 
floating, 235, 236. 

metallic, effect of water on, 218, 419. 
Viaduct, 

Crumlin, 756. 

Genesee, 756,757. 

Kinzua, 758. 

Lock-Ken, 599, 647. 

Portage, 756. 757. 

Verrugas, 758. 

Vibrating bodies, 364. 

Vibration, 364. 

Virtual 

head, 258, 280. 
velocities, 339. 

Vis viva, 308, 317. 

Voids 

in broken stone, 380, 678, 751. 
in concrete, 678. 
in rubble, 669, 741. 
in sand, 384, 677. 

Voussoir, 693. 


w. 

Wages, locomotive, 810, 815. 

Wagons, friction of, 3745. 

Wall, Walls, 
backing of, 683. 
battered, 685, Ac. 

bricks, number in a sq ft of, 669-671. 

cost, 667, 668. 
dam, 229, Ac, 282, 287. 
face, 683. 

foundations for, 633, Ac. 

incrustation of, 673, 678. 

offset, 683. 

plates, 524, 544, 614. 

of reservoirs, 287. 

retaining, 683. 

soap-wash for, 672. 

spandrel, 693, Ac. 

stability of, 229, 686. 

surcharged, 685, Ac. 

water, to render impervious, 672. 

to resist pressure of, 229, Ac. 
wharf, 236, 691. 
wing, 704. 

Walnut, 

strength, 436, 463, 493. 
weight, 385. 

Ward's flexible pipe-joint, 296. 

Warming by steam, surface required, 399. 
Warren truss, 558, 568. 

Washers, 406. 
lock-nut, 408. 

Washes for walls, 430, Ac, 672. 
Washington monument, concrete, 679. 
Waste, 

for locomotives, cost, 810,815. 
of water, in cities, 287. 
weir for reservoirs, 289. 

Watch, to regulate by star, 395. 

Water, 217. 

for boilers, top of 218. 

boiling, to measure heights by, 209. 

buoyancy, 234, Ac. 

brick-work, to render impervious, 67i 

in cement, quantity required, 675, 678. 

cisterns, 233, 800-803. 

column, 801. 

compensation, 290. 

composition of, 217. 

compressibility of, 217. 

concrete under, 680. 

corrosion by, 218, 645. 

dams for, 229,282. 

discharge. See Discharge. 

effect 

on cement, 673, Ac. 
on dynamite, 663. 
on iron, 218, 645. 
on lime, 669, 670. 
of zinc on, 219, 419. 
evaporation, 222. 
flow of. See Flow, 
foundations in, 634, Ac. 
freezing of, 217, 219. 
friction of, in pumping mains, 257. 
gates, 301. 
horse-power of, 2S0. 




INDEX. 

W a ter—W oo«I 


865 


Water—continued, 
jet, for pile-driving, 646. 
leakage, 222, 269, 282, 288. 
for locomotives, 218, 801-810, 815. 
masonry, to render impervious, 672. 
meters, 270. 
momentum of, 234. 
pipes, 290, &c. See also Pipes, 
in pipes. See Pipes, Velocity, Flow, 
Discharge, Pressure, &c. 
power, 280. 
pressure, 222, <£c, 239. 
in cylinders, 232. 
in pipes, 232, &c, 239, 257. 
running, 279 f, &c. 
still, 222. 

walls to resist, 229, Ac. 
quantity required in cities, 287. 
rain, 218, 385. 
ram, 234, 298, 303. 
reservoirs for, 287, Ac. 
resistance to moving bodies, 280. 
running, pressure of, 279/, &c. 
salt, effect on metals, 218, 645. 
scouring action, 279/. 
size of commercial measures by weight 
of, 391. 
stations, 800. 
storage of, 288. 
supply, 287, &c. 
tank, thicknesses, 803. 
traction on, 375. 

in tubes, flow of, 236, &c. See also Flow. 

velocity. See Velocity. 

walls, to render impervious, 672. 

to resist pressure of, 229, 236. 
way, contraction of, 703. 
weight of, 217, 385. 
in pipes one foot long. 246. 
size of commercial measures by, 391. 
wheel, 280. 

Wax, weight, 385. 

Way, 

permanent, 759. 
station, cost, 803. 

Wear 

of cars, 811. 
of locomotives, 811. 
of rails, 760. 
of ropes, 414. 
of ties, 759. 
of tools, 743. 
of wheels, 807, 812. 
of wire ropes, 414. 

Web 

of beams, 529. 
members of truss, 547. 
of riveted girders, 539, 540. 

Wedge, Wedges, 
mensuration of, 162. 
striking, for centers, 711, 712, 720. 
Weight, Weights, 381, &c. See also the 
article in question, 
of centers for arches, 719. 
defined, 306. 

on driving wheels, 546, 564, 805, &c. 

French, old, 393. 

and measures, 385, &c. 

56 


Weight, Weights—continued, 
metric, 381, &c, 393. 

Russian, 394. 

Spanish, 394. 

of substances, table, 381. See also the 
article in question. 

Weir, 264,4c. 

Well, Wells^ 
artesian, 627. 
boring, 626. 
contents, 1S7„ 

masonry, quantity in walls of, 158. 

Wellhouse process, 425a. 

Westing, 168. 

Westinghouse experiments, 374. 

Wet 

perimeter, 244. 
rot, 425. 

Wharf 
spikes. 762. 
walls, 236, 691. 

Wharton switch, 778. 

Wheel, Wheels, 
and axle, 339. 

barrows, loads ofi 745, 4c, 751. 

base, 805, 4c. 

of car, 812. 

cog, 342. 

driving, 805,4e. 

loads on, 546, 564, 805-807. 
of locomotives, 805, dec. 

loads on, 546, 564, 805-807. 
meters, 270. 

Persian, 379. 
and pinions, 342. 
skidding of, 374a. 
tire of, locomotive, 805, Ac. 
of cars, 812. 
of wagon, 380. 
train of, 342. 
tread of, 380, 765. 
tread-, 378, 641. 
water-, 280. 

Wheeled scrapers, 747. 

White effervescence on walls, 673, 678. 
lead paint, 429. 

-wash, 431. 

Whitworth standard screw thread, &c, 
406. 

Widths of bridges, 542, 609. 

Winch, 339,378. 

Wind, 216. 

effect on suspension bridges, 616. 
mills, 801. 

pressure on roofs, 216, 580, 681. 

Wing 

of frog, 780. 

-walls, 704. 

Wire, 410-412. 
brass, copper, 411. 
fence, 803. 
gauges, 410-412. 
iron, steel, 412. 
rope, 413. 
strength, 464. 

Wohler’s law, 435. 

Wood. See Timber, Wooden, 
board measure, table, 420. 







866 


INDEX. 

Wood—Zone, 


Wood—continued, 
cohesive strength, 463. 
compressibility, 434. 
compressive strength, 436. 
creosoting, 425. 
crushing strength, 436. 

De Yobon, rock-drill, 657. 
ductility, 4:t4. 
durability, 425. 
effect, on cement, 678. 
of lime; on, 670. 
of mortar on, 670. 
elastic limit of, 434. 
expansion by heat, 212. 
friction of, 373, Ac. 
fuel, 810, Ac. 
limit of elasticity, 434. 
modulus of elasticity, 434. 
preservation of, 425. 
shearing strength, 476. 
shingles, 429. 
specific gravity of, 381, Ac. 
strength, 436, 463, 476, 477, 493, 
stretch of, 434. 
tensile strength, 463. 
torsional strength, 477. 
transverse strength, constants for, 493. 
weight, 381, Ac. 

Wood’s frog, 785. 

Wooden 

beams, 493, Ac. See Beams, Wooden, 
bridges, 514. See also Truss, Trestle, 
Bridge, Ac. 
dams, 282. 
pillars, 458, Ac. 
pipes, 294. 
trestles, 755. 
turntables, 797. 

Work, 310, 316. 
of friction, 374/. 


W ork—continued, 
rate of, unit of, 311. 
unit of, 311. 
useful, 316. 

Working force, 310. 

World, railroad, miles in, 818. 

Worm fence, 803. 

Worms, sea, 425. 

Wrecking car, 750. 

Wrouglit-iron. See Iron, Wrought. 

Y. 

Yard, Yards, 385, 387. 
cubic, of earthwork, 732. 
equivalents of, 389. 

Yielding of centers for arches, 713, Ac, 
720. 

z. 

Zigzag riveting, 470. 

Zinc, 418. 

effect of cement, mortar, Ac, on, 670, 
673. 

of, on water, 419. 
of water on, 219. 
expansion l»y heat, 212. 
paint, 429. 
paint on, 403. 
price, 419. 
roofing, 418. 
sheets, 418. 

strength, compressive, 438. 
weight of, 385, 398, 400, 401, 410. 

Zone, Zones, 
circular. 146. 
of circular spindle, 167. 
parabolic, 152. 
spherical, 166. 


.1 




THE END, 











































































